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 = [5,-3,0,7,0,0,-2,-12,0,0,1,0,-7] 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: \(5\)
Coefficient field: 5.5.600268.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 6x^{2} + 6x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.22392\) 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)
 
\(f(q)\) \(=\) \(q-0.589830 q^{2} -1.65210 q^{4} +4.68977 q^{7} +2.15412 q^{8} +1.73991 q^{11} +0.0918459 q^{13} -2.76617 q^{14} +2.03364 q^{16} +0.766167 q^{17} +2.05417 q^{19} -1.02625 q^{22} -0.205912 q^{23} -0.0541735 q^{26} -7.74798 q^{28} -1.64807 q^{29} +9.69381 q^{31} -5.50774 q^{32} -0.451908 q^{34} -1.00000 q^{37} -1.21161 q^{38} +4.33046 q^{41} -2.82773 q^{43} -2.87451 q^{44} +0.121453 q^{46} -7.39605 q^{47} +14.9940 q^{49} -0.151739 q^{52} +12.9653 q^{53} +10.1023 q^{56} +0.972080 q^{58} +0.0376725 q^{59} +8.97375 q^{61} -5.71769 q^{62} -0.818654 q^{64} +12.2541 q^{67} -1.26579 q^{68} +9.89474 q^{71} -8.10730 q^{73} +0.589830 q^{74} -3.39370 q^{76} +8.15980 q^{77} +14.1760 q^{79} -2.55423 q^{82} -15.4664 q^{83} +1.66788 q^{86} +3.74798 q^{88} -13.0596 q^{89} +0.430737 q^{91} +0.340188 q^{92} +4.36241 q^{94} +0.973747 q^{97} -8.84389 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{2} + 7 q^{4} - 2 q^{7} - 12 q^{8} + q^{11} - 7 q^{13} - 11 q^{14} + 15 q^{16} + q^{17} + 13 q^{19} - 10 q^{22} - 6 q^{23} - 3 q^{26} - q^{28} - 8 q^{29} + 8 q^{31} - 27 q^{32} + 11 q^{34}+ \cdots + 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.589830 −0.417073 −0.208536 − 0.978015i \(-0.566870\pi\)
−0.208536 + 0.978015i \(0.566870\pi\)
\(3\) 0 0
\(4\) −1.65210 −0.826051
\(5\) 0 0
\(6\) 0 0
\(7\) 4.68977 1.77257 0.886284 − 0.463142i \(-0.153278\pi\)
0.886284 + 0.463142i \(0.153278\pi\)
\(8\) 2.15412 0.761595
\(9\) 0 0
\(10\) 0 0
\(11\) 1.73991 0.524604 0.262302 − 0.964986i \(-0.415518\pi\)
0.262302 + 0.964986i \(0.415518\pi\)
\(12\) 0 0
\(13\) 0.0918459 0.0254735 0.0127367 − 0.999919i \(-0.495946\pi\)
0.0127367 + 0.999919i \(0.495946\pi\)
\(14\) −2.76617 −0.739289
\(15\) 0 0
\(16\) 2.03364 0.508410
\(17\) 0.766167 0.185823 0.0929114 − 0.995674i \(-0.470383\pi\)
0.0929114 + 0.995674i \(0.470383\pi\)
\(18\) 0 0
\(19\) 2.05417 0.471260 0.235630 − 0.971843i \(-0.424285\pi\)
0.235630 + 0.971843i \(0.424285\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.02625 −0.218798
\(23\) −0.205912 −0.0429357 −0.0214678 − 0.999770i \(-0.506834\pi\)
−0.0214678 + 0.999770i \(0.506834\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.0541735 −0.0106243
\(27\) 0 0
\(28\) −7.74798 −1.46423
\(29\) −1.64807 −0.306039 −0.153019 − 0.988223i \(-0.548900\pi\)
−0.153019 + 0.988223i \(0.548900\pi\)
\(30\) 0 0
\(31\) 9.69381 1.74106 0.870529 − 0.492116i \(-0.163777\pi\)
