Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3025,2,Mod(1,3025)] 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("3025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3025.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +1.00000 q^{12} +3.46410 q^{13} -3.00000 q^{14} -5.00000 q^{16} -6.92820 q^{17} -3.46410 q^{18} +3.46410 q^{19} -1.73205 q^{21} -1.73205 q^{24} +6.00000 q^{26} -5.00000 q^{27} -1.73205 q^{28} -8.00000 q^{31} -5.19615 q^{32} -12.0000 q^{34} -2.00000 q^{36} +8.00000 q^{37} +6.00000 q^{38} +3.46410 q^{39} -12.1244 q^{41} -3.00000 q^{42} +8.66025 q^{43} -9.00000 q^{47} -5.00000 q^{48} -4.00000 q^{49} -6.92820 q^{51} +3.46410 q^{52} -6.00000 q^{53} -8.66025 q^{54} +3.00000 q^{56} +3.46410 q^{57} -12.0000 q^{59} -8.66025 q^{61} -13.8564 q^{62} +3.46410 q^{63} +1.00000 q^{64} +5.00000 q^{67} -6.92820 q^{68} -12.0000 q^{71} +3.46410 q^{72} +13.8564 q^{74} +3.46410 q^{76} +6.00000 q^{78} +10.3923 q^{79} +1.00000 q^{81} -21.0000 q^{82} +3.46410 q^{83} -1.73205 q^{84} +15.0000 q^{86} +3.00000 q^{89} -6.00000 q^{91} -8.00000 q^{93} -15.5885 q^{94} -5.19615 q^{96} +10.0000 q^{97} -6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9} + 2 q^{12} - 6 q^{14} - 10 q^{16} + 12 q^{26} - 10 q^{27} - 16 q^{31} - 24 q^{34} - 4 q^{36} + 16 q^{37} + 12 q^{38} - 6 q^{42} - 18 q^{47} - 10 q^{48} - 8 q^{49} - 12 q^{53}+ \cdots + 20 q^{97}+O(q^{100}) \) 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 \(0.290215\pi\)
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.73205 0.707107
\(7\) −1.73205 −0.654654 −0.327327 − 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) −1.73205 −0.612372
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 3.46410 0.960769 0.480384 − 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −6.92820 −1.68034 −0.840168 − 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) −3.46410 −0.816497
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) −1.73205 −0.377964
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.73205 −0.353553
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −5.00000 −0.962250
\(28\) −1.73205 −0.327327
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) −12.0000 −2.05798
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 6.00000 0.973329
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) −12.1244 −1.89351 −0.946753 − 0.321960i \(-0.895658\pi\)
−0.946753 + 0.321960i \(0.895658\pi\)
\(42\) −3.00000 −0.462910
\(43\) 8.66025 1.32068 0.660338 − 0.750968i \(-0.270413\pi\)
0.660338 + 0.750968i \(0.270413\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −5.00000 −0.721688
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) −6.92820 −0.970143
\(52\) 3.46410 0.480384
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −8.66025 −1.17851
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −8.66025 −1.10883 −0.554416 − 0.832240i \(-0.687058\pi\)
−0.554416 + 0.832240i \(0.687058\pi\)
\(62\) −13.8564 −1.75977
\(63\) 3.46410 0.436436
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −6.92820 −0.840168
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 3.46410 0.408248
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 13.8564 1.61077
\(75\) 0 0
\(76\) 3.46410 0.397360
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −21.0000 −2.31906
\(83\) 3.46410 0.380235 0.190117 − 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) −1.73205 −0.188982
\(85\) 0 0
\(86\) 15.0000 1.61749
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) −15.5885 −1.60783
\(95\) 0 0
\(96\) −5.19615 −0.530330
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −6.92820 −0.699854
\(99\) 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 3025.2.a.l.1.2 2
5.4 even 2 605.2.a.e.1.1 ✓ 2
11.10 odd 2 inner 3025.2.a.l.1.1 2
15.14 odd 2 5445.2.a.u.1.2 2
20.19 odd 2 9680.2.a.bu.1.1 2
55.4 even 10 605.2.g.i.511.1 8
55.9 even 10 605.2.g.i.81.2 8
55.14 even 10 605.2.g.i.251.1 8
55.19 odd 10 605.2.g.i.251.2 8
55.24 odd 10 605.2.g.i.81.1 8
55.29 odd 10 605.2.g.i.511.2 8
55.39 odd 10 605.2.g.i.366.1 8
55.49 even 10 605.2.g.i.366.2 8
55.54 odd 2 605.2.a.e.1.2 yes 2
165.164 even 2 5445.2.a.u.1.1 2
220.219 even 2 9680.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.e.1.1 ✓ 2 5.4 even 2
605.2.a.e.1.2 yes 2 55.54 odd 2
605.2.g.i.81.1 8 55.24 odd 10
605.2.g.i.81.2 8 55.9 even 10
605.2.g.i.251.1 8 55.14 even 10
605.2.g.i.251.2 8 55.19 odd 10
605.2.g.i.366.1 8 55.39 odd 10
605.2.g.i.366.2 8 55.49 even 10
605.2.g.i.511.1 8 55.4 even 10
605.2.g.i.511.2 8 55.29 odd 10
3025.2.a.l.1.1 2 11.10 odd 2 inner
3025.2.a.l.1.2 2 1.1 even 1 trivial
5445.2.a.u.1.1 2 165.164 even 2
5445.2.a.u.1.2 2 15.14 odd 2
9680.2.a.bu.1.1 2 20.19 odd 2
9680.2.a.bu.1.2 2 220.219 even 2