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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 23x^{2} - 9x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.74907\) of defining polynomial
Character \(\chi\) \(=\) 925.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.74907 q^{2} +2.56943 q^{3} +1.05925 q^{4} +4.49412 q^{6} +0.963428 q^{7} -1.64544 q^{8} +3.60198 q^{9} +2.88433 q^{11} +2.72168 q^{12} -1.65840 q^{13} +1.68510 q^{14} -4.99649 q^{16} +6.95992 q^{17} +6.30013 q^{18} -2.39544 q^{19} +2.47546 q^{21} +5.04490 q^{22} -0.0129671 q^{23} -4.22783 q^{24} -2.90067 q^{26} +1.54675 q^{27} +1.02051 q^{28} -7.63082 q^{29} +2.66242 q^{31} -5.44835 q^{32} +7.41108 q^{33} +12.1734 q^{34} +3.81541 q^{36} +1.00000 q^{37} -4.18979 q^{38} -4.26115 q^{39} +1.52856 q^{41} +4.32976 q^{42} +4.22095 q^{43} +3.05523 q^{44} -0.0226804 q^{46} -11.0761 q^{47} -12.8381 q^{48} -6.07181 q^{49} +17.8830 q^{51} -1.75667 q^{52} -9.51515 q^{53} +2.70538 q^{54} -1.58526 q^{56} -6.15492 q^{57} -13.3469 q^{58} -5.18662 q^{59} -0.854073 q^{61} +4.65677 q^{62} +3.47025 q^{63} +0.463427 q^{64} +12.9625 q^{66} +9.33337 q^{67} +7.37231 q^{68} -0.0333181 q^{69} +2.93088 q^{71} -5.92683 q^{72} -3.04319 q^{73} +1.74907 q^{74} -2.53737 q^{76} +2.77884 q^{77} -7.45306 q^{78} +8.24087 q^{79} -6.83167 q^{81} +2.67356 q^{82} -11.2635 q^{83} +2.62214 q^{84} +7.38275 q^{86} -19.6069 q^{87} -4.74597 q^{88} +8.75646 q^{89} -1.59775 q^{91} -0.0137354 q^{92} +6.84092 q^{93} -19.3729 q^{94} -13.9992 q^{96} -2.99392 q^{97} -10.6200 q^{98} +10.3893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - q^{3} + 7 q^{4} + 6 q^{6} + 10 q^{9} + 16 q^{11} + 11 q^{12} - q^{13} - 3 q^{14} + 3 q^{16} + 4 q^{17} - 23 q^{18} + 9 q^{19} + 2 q^{21} + q^{22} - q^{23} + 24 q^{26} - q^{27} + 7 q^{28}+ \cdots + 39 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.74907 1.23678 0.618390 − 0.785871i \(-0.287785\pi\)
0.618390 + 0.785871i \(0.287785\pi\)
\(3\) 2.56943 1.48346 0.741731 − 0.670697i \(-0.234005\pi\)
0.741731 + 0.670697i \(0.234005\pi\)
\(4\) 1.05925 0.529626
\(5\) 0 0
\(6\) 4.49412 1.83472
\(7\) 0.963428 0.364142 0.182071 − 0.983285i \(-0.441720\pi\)
0.182071 + 0.983285i \(0.441720\pi\)
\(8\) −1.64544 −0.581749
\(9\) 3.60198 1.20066
\(10\) 0 0
\(11\) 2.88433 0.869657 0.434829 − 0.900513i \(-0.356809\pi\)
0.434829 + 0.900513i \(0.356809\pi\)
\(12\) 2.72168 0.785680
\(13\) −1.65840 −0.459958 −0.229979 − 0.973196i \(-0.573866\pi\)
−0.229979 + 0.973196i \(0.573866\pi\)
\(14\) 1.68510 0.450363
\(15\) 0 0
\(16\) −4.99649 −1.24912
\(17\) 6.95992 1.68803 0.844014 − 0.536321i \(-0.180186\pi\)
0.844014 + 0.536321i \(0.180186\pi\)
\(18\) 6.30013 1.48495
\(19\) −2.39544 −0.549551 −0.274776 − 0.961508i \(-0.588604\pi\)
−0.274776 + 0.961508i \(0.588604\pi\)
\(20\) 0 0
\(21\) 2.47546 0.540190
\(22\) 5.04490 1.07558
\(23\) −0.0129671 −0.00270383 −0.00135191 − 0.999999i \(-0.500430\pi\)
−0.00135191 + 0.999999i \(0.500430\pi\)
\(24\) −4.22783 −0.863003
\(25\) 0 0
\(26\) −2.90067 −0.568867
\(27\) 1.54675 0.297673
\(28\) 1.02051 0.192859
\(29\) −7.63082 −1.41701 −0.708504 − 0.705707i \(-0.750629\pi\)
