Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,0,0,4,-6,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2775)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} -1.58579 q^{8} +2.82843 q^{11} +0.828427 q^{13} +0.828427 q^{14} +3.00000 q^{16} +7.65685 q^{17} +7.24264 q^{19} +1.17157 q^{22} -1.58579 q^{23} +0.343146 q^{26} -3.65685 q^{28} -0.828427 q^{29} +6.00000 q^{31} +4.41421 q^{32} +3.17157 q^{34} +1.00000 q^{37} +3.00000 q^{38} +10.6569 q^{41} +7.24264 q^{43} -5.17157 q^{44} -0.656854 q^{46} -3.17157 q^{47} -3.00000 q^{49} -1.51472 q^{52} -8.65685 q^{53} -3.17157 q^{56} -0.343146 q^{58} -10.4142 q^{59} -12.0000 q^{61} +2.48528 q^{62} -4.17157 q^{64} +3.65685 q^{67} -14.0000 q^{68} -15.3137 q^{71} +5.48528 q^{73} +0.414214 q^{74} -13.2426 q^{76} +5.65685 q^{77} -2.07107 q^{79} +4.41421 q^{82} -9.65685 q^{83} +3.00000 q^{86} -4.48528 q^{88} +12.8284 q^{89} +1.65685 q^{91} +2.89949 q^{92} -1.31371 q^{94} -8.48528 q^{97} -1.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} - 4 q^{13} - 4 q^{14} + 6 q^{16} + 4 q^{17} + 6 q^{19} + 8 q^{22} - 6 q^{23} + 12 q^{26} + 4 q^{28} + 4 q^{29} + 12 q^{31} + 6 q^{32} + 12 q^{34} + 2 q^{37} + 6 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 − 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.58579 −0.560660
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 − 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 0.828427 0.229764 0.114882 − 0.993379i \(-0.463351\pi\)
0.114882 + 0.993379i \(0.463351\pi\)
\(14\) 0.828427 0.221406
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 7.65685 1.85706 0.928530 − 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) 7.24264 1.66158 0.830788 − 0.556589i \(-0.187890\pi\)
0.830788 + 0.556589i \(0.187890\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.17157 0.249780
\(23\) −1.58579 −0.330659 −0.165330 − 0.986238i \(-0.552869\pi\)
−0.165330 + 0.986238i \(0.552869\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.343146 0.0672964
\(27\) 0 0
\(28\) −3.65685 −0.691080
\(29\) −0.828427 −0.153835 −0.0769175 − 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 4.41421 0.780330
\(33\) 0 0
\(34\) 3.17157 0.543920
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 3.00000 0.486664
\(39\) 0 0
\(40\) 0 0
\(41\) 10.6569 1.66432 0.832161 − 0.554535i \(-0.187104\pi\)
0.832161 + 0.554535i \(0.187104\pi\)
\(42\) 0 0
\(43\) 7.24264 1.10449 0.552246 − 0.833681i \(-0.313771\pi\)
0.552246 + 0.833681i \(0.313771\pi\)
\(44\) −5.17157 −0.779644
\(45\) 0 0
\(46\) −0.656854 −0.0968479
\(47\) −3.17157 −0.462621 −0.231311 − 0.972880i \(-0.574301\pi\)
−0.231311 + 0.972880i \(0.574301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −1.51472 −0.210054
\(53\) −8.65685 −1.18911 −0.594555 − 0.804055i \(-0.702672\pi\)
−0.594555 + 0.804055i \(0.702672\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.17157 −0.423819
\(57\) 0 0
\(58\) −0.343146 −0.0450572
\(59\) −10.4142 −1.35582 −0.677908 − 0.735147i \(-0.737113\pi\)
−0.677908 + 0.735147i \(0.737113\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 − 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 2.48528 0.315631
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) 3.65685 0.446756 0.223378 − 0.974732i \(-0.428292\pi\)
0.223378 + 0.974732i \(0.428292\pi\)
\(68\) −14.0000 −1.69775
\(69\) 0 0
\(70\) 0 0
\(71\) −15.3137 −1.81740 −0.908701 − 0.417447i \(-0.862925\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(72\) 0 0
\(73\) 5.48528 0.642004 0.321002 − 0.947079i \(-0.395980\pi\)
0.321002 + 0.947079i \(0.395980\pi\)
\(74\) 0.414214 0.0481513
\(75\) 0 0
\(76\) −13.2426 −1.51904
\(77\) 5.65685 0.644658
\(78\) 0 0
\(79\) −2.07107 −0.233013 −0.116507 − 0.993190i \(-0.537170\pi\)
−0.116507 + 0.993190i \(0.537170\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.41421 0.487468
\(83\) −9.65685 −1.05998 −0.529989 − 0.848005i \(-0.677804\pi\)
−0.529989 + 0.848005i \(0.677804\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.00000 0.323498
\(87\) 0 0
\(88\) −4.48528 −0.478133
\(89\) 12.8284 1.35981 0.679905 − 0.733300i \(-0.262021\pi\)
0.679905 + 0.733300i \(0.262021\pi\)
\(90\) 0 0
\(91\) 1.65685 0.173686
\(92\) 2.89949 0.302293
\(93\) 0 0
\(94\) −1.31371 −0.135499
\(95\) 0 0
\(96\) 0 0
\(97\) −8.48528 −0.861550 −0.430775 − 0.902459i \(-0.641760\pi\)
−0.430775 + 0.902459i \(0.641760\pi\)
\(98\) −1.24264 −0.125526
\(99\) 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 8325.2.a.bg.1.2 2
3.2 odd 2 2775.2.a.p.1.1 yes 2
5.4 even 2 8325.2.a.bn.1.1 2
15.14 odd 2 2775.2.a.k.1.2 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.k.1.2 ✓ 2 15.14 odd 2
2775.2.a.p.1.1 yes 2 3.2 odd 2
8325.2.a.bg.1.2 2 1.1 even 1 trivial
8325.2.a.bn.1.1 2 5.4 even 2