Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6624,2,Mod(1,6624)] 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("6624.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6624 = 2^{5} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,4,0,-4,0,0,0,0,0,-2,0,0,0,8,0,4,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.8929062989\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 736)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6624.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.41421 q^{5} -3.41421 q^{7} +4.24264 q^{11} +1.82843 q^{13} +5.41421 q^{17} +4.82843 q^{19} +1.00000 q^{23} +6.65685 q^{25} +5.00000 q^{29} -0.757359 q^{31} -11.6569 q^{35} -9.07107 q^{37} +12.6569 q^{41} +4.00000 q^{43} -3.24264 q^{47} +4.65685 q^{49} -13.3137 q^{53} +14.4853 q^{55} -6.82843 q^{59} +7.65685 q^{61} +6.24264 q^{65} -5.89949 q^{67} -6.41421 q^{71} +2.17157 q^{73} -14.4853 q^{77} +2.48528 q^{79} -14.7279 q^{83} +18.4853 q^{85} +8.00000 q^{89} -6.24264 q^{91} +16.4853 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 4 q^{7} - 2 q^{13} + 8 q^{17} + 4 q^{19} + 2 q^{23} + 2 q^{25} + 10 q^{29} - 10 q^{31} - 12 q^{35} - 4 q^{37} + 14 q^{41} + 8 q^{43} + 2 q^{47} - 2 q^{49} - 4 q^{53} + 12 q^{55} - 8 q^{59}+ \cdots + 16 q^{95}+O(q^{100}) \) 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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.41421 1.52688 0.763441 − 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 0 0
\(7\) −3.41421 −1.29045 −0.645226 − 0.763992i \(-0.723237\pi\)
−0.645226 + 0.763992i \(0.723237\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264 1.27920 0.639602 − 0.768706i \(-0.279099\pi\)
0.639602 + 0.768706i \(0.279099\pi\)
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 − 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.41421 1.31314 0.656570 − 0.754265i \(-0.272007\pi\)
0.656570 + 0.754265i \(0.272007\pi\)
\(18\) 0 0
\(19\) 4.82843 1.10772 0.553859 − 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −0.757359 −0.136026 −0.0680129 − 0.997684i \(-0.521666\pi\)
−0.0680129 + 0.997684i \(0.521666\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −11.6569 −1.97037
\(36\) 0 0
\(37\) −9.07107 −1.49127 −0.745637 − 0.666352i \(-0.767855\pi\)
−0.745637 + 0.666352i \(0.767855\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.6569 1.97667 0.988334 − 0.152300i \(-0.0486681\pi\)
0.988334 + 0.152300i \(0.0486681\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.24264 −0.472988 −0.236494 − 0.971633i \(-0.575998\pi\)
−0.236494 + 0.971633i \(0.575998\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.3137 −1.82878 −0.914389 − 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0 0
\(55\) 14.4853 1.95319
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.82843 −0.888985 −0.444493 − 0.895782i \(-0.646616\pi\)
−0.444493 + 0.895782i \(0.646616\pi\)
\(60\) 0 0
\(61\) 7.65685 0.980360 0.490180 − 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.24264 0.774304
\(66\) 0 0
\(67\) −5.89949 −0.720738 −0.360369 − 0.932810i \(-0.617349\pi\)
−0.360369 + 0.932810i \(0.617349\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.41421 −0.761227 −0.380614 − 0.924734i \(-0.624287\pi\)
−0.380614 + 0.924734i \(0.624287\pi\)
\(72\) 0 0
\(73\) 2.17157 0.254163 0.127082 − 0.991892i \(-0.459439\pi\)
0.127082 + 0.991892i \(0.459439\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.4853 −1.65075
\(78\) 0 0
\(79\) 2.48528 0.279616 0.139808 − 0.990179i \(-0.455351\pi\)
0.139808 + 0.990179i \(0.455351\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.7279 −1.61660 −0.808300 − 0.588771i \(-0.799612\pi\)
−0.808300 + 0.588771i \(0.799612\pi\)
\(84\) 0 0
\(85\) 18.4853 2.00501
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) −6.24264 −0.654407
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.4853 1.69135
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 − 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 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 6624.2.a.s.1.2 2
3.2 odd 2 736.2.a.d.1.2 yes 2
4.3 odd 2 6624.2.a.t.1.2 2
12.11 even 2 736.2.a.a.1.1 ✓ 2
24.5 odd 2 1472.2.a.o.1.1 2
24.11 even 2 1472.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.a.1.1 ✓ 2 12.11 even 2
736.2.a.d.1.2 yes 2 3.2 odd 2
1472.2.a.o.1.1 2 24.5 odd 2
1472.2.a.v.1.2 2 24.11 even 2
6624.2.a.s.1.2 2 1.1 even 1 trivial
6624.2.a.t.1.2 2 4.3 odd 2