Properties

Label 925.2.a.g.1.3
Level $925$
Weight $2$
Character 925.1
Self dual yes
Analytic conductor $7.386$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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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(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 = [5,-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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.65657.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.71737\) 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)
 
\(f(q)\) \(=\) \(q-0.531633 q^{2} -2.13508 q^{3} -1.71737 q^{4} +1.13508 q^{6} +2.93944 q^{7} +1.97628 q^{8} +1.55856 q^{9} -5.92562 q^{11} +3.66671 q^{12} -1.35031 q^{13} -1.56270 q^{14} +2.38408 q^{16} +6.16935 q^{17} -0.828584 q^{18} +5.46850 q^{19} -6.27593 q^{21} +3.15026 q^{22} +1.87203 q^{23} -4.21950 q^{24} +0.717868 q^{26} +3.07758 q^{27} -5.04809 q^{28} -7.96110 q^{29} +0.229916 q^{31} -5.22001 q^{32} +12.6517 q^{33} -3.27983 q^{34} -2.67662 q^{36} +1.00000 q^{37} -2.90724 q^{38} +2.88301 q^{39} -3.07758 q^{41} +3.33649 q^{42} -8.13251 q^{43} +10.1765 q^{44} -0.995233 q^{46} -9.06340 q^{47} -5.09020 q^{48} +1.64029 q^{49} -13.1721 q^{51} +2.31897 q^{52} +2.63638 q^{53} -1.63615 q^{54} +5.80914 q^{56} -11.6757 q^{57} +4.23238 q^{58} -11.7344 q^{59} +4.90190 q^{61} -0.122231 q^{62} +4.58129 q^{63} -1.99303 q^{64} -6.72605 q^{66} -4.03427 q^{67} -10.5950 q^{68} -3.99693 q^{69} -9.03377 q^{71} +3.08015 q^{72} -16.0541 q^{73} -0.531633 q^{74} -9.39142 q^{76} -17.4180 q^{77} -1.53271 q^{78} -5.53201 q^{79} -11.2466 q^{81} +1.63615 q^{82} +1.13714 q^{83} +10.7781 q^{84} +4.32351 q^{86} +16.9976 q^{87} -11.7107 q^{88} +8.14306 q^{89} -3.96914 q^{91} -3.21496 q^{92} -0.490890 q^{93} +4.81841 q^{94} +11.1451 q^{96} +17.9941 q^{97} -0.872030 q^{98} -9.23545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + q^{3} + q^{4} - 6 q^{6} - 16 q^{11} + 5 q^{12} - 3 q^{13} - 5 q^{14} - 11 q^{16} + 2 q^{17} - 13 q^{18} - 11 q^{19} - 14 q^{21} + 19 q^{22} - 7 q^{23} + 6 q^{24} - 4 q^{26} + 13 q^{27}+ \cdots - 21 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.531633 −0.375921 −0.187961 − 0.982177i \(-0.560188\pi\)
−0.187961 + 0.982177i \(0.560188\pi\)
\(3\) −2.13508 −1.23269 −0.616344 − 0.787477i \(-0.711387\pi\)
−0.616344 + 0.787477i \(0.711387\pi\)
\(4\) −1.71737 −0.858683
\(5\) 0 0
\(6\) 1.13508 0.463394
\(7\) 2.93944 1.11100 0.555501 − 0.831516i \(-0.312526\pi\)
0.555501 + 0.831516i \(0.312526\pi\)
\(8\) 1.97628 0.698719
\(9\) 1.55856 0.519521
\(10\) 0 0
\(11\) −5.92562 −1.78664 −0.893321 − 0.449419i \(-0.851631\pi\)
−0.893321 + 0.449419i \(0.851631\pi\)
\(12\) 3.66671 1.05849
\(13\) −1.35031 −0.374508 −0.187254 − 0.982312i \(-0.559959\pi\)
−0.187254 + 0.982312i \(0.559959\pi\)
\(14\) −1.56270 −0.417650
\(15\) 0 0
\(16\) 2.38408 0.596020
\(17\) 6.16935 1.49629 0.748144 − 0.663537i \(-0.230945\pi\)
0.748144 + 0.663537i \(0.230945\pi\)
\(18\) −0.828584 −0.195299
\(19\) 5.46850 1.25456 0.627280 − 0.778793i \(-0.284168\pi\)
0.627280 + 0.778793i \(0.284168\pi\)
\(20\) 0 0
\(21\) −6.27593 −1.36952
\(22\) 3.15026 0.671637
\(23\) 1.87203 0.390345 0.195173 − 0.980769i \(-0.437473\pi\)
0.195173 + 0.980769i \(0.437473\pi\)
\(24\) −4.21950 −0.861303