0.870529 + 0.492116i \(0.163777\pi\)
\(32\) −5.50774 −0.973639
\(33\) 0 0
\(34\) −0.451908 −0.0775016
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −1.21161 −0.196549
\(39\) 0 0
\(40\) 0 0
\(41\) 4.33046 0.676303 0.338152 − 0.941092i \(-0.390198\pi\)
0.338152 + 0.941092i \(0.390198\pi\)
\(42\) 0 0
\(43\) −2.82773 −0.431224 −0.215612 − 0.976479i \(-0.569175\pi\)
−0.215612 + 0.976479i \(0.569175\pi\)
\(44\) −2.87451 −0.433349
\(45\) 0 0
\(46\) 0.121453 0.0179073
\(47\) −7.39605 −1.07882 −0.539412 − 0.842042i \(-0.681354\pi\)
−0.539412 + 0.842042i \(0.681354\pi\)
\(48\) 0 0
\(49\) 14.9940 2.14200
\(50\) 0 0
\(51\) 0 0
\(52\) −0.151739 −0.0210424
\(53\) 12.9653 1.78092 0.890461 − 0.455059i \(-0.150382\pi\)
0.890461 + 0.455059i \(0.150382\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 10.1023 1.34998
\(57\) 0 0
\(58\) 0.972080 0.127640
\(59\) 0.0376725 0.00490454 0.00245227 − 0.999997i \(-0.499219\pi\)
0.00245227 + 0.999997i \(0.499219\pi\)
\(60\) 0 0
\(61\) 8.97375 1.14897 0.574485 − 0.818515i \(-0.305202\pi\)
0.574485 + 0.818515i \(0.305202\pi\)
\(62\) −5.71769 −0.726148
\(63\) 0 0
\(64\) −0.818654 −0.102332
\(65\) 0 0
\(66\) 0 0
\(67\) 12.2541 1.49707 0.748536 − 0.663094i \(-0.230757\pi\)
0.748536 + 0.663094i \(0.230757\pi\)
\(68\) −1.26579 −0.153499
\(69\) 0 0
\(70\) 0 0
\(71\) 9.89474 1.17429 0.587145 − 0.809482i \(-0.300252\pi\)
0.587145 + 0.809482i \(0.300252\pi\)
\(72\) 0 0
\(73\) −8.10730 −0.948887 −0.474444 − 0.880286i \(-0.657351\pi\)
−0.474444 + 0.880286i \(0.657351\pi\)
\(74\) 0.589830 0.0685663
\(75\) 0 0
\(76\) −3.39370 −0.389284
\(77\) 8.15980 0.929896
\(78\) 0 0
\(79\) 14.1760 1.59492 0.797462 − 0.603369i \(-0.206175\pi\)
0.797462 + 0.603369i \(0.206175\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.55423 −0.282068
\(83\) −15.4664 −1.69766 −0.848830 − 0.528666i \(-0.822692\pi\)
−0.848830 + 0.528666i \(0.822692\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.66788 0.179852
\(87\) 0 0
\(88\) 3.74798 0.399536
\(89\) −13.0596 −1.38431 −0.692156 − 0.721748i \(-0.743339\pi\)
−0.692156 + 0.721748i \(0.743339\pi\)
\(90\) 0 0
\(91\) 0.430737 0.0451535
\(92\) 0.340188 0.0354670
\(93\) 0 0
\(94\) 4.36241 0.449948
\(95\) 0 0
\(96\) 0 0
\(97\) 0.973747 0.0988690 0.0494345 − 0.998777i \(-0.484258\pi\)
0.0494345 + 0.998777i \(0.484258\pi\)
\(98\) −8.84389 −0.893368
\(99\) 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.bx.1.3 5
3.2 odd 2 2775.2.a.bd.1.3 5
5.4 even 2 1665.2.a.s.1.3 5
15.14 odd 2 555.2.a.j.1.3 ✓ 5
60.59 even 2 8880.2.a.cm.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.a.j.1.3 ✓ 5 15.14 odd 2
1665.2.a.s.1.3 5 5.4 even 2
2775.2.a.bd.1.3 5 3.2 odd 2
8325.2.a.bx.1.3 5 1.1 even 1 trivial
8880.2.a.cm.1.5 5 60.59 even 2