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 149.6
Root \(-0.728950i\) of defining polynomial
Character \(\chi\) \(=\) 925.149
Dual form 925.2.b.f.149.5

$q$-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)
 
\(f(q)\) \(=\) \(q+0.728950i q^{2} -2.62871i q^{3} +1.46863 q^{4} +1.91620 q^{6} +2.55244i q^{7} +2.52846i q^{8} -3.91009 q^{9} +2.46863 q^{11} -3.86060i q^{12} -1.55854i q^{13} -1.86060 q^{14} +1.09414 q^{16} -6.83662i q^{17} -2.85026i q^{18} +7.66011 q^{19} +6.70960 q^{21} +1.79951i q^{22} +7.50003i q^{23} +6.64658 q^{24} +1.13610 q^{26} +2.39236i q^{27} +3.74859i q^{28} -3.25741 q^{29} +0.658785 q^{31} +5.85449i q^{32} -6.48930i q^{33} +4.98356 q^{34} -5.74248 q^{36} +1.00000i q^{37} +5.58384i q^{38} -4.09694 q^{39} +2.46863 q^{41} +4.89097i q^{42} -10.9579i q^{43} +3.62551 q^{44} -5.46715 q^{46} +3.11521i q^{47} -2.87617i q^{48} +0.485072 q^{49} -17.9715 q^{51} -2.28892i q^{52} -8.64184i q^{53} -1.74391 q^{54} -6.45373 q^{56} -20.1362i q^{57} -2.37449i q^{58} +6.23634 q^{59} +3.27808 q^{61} +0.480222i q^{62} -9.98026i q^{63} -2.07935 q^{64} +4.73038 q^{66} +1.47764i q^{67} -10.0405i q^{68} +19.7154 q^{69} -8.06686 q^{71} -9.88651i q^{72} +4.96199i q^{73} -0.728950 q^{74} +11.2499 q^{76} +6.30102i q^{77} -2.98647i q^{78} -12.8206 q^{79} -5.44146 q^{81} +1.79951i q^{82} -1.14934i q^{83} +9.85393 q^{84} +7.98779 q^{86} +8.56277i q^{87} +6.24184i q^{88} -11.5207 q^{89} +3.97807 q^{91} +11.0148i q^{92} -1.73175i q^{93} -2.27083 q^{94} +15.3897 q^{96} +17.2929i q^{97} +0.353594i q^{98} -9.65257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.728950i 0.515446i 0.966219 + 0.257723i \(0.0829722\pi\)
−0.966219 + 0.257723i \(0.917028\pi\)
\(3\) − 2.62871i − 1.51768i −0.651275 − 0.758842i \(-0.725766\pi\)
0.651275 − 0.758842i \(-0.274234\pi\)
\(4\) 1.46863 0.734316
\(5\) 0 0
\(6\) 1.91620 0.782284
\(7\) 2.55244i 0.964730i 0.875970 + 0.482365i \(0.160222\pi\)
−0.875970 + 0.482365i \(0.839778\pi\)
\(8\) 2.52846i 0.893946i
\(9\) −3.91009 −1.30336
\(10\) 0 0
\(11\) 2.46863 0.744320 0.372160 − 0.928169i \(-0.378617\pi\)
0.372160 + 0.928169i \(0.378617\pi\)
\(12\) − 3.86060i − 1.11446i
\(13\) − 1.55854i − 0.432261i −0.976364 − 0.216131i \(-0.930656\pi\)
0.976364 − 0.216131i \(-0.0693437\pi\)
\(14\) −1.86060 −0.497266
\(15\) 0 0
\(16\) 1.09414 0.273535
\(17\) − 6.83662i − 1.65812i −0.559156 − 0.829062i \(-0.688875\pi\)
0.559156 − 0.829062i \(-0.311125\pi\)
\(18\) − 2.85026i − 0.671813i
\(19\) 7.66011 1.75735 0.878675 − 0.477421i \(-0.158428\pi\)
0.878675 + 0.477421i \(0.158428\pi\)
\(20\) 0 0
\(21\) 6.70960 1.46415
\(22\) 1.79951i 0.383657i
\(23\) 7.50003i 1.56387i 0.623363 + 0.781933i \(0.285766\pi\)
−0.623363 + 0.781933i \(0.714234\pi\)
\(24\) 6.64658 1.35673
\(25\) 0 0
\(26\) 1.13610 0.222807
\(27\) 2.39236i 0.460410i
\(28\) 3.74859i 0.708416i
\(29\) −3.25741 −0.604886 −0.302443 − 0.953167i \(-0.597802\pi\)
−0.302443 + 0.953167i \(0.597802\pi\)
\(30\) 0 0
\(31\) 0.658785 0.118321 0.0591607 − 0.998248i \(-0.481158\pi\)
0.0591607 + 0.998248i \(0.481158\pi\)
\(32\) 5.85449i 1.03494i
\(33\) − 6.48930i − 1.12964i
\(34\) 4.98356 0.854673
\(35\) 0 0