\(25\) 0 0
\(26\) 0.717868 0.140786
\(27\) 3.07758 0.592281
\(28\) −5.04809 −0.953999
\(29\) −7.96110 −1.47834 −0.739169 − 0.673520i \(-0.764782\pi\)
−0.739169 + 0.673520i \(0.764782\pi\)
\(30\) 0 0
\(31\) 0.229916 0.0412942 0.0206471 − 0.999787i \(-0.493427\pi\)
0.0206471 + 0.999787i \(0.493427\pi\)
\(32\) −5.22001 −0.922775
\(33\) 12.6517 2.20237
\(34\) −3.27983 −0.562487
\(35\) 0 0
\(36\) −2.67662 −0.446104
\(37\) 1.00000 0.164399
\(38\) −2.90724 −0.471616
\(39\) 2.88301 0.461651
\(40\) 0 0
\(41\) −3.07758 −0.480638 −0.240319 − 0.970694i \(-0.577252\pi\)
−0.240319 + 0.970694i \(0.577252\pi\)
\(42\) 3.33649 0.514832
\(43\) −8.13251 −1.24020 −0.620098 − 0.784524i \(-0.712907\pi\)
−0.620098 + 0.784524i \(0.712907\pi\)
\(44\) 10.1765 1.53416
\(45\) 0 0
\(46\) −0.995233 −0.146739
\(47\) −9.06340 −1.32203 −0.661017 − 0.750371i \(-0.729875\pi\)
−0.661017 + 0.750371i \(0.729875\pi\)
\(48\) −5.09020 −0.734706
\(49\) 1.64029 0.234326
\(50\) 0 0
\(51\) −13.1721 −1.84446
\(52\) 2.31897 0.321583
\(53\) 2.63638 0.362135 0.181067 − 0.983471i \(-0.442045\pi\)
0.181067 + 0.983471i \(0.442045\pi\)
\(54\) −1.63615 −0.222651
\(55\) 0 0
\(56\) 5.80914 0.776278
\(57\) −11.6757 −1.54648
\(58\) 4.23238 0.555739
\(59\) −11.7344 −1.52769 −0.763843 − 0.645402i \(-0.776690\pi\)
−0.763843 + 0.645402i \(0.776690\pi\)
\(60\) 0 0
\(61\) 4.90190 0.627624 0.313812 − 0.949485i \(-0.398394\pi\)
0.313812 + 0.949485i \(0.398394\pi\)
\(62\) −0.122231 −0.0155234
\(63\) 4.58129 0.577189
\(64\) −1.99303 −0.249128
\(65\) 0 0
\(66\) −6.72605 −0.827919
\(67\) −4.03427 −0.492865 −0.246432 − 0.969160i \(-0.579258\pi\)
−0.246432 + 0.969160i \(0.579258\pi\)
\(68\) −10.5950 −1.28484
\(69\) −3.99693 −0.481174
\(70\) 0 0
\(71\) −9.03377 −1.07211 −0.536056 − 0.844183i \(-0.680086\pi\)
−0.536056 + 0.844183i \(0.680086\pi\)
\(72\) 3.08015 0.362999
\(73\) −16.0541 −1.87899 −0.939497 − 0.342557i \(-0.888707\pi\)
−0.939497 + 0.342557i \(0.888707\pi\)
\(74\) −0.531633 −0.0618011
\(75\) 0 0
\(76\) −9.39142 −1.07727
\(77\) −17.4180 −1.98496
\(78\) −1.53271 −0.173545
\(79\) −5.53201 −0.622399 −0.311200 − 0.950345i \(-0.600731\pi\)
−0.311200 + 0.950345i \(0.600731\pi\)
\(80\) 0 0
\(81\) −11.2466 −1.24962
\(82\) 1.63615 0.180682
\(83\) 1.13714 0.124818 0.0624088 − 0.998051i \(-0.480122\pi\)
0.0624088 + 0.998051i \(0.480122\pi\)
\(84\) 10.7781 1.17598
\(85\) 0 0
\(86\) 4.32351 0.466217
\(87\) 16.9976 1.82233
\(88\) −11.7107 −1.24836
\(89\) 8.14306 0.863163 0.431581 − 0.902074i \(-0.357956\pi\)
0.431581 + 0.902074i \(0.357956\pi\)
\(90\) 0 0
\(91\) −3.96914 −0.416079
\(92\) −3.21496 −0.335183
\(93\) −0.490890 −0.0509029
\(94\) 4.81841 0.496981
\(95\) 0 0
\(96\) 11.1451 1.13749
\(97\) 17.9941 1.82702 0.913511 − 0.406815i \(-0.133361\pi\)
0.913511 + 0.406815i \(0.133361\pi\)
\(98\) −0.872030 −0.0880883
\(99\) −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.a.g.1.3 ✓ 5
3.2 odd 2 8325.2.a.cd.1.3 5
5.2 odd 4 925.2.b.h.149.5 10
5.3 odd 4 925.2.b.h.149.6 10
5.4 even 2 925.2.a.i.1.3 yes 5
15.14 odd 2 8325.2.a.cb.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.g.1.3 ✓ 5 1.1 even 1 trivial
925.2.a.i.1.3 yes 5 5.4 even 2
925.2.b.h.149.5 10 5.2 odd 4
925.2.b.h.149.6 10 5.3 odd 4
8325.2.a.cb.1.3 5 15.14 odd 2
8325.2.a.cd.1.3 5 3.2 odd 2