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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,-2,0,-12,0,0,0,0,-32] 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: 10.0.4414301848576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 11x^{8} + 43x^{6} + 72x^{4} + 49x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.6
Root \(0.531633i\) of defining polynomial
Character \(\chi\) \(=\) 925.149
Dual form 925.2.b.h.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)
 
\(f(q)\) \(=\) \(q+0.531633i q^{2} -2.13508i q^{3} +1.71737 q^{4} +1.13508 q^{6} -2.93944i q^{7} +1.97628i q^{8} -1.55856 q^{9} -5.92562 q^{11} -3.66671i q^{12} -1.35031i q^{13} +1.56270 q^{14} +2.38408 q^{16} -6.16935i q^{17} -0.828584i q^{18} -5.46850 q^{19} -6.27593 q^{21} -3.15026i q^{22} +1.87203i q^{23} +4.21950 q^{24} +0.717868 q^{26} -3.07758i q^{27} -5.04809i q^{28} +7.96110 q^{29} +0.229916 q^{31} +5.22001i q^{32} +12.6517i q^{33} +3.27983 q^{34} -2.67662 q^{36} -1.00000i q^{37} -2.90724i q^{38} -2.88301 q^{39} -3.07758 q^{41} -3.33649i q^{42} -8.13251i q^{43} -10.1765 q^{44} -0.995233 q^{46} +9.06340i q^{47} -5.09020i q^{48} -1.64029 q^{49} -13.1721 q^{51} -2.31897i q^{52} +2.63638i q^{53} +1.63615 q^{54} +5.80914 q^{56} +11.6757i q^{57} +4.23238i q^{58} +11.7344 q^{59} +4.90190 q^{61} +0.122231i q^{62} +4.58129i q^{63} +1.99303 q^{64} -6.72605 q^{66} +4.03427i q^{67} -10.5950i q^{68} +3.99693 q^{69} -9.03377 q^{71} -3.08015i q^{72} -16.0541i q^{73} +0.531633 q^{74} -9.39142 q^{76} +17.4180i q^{77} -1.53271i q^{78} +5.53201 q^{79} -11.2466 q^{81} -1.63615i q^{82} +1.13714i q^{83} -10.7781 q^{84} +4.32351 q^{86} -16.9976i q^{87} -11.7107i q^{88} -8.14306 q^{89} -3.96914 q^{91} +3.21496i q^{92} -0.490890i q^{93} -4.81841 q^{94} +11.1451 q^{96} -17.9941i q^{97} -0.872030i q^{98} +9.23545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{4} - 12 q^{6} - 32 q^{11} + 10 q^{14} - 22 q^{16} + 22 q^{19} - 28 q^{21} - 12 q^{24} - 8 q^{26} + 30 q^{29} - 26 q^{31} - 26 q^{34} - 22 q^{36} + 20 q^{39} - 26 q^{41} + 12 q^{44} - 6 q^{46}+ \cdots + 42 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.531633i 0.375921i 0.982177 + 0.187961i \(0.0601878\pi\)
−0.982177 + 0.187961i \(0.939812\pi\)
\(3\) − 2.13508i − 1.23269i −0.787477 − 0.616344i \(-0.788613\pi\)
0.787477 − 0.616344i \(-0.211387\pi\)
\(4\) 1.71737 0.858683
\(5\) 0 0
\(6\) 1.13508 0.463394
\(7\) − 2.93944i − 1.11100i −0.831516 − 0.555501i \(-0.812526\pi\)
0.831516 − 0.555501i \(-0.187474\pi\)
\(8\) 1.97628i 0.698719i
\(9\) −1.55856 −0.519521
\(10\) 0 0
\(11\) −5.92562 −1.78664 −0.893321 − 0.449419i \(-0.851631\pi\)
−0.893321 + 0.449419i \(0.851631\pi\)
\(12\) − 3.66671i − 1.05849i
\(13\) − 1.35031i − 0.374508i −0.982312 − 0.187254i \(-0.940041\pi\)
0.982312 − 0.187254i \(-0.0599587\pi\)
\(14\) 1.56270 0.417650
\(15\) 0 0
\(16\) 2.38408 0.596020
\(17\) − 6.16935i − 1.49629i −0.663537 − 0.748144i \(-0.730945\pi\)
0.663537 − 0.748144i \(-0.269055\pi\)
\(18\) − 0.828584i − 0.195299i
\(19\) −5.46850 −1.25456 −0.627280 − 0.778793i \(-0.715832\pi\)
−0.627280 + 0.778793i \(0.715832\pi\)
\(20\) 0 0
\(21\) −6.27593 −1.36952
\(22\) − 3.15026i − 0.671637i
\(23\) 1.87203i 0.390345i 0.980769 + 0.195173i \(0.0625267\pi\)
−0.980769 + 0.195173i \(0.937473\pi\)
\(24\) 4.21950 0.861303
\(25\) 0 0
\(26\) 0.717868 0.140786