\(36\) −5.74248 −0.957080
\(37\) 1.00000i 0.164399i
\(38\) 5.58384i 0.905818i
\(39\) −4.09694 −0.656036
\(40\) 0 0
\(41\) 2.46863 0.385535 0.192768 − 0.981244i \(-0.438254\pi\)
0.192768 + 0.981244i \(0.438254\pi\)
\(42\) 4.89097i 0.754692i
\(43\) − 10.9579i − 1.67107i −0.549438 − 0.835535i \(-0.685158\pi\)
0.549438 − 0.835535i \(-0.314842\pi\)
\(44\) 3.62551 0.546566
\(45\) 0 0
\(46\) −5.46715 −0.806088
\(47\) 3.11521i 0.454400i 0.973848 + 0.227200i \(0.0729571\pi\)
−0.973848 + 0.227200i \(0.927043\pi\)
\(48\) − 2.87617i − 0.415140i
\(49\) 0.485072 0.0692960
\(50\) 0 0
\(51\) −17.9715 −2.51651
\(52\) − 2.28892i − 0.317416i
\(53\) − 8.64184i − 1.18705i −0.804816 − 0.593524i \(-0.797736\pi\)
0.804816 − 0.593524i \(-0.202264\pi\)
\(54\) −1.74391 −0.237317
\(55\) 0 0
\(56\) −6.45373 −0.862416
\(57\) − 20.1362i − 2.66710i
\(58\) − 2.37449i − 0.311786i
\(59\) 6.23634 0.811903 0.405951 − 0.913895i \(-0.366940\pi\)
0.405951 + 0.913895i \(0.366940\pi\)
\(60\) 0 0
\(61\) 3.27808 0.419716 0.209858 − 0.977732i \(-0.432700\pi\)
0.209858 + 0.977732i \(0.432700\pi\)
\(62\) 0.480222i 0.0609882i
\(63\) − 9.98026i − 1.25739i
\(64\) −2.07935 −0.259919
\(65\) 0 0
\(66\) 4.73038 0.582270
\(67\) 1.47764i 0.180523i 0.995918 + 0.0902615i \(0.0287703\pi\)
−0.995918 + 0.0902615i \(0.971230\pi\)
\(68\) − 10.0405i − 1.21759i
\(69\) 19.7154 2.37345
\(70\) 0 0
\(71\) −8.06686 −0.957360 −0.478680 − 0.877989i \(-0.658885\pi\)
−0.478680 + 0.877989i \(0.658885\pi\)
\(72\) − 9.88651i − 1.16514i
\(73\) 4.96199i 0.580757i 0.956912 + 0.290379i \(0.0937812\pi\)
−0.956912 + 0.290379i \(0.906219\pi\)
\(74\) −0.728950 −0.0847388
\(75\) 0 0
\(76\) 11.2499 1.29045
\(77\) 6.30102i 0.718068i
\(78\) − 2.98647i − 0.338151i
\(79\) −12.8206 −1.44243 −0.721214 − 0.692713i \(-0.756415\pi\)
−0.721214 + 0.692713i \(0.756415\pi\)
\(80\) 0 0
\(81\) −5.44146 −0.604607
\(82\) 1.79951i 0.198723i
\(83\) − 1.14934i − 0.126157i −0.998009 − 0.0630784i \(-0.979908\pi\)
0.998009 − 0.0630784i \(-0.0200918\pi\)
\(84\) 9.85393 1.07515
\(85\) 0 0
\(86\) 7.98779 0.861346
\(87\) 8.56277i 0.918026i
\(88\) 6.24184i 0.665382i
\(89\) −11.5207 −1.22119 −0.610596 − 0.791942i \(-0.709070\pi\)
−0.610596 + 0.791942i \(0.709070\pi\)
\(90\) 0 0
\(91\) 3.97807 0.417015
\(92\) 11.0148i 1.14837i
\(93\) − 1.73175i − 0.179574i
\(94\) −2.27083 −0.234218
\(95\) 0 0
\(96\) 15.3897 1.57071
\(97\) 17.2929i 1.75583i 0.478815 + 0.877916i \(0.341067\pi\)
−0.478815 + 0.877916i \(0.658933\pi\)
\(98\) 0.353594i 0.0357183i
\(99\) −9.65257 −0.970120
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.149.6 10
5.2 odd 4 925.2.a.f.1.3 5
5.3 odd 4 185.2.a.e.1.3 ✓ 5
5.4 even 2 inner 925.2.b.f.149.5 10
15.2 even 4 8325.2.a.ch.1.3 5
15.8 even 4 1665.2.a.p.1.3 5
20.3 even 4 2960.2.a.w.1.1 5
35.13 even 4 9065.2.a.k.1.3 5
185.73 odd 4 6845.2.a.f.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.3 ✓ 5 5.3 odd 4
925.2.a.f.1.3 5 5.2 odd 4
925.2.b.f.149.5 10 5.4 even 2 inner
925.2.b.f.149.6 10 1.1 even 1 trivial
1665.2.a.p.1.3 5 15.8 even 4
2960.2.a.w.1.1 5 20.3 even 4
6845.2.a.f.1.3 5 185.73 odd 4
8325.2.a.ch.1.3 5 15.2 even 4
9065.2.a.k.1.3 5 35.13 even 4