−0.708504 + 0.705707i \(0.750629\pi\)
\(30\) 0 0
\(31\) 2.66242 0.478186 0.239093 − 0.970997i \(-0.423150\pi\)
0.239093 + 0.970997i \(0.423150\pi\)
\(32\) −5.44835 −0.963141
\(33\) 7.41108 1.29010
\(34\) 12.1734 2.08772
\(35\) 0 0
\(36\) 3.81541 0.635901
\(37\) 1.00000 0.164399
\(38\) −4.18979 −0.679674
\(39\) −4.26115 −0.682331
\(40\) 0 0
\(41\) 1.52856 0.238721 0.119360 − 0.992851i \(-0.461916\pi\)
0.119360 + 0.992851i \(0.461916\pi\)
\(42\) 4.32976 0.668097
\(43\) 4.22095 0.643689 0.321845 − 0.946792i \(-0.395697\pi\)
0.321845 + 0.946792i \(0.395697\pi\)
\(44\) 3.05523 0.460593
\(45\) 0 0
\(46\) −0.0226804 −0.00334404
\(47\) −11.0761 −1.61562 −0.807808 − 0.589446i \(-0.799346\pi\)
−0.807808 + 0.589446i \(0.799346\pi\)
\(48\) −12.8381 −1.85303
\(49\) −6.07181 −0.867401
\(50\) 0 0
\(51\) 17.8830 2.50413
\(52\) −1.75667 −0.243606
\(53\) −9.51515 −1.30701 −0.653503 − 0.756924i \(-0.726701\pi\)
−0.653503 + 0.756924i \(0.726701\pi\)
\(54\) 2.70538 0.368156
\(55\) 0 0
\(56\) −1.58526 −0.211839
\(57\) −6.15492 −0.815238
\(58\) −13.3469 −1.75253
\(59\) −5.18662 −0.675240 −0.337620 − 0.941282i \(-0.609622\pi\)
−0.337620 + 0.941282i \(0.609622\pi\)
\(60\) 0 0
\(61\) −0.854073 −0.109353 −0.0546764 − 0.998504i \(-0.517413\pi\)
−0.0546764 + 0.998504i \(0.517413\pi\)
\(62\) 4.65677 0.591411
\(63\) 3.47025 0.437210
\(64\) 0.463427 0.0579284
\(65\) 0 0
\(66\) 12.9625 1.59558
\(67\) 9.33337 1.14025 0.570126 − 0.821557i \(-0.306894\pi\)
0.570126 + 0.821557i \(0.306894\pi\)
\(68\) 7.37231 0.894024
\(69\) −0.0333181 −0.00401102
\(70\) 0 0
\(71\) 2.93088 0.347831 0.173916 − 0.984761i \(-0.444358\pi\)
0.173916 + 0.984761i \(0.444358\pi\)
\(72\) −5.92683 −0.698483
\(73\) −3.04319 −0.356179 −0.178089 − 0.984014i \(-0.556992\pi\)
−0.178089 + 0.984014i \(0.556992\pi\)
\(74\) 1.74907 0.203325
\(75\) 0 0
\(76\) −2.53737 −0.291057
\(77\) 2.77884 0.316678
\(78\) −7.45306 −0.843893
\(79\) 8.24087 0.927171 0.463585 − 0.886052i \(-0.346563\pi\)
0.463585 + 0.886052i \(0.346563\pi\)
\(80\) 0 0
\(81\) −6.83167 −0.759075
\(82\) 2.67356 0.295245
\(83\) −11.2635 −1.23633 −0.618166 − 0.786047i \(-0.712124\pi\)
−0.618166 + 0.786047i \(0.712124\pi\)
\(84\) 2.62214 0.286099
\(85\) 0 0
\(86\) 7.38275 0.796102
\(87\) −19.6069 −2.10208
\(88\) −4.74597 −0.505923
\(89\) 8.75646 0.928183 0.464092 − 0.885787i \(-0.346381\pi\)
0.464092 + 0.885787i \(0.346381\pi\)
\(90\) 0 0
\(91\) −1.59775 −0.167490
\(92\) −0.0137354 −0.00143202
\(93\) 6.84092 0.709370
\(94\) −19.3729 −1.99816
\(95\) 0 0
\(96\) −13.9992 −1.42878
\(97\) −2.99392 −0.303987 −0.151993 − 0.988382i \(-0.548569\pi\)
−0.151993 + 0.988382i \(0.548569\pi\)
\(98\) −10.6200 −1.07278
\(99\) 10.3893 1.04416
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 925.2.a.k.1.6 yes 7
3.2 odd 2 8325.2.a.cm.1.2 7
5.2 odd 4 925.2.b.i.149.11 14
5.3 odd 4 925.2.b.i.149.4 14
5.4 even 2 925.2.a.j.1.2 ✓ 7
15.14 odd 2 8325.2.a.cn.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.j.1.2 ✓ 7 5.4 even 2
925.2.a.k.1.6 yes 7 1.1 even 1 trivial
925.2.b.i.149.4 14 5.3 odd 4
925.2.b.i.149.11 14 5.2 odd 4
8325.2.a.cm.1.2 7 3.2 odd 2
8325.2.a.cn.1.6 7 15.14 odd 2