\(27\) − 3.07758i − 0.592281i
\(28\) − 5.04809i − 0.953999i
\(29\) 7.96110 1.47834 0.739169 − 0.673520i \(-0.235218\pi\)
0.739169 + 0.673520i \(0.235218\pi\)
\(30\) 0 0
\(31\) 0.229916 0.0412942 0.0206471 − 0.999787i \(-0.493427\pi\)
0.0206471 + 0.999787i \(0.493427\pi\)
\(32\) 5.22001i 0.922775i
\(33\) 12.6517i 2.20237i
\(34\) 3.27983 0.562487
\(35\) 0 0
\(36\) −2.67662 −0.446104
\(37\) − 1.00000i − 0.164399i
\(38\) − 2.90724i − 0.471616i
\(39\) −2.88301 −0.461651
\(40\) 0 0
\(41\) −3.07758 −0.480638 −0.240319 − 0.970694i \(-0.577252\pi\)
−0.240319 + 0.970694i \(0.577252\pi\)
\(42\) − 3.33649i − 0.514832i
\(43\) − 8.13251i − 1.24020i −0.784524 − 0.620098i \(-0.787093\pi\)
0.784524 − 0.620098i \(-0.212907\pi\)
\(44\) −10.1765 −1.53416
\(45\) 0 0
\(46\) −0.995233 −0.146739
\(47\) 9.06340i 1.32203i 0.750371 + 0.661017i \(0.229875\pi\)
−0.750371 + 0.661017i \(0.770125\pi\)
\(48\) − 5.09020i − 0.734706i
\(49\) −1.64029 −0.234326
\(50\) 0 0
\(51\) −13.1721 −1.84446
\(52\) − 2.31897i − 0.321583i
\(53\) 2.63638i 0.362135i 0.983471 + 0.181067i \(0.0579552\pi\)
−0.983471 + 0.181067i \(0.942045\pi\)
\(54\) 1.63615 0.222651
\(55\) 0 0
\(56\) 5.80914 0.776278
\(57\) 11.6757i 1.54648i
\(58\) 4.23238i 0.555739i
\(59\) 11.7344 1.52769 0.763843 − 0.645402i \(-0.223310\pi\)
0.763843 + 0.645402i \(0.223310\pi\)
\(60\) 0 0
\(61\) 4.90190 0.627624 0.313812 − 0.949485i \(-0.398394\pi\)
0.313812 + 0.949485i \(0.398394\pi\)
\(62\) 0.122231i 0.0155234i
\(63\) 4.58129i 0.577189i
\(64\) 1.99303 0.249128
\(65\) 0 0
\(66\) −6.72605 −0.827919
\(67\) 4.03427i 0.492865i 0.969160 + 0.246432i \(0.0792584\pi\)
−0.969160 + 0.246432i \(0.920742\pi\)
\(68\) − 10.5950i − 1.28484i
\(69\) 3.99693 0.481174
\(70\) 0 0
\(71\) −9.03377 −1.07211 −0.536056 − 0.844183i \(-0.680086\pi\)
−0.536056 + 0.844183i \(0.680086\pi\)
\(72\) − 3.08015i − 0.362999i
\(73\) − 16.0541i − 1.87899i −0.342557 − 0.939497i \(-0.611293\pi\)
0.342557 − 0.939497i \(-0.388707\pi\)
\(74\) 0.531633 0.0618011
\(75\) 0 0
\(76\) −9.39142 −1.07727
\(77\) 17.4180i 1.98496i
\(78\) − 1.53271i − 0.173545i
\(79\) 5.53201 0.622399 0.311200 − 0.950345i \(-0.399269\pi\)
0.311200 + 0.950345i \(0.399269\pi\)
\(80\) 0 0
\(81\) −11.2466 −1.24962
\(82\) − 1.63615i − 0.180682i
\(83\) 1.13714i 0.124818i 0.998051 + 0.0624088i \(0.0198783\pi\)
−0.998051 + 0.0624088i \(0.980122\pi\)
\(84\) −10.7781 −1.17598
\(85\) 0 0
\(86\) 4.32351 0.466217
\(87\) − 16.9976i − 1.82233i
\(88\) − 11.7107i − 1.24836i
\(89\) −8.14306 −0.863163 −0.431581 − 0.902074i \(-0.642044\pi\)
−0.431581 + 0.902074i \(0.642044\pi\)
\(90\) 0 0
\(91\) −3.96914 −0.416079
\(92\) 3.21496i 0.335183i
\(93\) − 0.490890i − 0.0509029i
\(94\) −4.81841 −0.496981
\(95\) 0 0
\(96\) 11.1451 1.13749
\(97\) − 17.9941i − 1.82702i −0.406815 − 0.913511i \(-0.633361\pi\)
0.406815 − 0.913511i \(-0.366639\pi\)
\(98\) − 0.872030i − 0.0880883i
\(99\) 9.23545 0.928198
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.h.149.6 10
5.2 odd 4 925.2.a.g.1.3 ✓ 5
5.3 odd 4 925.2.a.i.1.3 yes 5
5.4 even 2 inner 925.2.b.h.149.5 10
15.2 even 4 8325.2.a.cd.1.3 5
15.8 even 4 8325.2.a.cb.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.g.1.3 ✓ 5 5.2 odd 4
925.2.a.i.1.3 yes 5 5.3 odd 4
925.2.b.h.149.5 10 5.4 even 2 inner
925.2.b.h.149.6 10 1.1 even 1 trivial
8325.2.a.cb.1.3 5 15.8 even 4
8325.2.a.cd.1.3 5 15.2 even 4