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 1.1
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)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{7} -4.41421 q^{8} -2.82843 q^{11} -4.82843 q^{13} -4.82843 q^{14} +3.00000 q^{16} -3.65685 q^{17} -1.24264 q^{19} +6.82843 q^{22} -4.41421 q^{23} +11.6569 q^{26} +7.65685 q^{28} +4.82843 q^{29} +6.00000 q^{31} +1.58579 q^{32} +8.82843 q^{34} +1.00000 q^{37} +3.00000 q^{38} -0.656854 q^{41} -1.24264 q^{43} -10.8284 q^{44} +10.6569 q^{46} -8.82843 q^{47} -3.00000 q^{49} -18.4853 q^{52} +2.65685 q^{53} -8.82843 q^{56} -11.6569 q^{58} -7.58579 q^{59} -12.0000 q^{61} -14.4853 q^{62} -9.82843 q^{64} -7.65685 q^{67} -14.0000 q^{68} +7.31371 q^{71} -11.4853 q^{73} -2.41421 q^{74} -4.75736 q^{76} -5.65685 q^{77} +12.0711 q^{79} +1.58579 q^{82} +1.65685 q^{83} +3.00000 q^{86} +12.4853 q^{88} +7.17157 q^{89} -9.65685 q^{91} -16.8995 q^{92} +21.3137 q^{94} +8.48528 q^{97} +7.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} - 4 q^{13} - 4 q^{14} + 6 q^{16} + 4 q^{17} + 6 q^{19} + 8 q^{22} - 6 q^{23} + 12 q^{26} + 4 q^{28} + 4 q^{29} + 12 q^{31} + 6 q^{32} + 12 q^{34} + 2 q^{37} + 6 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 − 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −4.41421 −1.56066
\(9\) 0 0
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 − 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) −4.82843 −1.33916 −0.669582 − 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) −4.82843 −1.29045
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −3.65685 −0.886917 −0.443459 − 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) −1.24264 −0.285081 −0.142541 − 0.989789i \(-0.545527\pi\)
−0.142541 + 0.989789i \(0.545527\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.82843 1.45583
\(23\) −4.41421 −0.920427 −0.460214 − 0.887808i \(-0.652227\pi\)
−0.460214 + 0.887808i \(0.652227\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 11.6569 2.28610
\(27\) 0 0
\(28\) 7.65685 1.44701
\(29\) 4.82843 0.896616 0.448308 − 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.58579 0.280330
\(33\) 0 0
\(34\) 8.82843 1.51406
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 3.00000 0.486664
\(39\) 0 0
\(40\) 0 0
\(41\) −0.656854 −0.102583 −0.0512917 − 0.998684i \(-0.516334\pi\)
−0.0512917 + 0.998684i \(0.516334\pi\)
\(42\) 0 0
\(43\) −1.24264 −0.189501 −0.0947505 − 0.995501i \(-0.530205\pi\)
−0.0947505 + 0.995501i \(0.530205\pi\)
\(44\) −10.8284 −1.63245
\(45\) 0 0
\(46\) 10.6569 1.57127
\(47\) −8.82843 −1.28776 −0.643879 − 0.765127i \(-0.722676\pi\)
−0.643879 + 0.765127i \(0.722676\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −18.4853 −2.56345
\(53\) 2.65685 0.364947 0.182473 − 0.983211i \(-0.441590\pi\)
0.182473 + 0.983211i \(0.441590\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −8.82843 −1.17975
\(57\) 0 0
\(58\) −11.6569 −1.53062
\(59\) −7.58579 −0.987585 −0.493793 − 0.869580i \(-0.664390\pi\)
−0.493793 + 0.869580i \(0.664390\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 − 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) −14.4853 −1.83963
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 0 0
\(67\) −7.65685 −0.935434 −0.467717 − 0.883878i \(-0.654923\pi\)
−0.467717 + 0.883878i \(0.654923\pi\)
\(68\) −14.0000 −1.69775
\(69\) 0 0
\(70\) 0 0
\(71\) 7.31371 0.867978 0.433989 − 0.900918i \(-0.357106\pi\)
0.433989 + 0.900918i \(0.357106\pi\)
\(72\) 0 0
\(73\) −11.4853 −1.34425 −0.672125 − 0.740437i \(-0.734618\pi\)
−0.672125 + 0.740437i \(0.734618\pi\)
\(74\) −2.41421 −0.280647
\(75\) 0 0
\(76\) −4.75736 −0.545707
\(77\) −5.65685 −0.644658
\(78\) 0 0
\(79\) 12.0711 1.35810 0.679051 − 0.734091i \(-0.262392\pi\)
0.679051 + 0.734091i \(0.262392\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.58579 0.175121
\(83\) 1.65685 0.181863 0.0909317 − 0.995857i \(-0.471016\pi\)
0.0909317 + 0.995857i \(0.471016\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.00000 0.323498
\(87\) 0 0
\(88\) 12.4853 1.33094
\(89\) 7.17157 0.760185 0.380093 − 0.924948i \(-0.375892\pi\)
0.380093 + 0.924948i \(0.375892\pi\)
\(90\) 0 0
\(91\) −9.65685 −1.01231
\(92\) −16.8995 −1.76189
\(93\) 0 0
\(94\) 21.3137 2.19834
\(95\) 0 0
\(96\) 0 0
\(97\) 8.48528 0.861550 0.430775 − 0.902459i \(-0.358240\pi\)
0.430775 + 0.902459i \(0.358240\pi\)
\(98\) 7.24264 0.731617
\(99\) 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.1.1 2
3.2 odd 2 2775.2.a.p.1.2 yes 2
5.4 even 2 8325.2.a.bn.1.2 2
15.14 odd 2 2775.2.a.k.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.k.1.1 ✓ 2 15.14 odd 2
2775.2.a.p.1.2 yes 2 3.2 odd 2
8325.2.a.bg.1.1 2 1.1 even 1 trivial
8325.2.a.bn.1.2 2 5.4 even 2