Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [725,2,Mod(1,725)] 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("725.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.52434\) of defining polynomial
Character \(\chi\) \(=\) 725.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} -2.52434 q^{3} +1.00000 q^{4} -4.37228 q^{6} -1.58457 q^{7} -1.73205 q^{8} +3.37228 q^{9} +6.37228 q^{11} -2.52434 q^{12} +0.939764 q^{13} -2.74456 q^{14} -5.00000 q^{16} +5.04868 q^{17} +5.84096 q^{18} +4.00000 q^{19} +4.00000 q^{21} +11.0371 q^{22} +3.46410 q^{23} +4.37228 q^{24} +1.62772 q^{26} -0.939764 q^{27} -1.58457 q^{28} -1.00000 q^{29} +2.37228 q^{31} -5.19615 q^{32} -16.0858 q^{33} +8.74456 q^{34} +3.37228 q^{36} -10.0974 q^{37} +6.92820 q^{38} -2.37228 q^{39} +6.74456 q^{41} +6.92820 q^{42} +5.69349 q^{43} +6.37228 q^{44} +6.00000 q^{46} +5.69349 q^{47} +12.6217 q^{48} -4.48913 q^{49} -12.7446 q^{51} +0.939764 q^{52} +0.939764 q^{53} -1.62772 q^{54} +2.74456 q^{56} -10.0974 q^{57} -1.73205 q^{58} +0.744563 q^{59} +6.00000 q^{61} +4.10891 q^{62} -5.34363 q^{63} +1.00000 q^{64} -27.8614 q^{66} -8.51278 q^{67} +5.04868 q^{68} -8.74456 q^{69} -4.74456 q^{71} -5.84096 q^{72} -6.92820 q^{73} -17.4891 q^{74} +4.00000 q^{76} -10.0974 q^{77} -4.10891 q^{78} +5.62772 q^{79} -7.74456 q^{81} +11.6819 q^{82} +16.7306 q^{83} +4.00000 q^{84} +9.86141 q^{86} +2.52434 q^{87} -11.0371 q^{88} +10.7446 q^{89} -1.48913 q^{91} +3.46410 q^{92} -5.98844 q^{93} +9.86141 q^{94} +13.1168 q^{96} -6.92820 q^{97} -7.77539 q^{98} +21.4891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 14 q^{11} + 12 q^{14} - 20 q^{16} + 16 q^{19} + 16 q^{21} + 6 q^{24} + 18 q^{26} - 4 q^{29} - 2 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} + 4 q^{41} + 14 q^{44} + 24 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 − 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −2.52434 −1.45743 −0.728714 − 0.684819i \(-0.759881\pi\)
−0.728714 + 0.684819i \(0.759881\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −4.37228 −1.78498
\(7\) −1.58457 −0.598913 −0.299456 − 0.954110i \(-0.596805\pi\)
−0.299456 + 0.954110i \(0.596805\pi\)
\(8\) −1.73205 −0.612372
\(9\) 3.37228 1.12409
\(10\) 0 0
\(11\) 6.37228 1.92132 0.960658 − 0.277736i \(-0.0895839\pi\)
0.960658 + 0.277736i \(0.0895839\pi\)
\(12\) −2.52434 −0.728714
\(13\) 0.939764 0.260644 0.130322 − 0.991472i \(-0.458399\pi\)
0.130322 + 0.991472i \(0.458399\pi\)
\(14\) −2.74456 −0.733515
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 5.04868 1.22448 0.612242 − 0.790671i \(-0.290268\pi\)
0.612242 + 0.790671i \(0.290268\pi\)
\(18\) 5.84096 1.37673
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 11.0371 2.35312
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 4.37228 0.892488
\(25\) 0 0
\(26\) 1.62772 0.319222
\(27\) −0.939764 −0.180858
\(28\) −1.58457 −0.299456
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 2.37228 0.426074 0.213037 − 0.977044i \(-0.431664\pi\)
0.213037 + 0.977044i \(0.431664\pi\)
\(32\) −5.19615 −0.918559
\(33\) −16.0858 −2.80018
\(34\) 8.74456 1.49968
\(35\) 0 0
\(36\) 3.37228 0.562047
