Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(149,925)] 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("925.149"); 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([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-20,0,-12,0,0,-12,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.60703296077824.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 20x^{8} + 142x^{6} + 420x^{4} + 457x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.10
Root \(-2.72362i\) of defining polynomial
Character \(\chi\) \(=\) 925.149
Dual form 925.2.b.f.149.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.72362i q^{2} +2.29298i q^{3} -5.41809 q^{4} -6.24519 q^{6} +3.82710i q^{7} -9.30957i q^{8} -2.25774 q^{9} -4.41809 q^{11} -12.4236i q^{12} +3.67583i q^{13} -10.4236 q^{14} +14.5195 q^{16} -2.28688i q^{17} -6.14922i q^{18} +2.39037 q^{19} -8.77545 q^{21} -12.0332i q^{22} +0.265251i q^{23} +21.3466 q^{24} -10.0116 q^{26} +1.70198i q^{27} -20.7356i q^{28} +6.58595 q^{29} +2.34076 q^{31} +20.9265i q^{32} -10.1306i q^{33} +6.22860 q^{34} +12.2326 q^{36} +1.00000i q^{37} +6.51044i q^{38} -8.42859 q^{39} -4.41809 q^{41} -23.9010i q^{42} -7.71249i q^{43} +23.9376 q^{44} -0.722443 q^{46} +10.9285i q^{47} +33.2929i q^{48} -7.64669 q^{49} +5.24377 q^{51} -19.9160i q^{52} +0.109574i q^{53} -4.63555 q^{54} +35.6286 q^{56} +5.48105i q^{57} +17.9376i q^{58} +2.00504 q^{59} +3.96271 q^{61} +6.37534i q^{62} -8.64059i q^{63} -27.9567 q^{64} +27.5918 q^{66} +6.80664i q^{67} +12.3905i q^{68} -0.608215 q^{69} -5.79485 q^{71} +21.0186i q^{72} +0.140654i q^{73} -2.72362 q^{74} -12.9512 q^{76} -16.9085i q^{77} -22.9563i q^{78} +6.62418 q^{79} -10.6758 q^{81} -12.0332i q^{82} -13.9904i q^{83} +47.5462 q^{84} +21.0059 q^{86} +15.1014i q^{87} +41.1305i q^{88} -14.8139 q^{89} -14.0678 q^{91} -1.43716i q^{92} +5.36731i q^{93} -29.7651 q^{94} -47.9839 q^{96} -8.94394i q^{97} -20.8266i q^{98} +9.97490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72362i 1.92589i 0.269701 + 0.962944i \(0.413075\pi\)
−0.269701 + 0.962944i \(0.586925\pi\)
\(3\) 2.29298i 1.32385i 0.749570 + 0.661925i \(0.230260\pi\)
−0.749570 + 0.661925i \(0.769740\pi\)
\(4\) −5.41809 −2.70905
\(5\) 0 0
\(6\) −6.24519 −2.54959
\(7\) 3.82710i 1.44651i 0.690582 + 0.723254i \(0.257354\pi\)
−0.690582 + 0.723254i \(0.742646\pi\)
\(8\) − 9.30957i − 3.29143i
\(9\) −2.25774 −0.752580
\(10\) 0 0
\(11\) −4.41809 −1.33210 −0.666052 − 0.745905i \(-0.732017\pi\)
−0.666052 + 0.745905i \(0.732017\pi\)
\(12\) − 12.4236i − 3.58637i
\(13\) 3.67583i 1.01949i 0.860325 + 0.509746i \(0.170261\pi\)
−0.860325 + 0.509746i \(0.829739\pi\)
\(14\) −10.4236 −2.78581
\(15\) 0 0
\(16\) 14.5195 3.62988
\(17\) − 2.28688i − 0.554651i −0.960776 − 0.277325i \(-0.910552\pi\)
0.960776 − 0.277325i \(-0.0894480\pi\)
\(18\) − 6.14922i − 1.44938i
\(19\) 2.39037 0.548387 0.274194 − 0.961674i \(-0.411589\pi\)
0.274194 + 0.961674i \(0.411589\pi\)
\(20\) 0 0
\(21\) −8.77545 −1.91496
\(22\) − 12.0332i − 2.56548i
\(23\) 0.265251i 0.0553087i 0.999618 + 0.0276544i \(0.00880378\pi\)
−0.999618 + 0.0276544i \(0.991196\pi\)
\(24\) 21.3466 4.35736
\(25\) 0 0
\(26\) −10.0116 −1.96343
\(27\) 1.70198i 0.327547i
\(28\) − 20.7356i − 3.91865i
\(29\) 6.58595 1.22298 0.611490 − 0.791252i \(-0.290570\pi\)
0.611490 + 0.791252i \(0.290570\pi\)
