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Properties

Label 1425.2.a.l.1.1
Level $1425$
Weight $2$
Character 1425.1
Self dual yes
Analytic conductor $11.379$
Analytic rank $1$
Dimension $2$
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] = [1425,2,Mod(1,1425)] 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("1425.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,2,2,0,-2,-4,-6,2,0,0] 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: yes
Analytic conductor: \(11.3786822880\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1425.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} -3.41421 q^{7} -4.41421 q^{8} +1.00000 q^{9} -1.41421 q^{11} +3.82843 q^{12} -2.58579 q^{13} +8.24264 q^{14} +3.00000 q^{16} +6.82843 q^{17} -2.41421 q^{18} +1.00000 q^{19} -3.41421 q^{21} +3.41421 q^{22} +3.65685 q^{23} -4.41421 q^{24} +6.24264 q^{26} +1.00000 q^{27} -13.0711 q^{28} +5.07107 q^{29} -10.4853 q^{31} +1.58579 q^{32} -1.41421 q^{33} -16.4853 q^{34} +3.82843 q^{36} +3.07107 q^{37} -2.41421 q^{38} -2.58579 q^{39} -4.58579 q^{41} +8.24264 q^{42} -3.41421 q^{43} -5.41421 q^{44} -8.82843 q^{46} -11.6569 q^{47} +3.00000 q^{48} +4.65685 q^{49} +6.82843 q^{51} -9.89949 q^{52} -4.00000 q^{53} -2.41421 q^{54} +15.0711 q^{56} +1.00000 q^{57} -12.2426 q^{58} -8.48528 q^{59} -5.65685 q^{61} +25.3137 q^{62} -3.41421 q^{63} -9.82843 q^{64} +3.41421 q^{66} -12.0000 q^{67} +26.1421 q^{68} +3.65685 q^{69} +12.4853 q^{71} -4.41421 q^{72} +2.00000 q^{73} -7.41421 q^{74} +3.82843 q^{76} +4.82843 q^{77} +6.24264 q^{78} +11.3137 q^{79} +1.00000 q^{81} +11.0711 q^{82} -6.48528 q^{83} -13.0711 q^{84} +8.24264 q^{86} +5.07107 q^{87} +6.24264 q^{88} -14.7279 q^{89} +8.82843 q^{91} +14.0000 q^{92} -10.4853 q^{93} +28.1421 q^{94} +1.58579 q^{96} -4.24264 q^{97} -11.2426 q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 8 q^{13} + 8 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} - 4 q^{21} + 4 q^{22} - 4 q^{23} - 6 q^{24} + 4 q^{26}+ \cdots - 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 − 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) −3.41421 −1.29045 −0.645226 − 0.763992i \(-0.723237\pi\)
−0.645226 + 0.763992i \(0.723237\pi\)
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 − 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 3.82843 1.10517
\(13\) −2.58579 −0.717168 −0.358584 − 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(14\) 8.24264 2.20294
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 − 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) −2.41421 −0.569036
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.41421 −0.745042
\(22\) 3.41421 0.727913
\(23\) 3.65685 0.762507 0.381253 − 0.924471i \(-0.375493\pi\)
0.381253 + 0.924471i \(0.375493\pi\)
\(24\) −4.41421 −0.901048
\(25\) 0 0
\(26\) 6.24264 1.22428
\(27\) 1.00000 0.192450
\(28\) −13.0711 −2.47020
\(29\) 5.07107 0.941674 0.470837 − 0.882220i \(-0.343952\pi\)
0.470837 + 0.882220i \(0.343952\pi\)
\(30\) 0 0
\(31\) −10.4853 −1.88321 −0.941606 − 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) 1.58579 0.280330
\(33\) −1.41421 −0.246183
\(34\) −16.4853 −2.82720
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 3.07107 0.504880 0.252440 − 0.967612i \(-0.418767\pi\)
0.252440 + 0.967612i \(0.418767\pi\)
\(38\) −2.41421 −0.391637
\(39\) −2.58579 −0.414057
\(40\) 0 0
\(41\) −4.58579 −0.716180 −0.358090 − 0.933687i \(-0.616572\pi\)
−0.358090 + 0.933687i \(0.616572\pi\)
\(42\) 8.24264 1.27187
\(43\) −3.41421 −0.520663 −0.260331 − 0.965519i \(-0.583832\pi\)
−0.260331 + 0.965519i \(0.583832\pi\)
\(44\) −5.41421 −0.816223
\(45\) 0 0
\(46\) −8.82843 −1.30168
\(47\) −11.6569 −1.70033 −0.850163 − 0.526519i \(-0.823497\pi\)
−0.850163 + 0.526519i \(0.823497\pi\)
\(48\) 3.00000 0.433013
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 6.82843 0.956171
\(52\) −9.89949 −1.37281
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −2.41421 −0.328533
\(55\) 0 0
\(56\) 15.0711 2.01396
\(57\) 1.00000 0.132453
\(58\) −12.2426 −1.60754
\(59\) −8.48528 −1.10469 −0.552345 − 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 − 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 25.3137 3.21484
\(63\) −3.41421 −0.430150
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 3.41421 0.420261
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 26.1421 3.17020
\(69\) 3.65685 0.440234
\(70\) 0 0
\(71\) 12.4853 1.48173 0.740865 − 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) −4.41421 −0.520220
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −7.41421 −0.861885
\(75\) 0 0
\(76\) 3.82843 0.439151
\(77\) 4.82843 0.550250
\(78\) 6.24264 0.706840
\(79\) 11.3137 1.27289 0.636446 − 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 11.0711 1.22259
\(83\) −6.48528 −0.711852 −0.355926 − 0.934514i \(-0.615835\pi\)
−0.355926 + 0.934514i \(0.615835\pi\)
\(84\) −13.0711 −1.42617
\(85\) 0 0
\(86\) 8.24264 0.888827
\(87\) 5.07107 0.543676
\(88\) 6.24264 0.665468
\(89\) −14.7279 −1.56116 −0.780578 − 0.625058i \(-0.785075\pi\)
−0.780578 + 0.625058i \(0.785075\pi\)
\(90\) 0 0
\(91\) 8.82843 0.925471
\(92\) 14.0000 1.45960
\(93\) −10.4853 −1.08727
\(94\) 28.1421 2.90264
\(95\) 0 0
\(96\) 1.58579 0.161849
\(97\) −4.24264 −0.430775 −0.215387 − 0.976529i \(-0.569101\pi\)
−0.215387 + 0.976529i \(0.569101\pi\)
\(98\) −11.2426 −1.13568
\(99\) −1.41421 −0.142134
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 1425.2.a.l.1.1 2
3.2 odd 2 4275.2.a.x.1.2 2
5.2 odd 4 1425.2.c.j.799.1 4
5.3 odd 4 1425.2.c.j.799.4 4
5.4 even 2 285.2.a.f.1.2 ✓ 2
15.14 odd 2 855.2.a.e.1.1 2
20.19 odd 2 4560.2.a.bj.1.1 2
95.94 odd 2 5415.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.2 ✓ 2 5.4 even 2
855.2.a.e.1.1 2 15.14 odd 2
1425.2.a.l.1.1 2 1.1 even 1 trivial
1425.2.c.j.799.1 4 5.2 odd 4
1425.2.c.j.799.4 4 5.3 odd 4
4275.2.a.x.1.2 2 3.2 odd 2
4560.2.a.bj.1.1 2 20.19 odd 2
5415.2.a.p.1.1 2 95.94 odd 2