\(37\) −10.0974 −1.65999 −0.829997 − 0.557768i \(-0.811658\pi\)
−0.829997 + 0.557768i \(0.811658\pi\)
\(38\) 6.92820 1.12390
\(39\) −2.37228 −0.379869
\(40\) 0 0
\(41\) 6.74456 1.05332 0.526662 − 0.850075i \(-0.323443\pi\)
0.526662 + 0.850075i \(0.323443\pi\)
\(42\) 6.92820 1.06904
\(43\) 5.69349 0.868248 0.434124 − 0.900853i \(-0.357058\pi\)
0.434124 + 0.900853i \(0.357058\pi\)
\(44\) 6.37228 0.960658
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 5.69349 0.830480 0.415240 − 0.909712i \(-0.363698\pi\)
0.415240 + 0.909712i \(0.363698\pi\)
\(48\) 12.6217 1.82178
\(49\) −4.48913 −0.641304
\(50\) 0 0
\(51\) −12.7446 −1.78460
\(52\) 0.939764 0.130322
\(53\) 0.939764 0.129086 0.0645432 − 0.997915i \(-0.479441\pi\)
0.0645432 + 0.997915i \(0.479441\pi\)
\(54\) −1.62772 −0.221504
\(55\) 0 0
\(56\) 2.74456 0.366758
\(57\) −10.0974 −1.33743
\(58\) −1.73205 −0.227429
\(59\) 0.744563 0.0969338 0.0484669 − 0.998825i \(-0.484566\pi\)
0.0484669 + 0.998825i \(0.484566\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.10891 0.521832
\(63\) −5.34363 −0.673234
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −27.8614 −3.42950
\(67\) −8.51278 −1.04000 −0.520001 − 0.854166i \(-0.674068\pi\)
−0.520001 + 0.854166i \(0.674068\pi\)
\(68\) 5.04868 0.612242
\(69\) −8.74456 −1.05272
\(70\) 0 0
\(71\) −4.74456 −0.563076 −0.281538 − 0.959550i \(-0.590845\pi\)
−0.281538 + 0.959550i \(0.590845\pi\)
\(72\) −5.84096 −0.688364
\(73\) −6.92820 −0.810885 −0.405442 − 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) −17.4891 −2.03307
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −10.0974 −1.15070
\(78\) −4.10891 −0.465243
\(79\) 5.62772 0.633168 0.316584 − 0.948565i \(-0.397464\pi\)
0.316584 + 0.948565i \(0.397464\pi\)
\(80\) 0 0
\(81\) −7.74456 −0.860507
\(82\) 11.6819 1.29005
\(83\) 16.7306 1.83642 0.918211 − 0.396092i \(-0.129634\pi\)
0.918211 + 0.396092i \(0.129634\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 9.86141 1.06338
\(87\) 2.52434 0.270637
\(88\) −11.0371 −1.17656
\(89\) 10.7446 1.13892 0.569461 − 0.822019i \(-0.307152\pi\)
0.569461 + 0.822019i \(0.307152\pi\)
\(90\) 0 0
\(91\) −1.48913 −0.156103
\(92\) 3.46410 0.361158
\(93\) −5.98844 −0.620972
\(94\) 9.86141 1.01713
\(95\) 0 0
\(96\) 13.1168 1.33873
\(97\) −6.92820 −0.703452 −0.351726 − 0.936103i \(-0.614405\pi\)
−0.351726 + 0.936103i \(0.614405\pi\)
\(98\) −7.77539 −0.785433
\(99\) 21.4891 2.15974
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 725.2.a.g.1.3 4
3.2 odd 2 6525.2.a.bk.1.1 4
5.2 odd 4 145.2.b.a.59.4 yes 4
5.3 odd 4 145.2.b.a.59.1 ✓ 4
5.4 even 2 inner 725.2.a.g.1.2 4
15.2 even 4 1305.2.c.e.784.2 4
15.8 even 4 1305.2.c.e.784.4 4
15.14 odd 2 6525.2.a.bk.1.4 4
20.3 even 4 2320.2.d.c.929.4 4
20.7 even 4 2320.2.d.c.929.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.a.59.1 ✓ 4 5.3 odd 4
145.2.b.a.59.4 yes 4 5.2 odd 4
725.2.a.g.1.2 4 5.4 even 2 inner
725.2.a.g.1.3 4 1.1 even 1 trivial
1305.2.c.e.784.2 4 15.2 even 4
1305.2.c.e.784.4 4 15.8 even 4
2320.2.d.c.929.1 4 20.7 even 4
2320.2.d.c.929.4 4 20.3 even 4
6525.2.a.bk.1.1 4 3.2 odd 2
6525.2.a.bk.1.4 4 15.14 odd 2