\(30\) 0 0
\(31\) 2.34076 0.420413 0.210207 − 0.977657i \(-0.432586\pi\)
0.210207 + 0.977657i \(0.432586\pi\)
\(32\) 20.9265i 3.69931i
\(33\) − 10.1306i − 1.76351i
\(34\) 6.22860 1.06820
\(35\) 0 0
\(36\) 12.2326 2.03877
\(37\) 1.00000i 0.164399i
\(38\) 6.51044i 1.05613i
\(39\) −8.42859 −1.34965
\(40\) 0 0
\(41\) −4.41809 −0.689990 −0.344995 − 0.938605i \(-0.612119\pi\)
−0.344995 + 0.938605i \(0.612119\pi\)
\(42\) − 23.9010i − 3.68800i
\(43\) − 7.71249i − 1.17614i −0.808809 − 0.588072i \(-0.799887\pi\)
0.808809 − 0.588072i \(-0.200113\pi\)
\(44\) 23.9376 3.60873
\(45\) 0 0
\(46\) −0.722443 −0.106518
\(47\) 10.9285i 1.59409i 0.603920 + 0.797045i \(0.293605\pi\)
−0.603920 + 0.797045i \(0.706395\pi\)
\(48\) 33.2929i 4.80542i
\(49\) −7.64669 −1.09238
\(50\) 0 0
\(51\) 5.24377 0.734275
\(52\) − 19.9160i − 2.76185i
\(53\) 0.109574i 0.0150512i 0.999972 + 0.00752559i \(0.00239549\pi\)
−0.999972 + 0.00752559i \(0.997605\pi\)
\(54\) −4.63555 −0.630819
\(55\) 0 0
\(56\) 35.6286 4.76108
\(57\) 5.48105i 0.725983i
\(58\) 17.9376i 2.35532i
\(59\) 2.00504 0.261034 0.130517 − 0.991446i \(-0.458336\pi\)
0.130517 + 0.991446i \(0.458336\pi\)
\(60\) 0 0
\(61\) 3.96271 0.507374 0.253687 − 0.967286i \(-0.418357\pi\)
0.253687 + 0.967286i \(0.418357\pi\)
\(62\) 6.37534i 0.809669i
\(63\) − 8.64059i − 1.08861i
\(64\) −27.9567 −3.49458
\(65\) 0 0
\(66\) 27.5918 3.39632
\(67\) 6.80664i 0.831563i 0.909464 + 0.415782i \(0.136492\pi\)
−0.909464 + 0.415782i \(0.863508\pi\)
\(68\) 12.3905i 1.50257i
\(69\) −0.608215 −0.0732205
\(70\) 0 0
\(71\) −5.79485 −0.687722 −0.343861 − 0.939020i \(-0.611735\pi\)
−0.343861 + 0.939020i \(0.611735\pi\)
\(72\) 21.0186i 2.47706i
\(73\) 0.140654i 0.0164623i 0.999966 + 0.00823116i \(0.00262009\pi\)
−0.999966 + 0.00823116i \(0.997380\pi\)
\(74\) −2.72362 −0.316614
\(75\) 0 0
\(76\) −12.9512 −1.48561
\(77\) − 16.9085i − 1.92690i
\(78\) − 22.9563i − 2.59928i
\(79\) 6.62418 0.745278 0.372639 − 0.927976i \(-0.378453\pi\)
0.372639 + 0.927976i \(0.378453\pi\)
\(80\) 0 0
\(81\) −10.6758 −1.18620
\(82\) − 12.0332i − 1.32884i
\(83\) − 13.9904i − 1.53565i −0.640660 − 0.767825i \(-0.721339\pi\)
0.640660 − 0.767825i \(-0.278661\pi\)
\(84\) 47.5462 5.18771
\(85\) 0 0
\(86\) 21.0059 2.26512
\(87\) 15.1014i 1.61904i
\(88\) 41.1305i 4.38453i
\(89\) −14.8139 −1.57027 −0.785136 − 0.619323i \(-0.787407\pi\)
−0.785136 + 0.619323i \(0.787407\pi\)
\(90\) 0 0
\(91\) −14.0678 −1.47470
\(92\) − 1.43716i − 0.149834i
\(93\) 5.36731i 0.556565i
\(94\) −29.7651 −3.07004
\(95\) 0 0
\(96\) −47.9839 −4.89734
\(97\) − 8.94394i − 0.908119i −0.890971 − 0.454060i \(-0.849975\pi\)
0.890971 − 0.454060i \(-0.150025\pi\)
\(98\) − 20.8266i − 2.10381i
\(99\) 9.97490 1.00252
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.b.f.149.10 10
5.2 odd 4 925.2.a.f.1.1 5
5.3 odd 4 185.2.a.e.1.5 ✓ 5
5.4 even 2 inner 925.2.b.f.149.1 10
15.2 even 4 8325.2.a.ch.1.5 5
15.8 even 4 1665.2.a.p.1.1 5
20.3 even 4 2960.2.a.w.1.5 5
35.13 even 4 9065.2.a.k.1.5 5
185.73 odd 4 6845.2.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.5 ✓ 5 5.3 odd 4
925.2.a.f.1.1 5 5.2 odd 4
925.2.b.f.149.1 10 5.4 even 2 inner
925.2.b.f.149.10 10 1.1 even 1 trivial
1665.2.a.p.1.1 5 15.8 even 4
2960.2.a.w.1.5 5 20.3 even 4
6845.2.a.f.1.1 5 185.73 odd 4
8325.2.a.ch.1.5 5 15.2 even 4
9065.2.a.k.1.5 5 35.13 even 4