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Properties

Label 1421.2.b.l.1275.4
Level $1421$
Weight $2$
Character 1421.1275
Analytic conductor $11.347$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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] = [1421,2,Mod(1275,1421)] 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("1421.1275"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1421, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1421.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,-20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3467421272\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 61 x^{18} - 256 x^{17} + 871 x^{16} - 2474 x^{15} + 5887 x^{14} - 11788 x^{13} + \cdots + 784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1275.4
Root \(-1.24023 - 2.46921i\) of defining polynomial
Character \(\chi\) \(=\) 1421.1275
Dual form 1421.2.b.l.1275.17

$q$-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)
 
\(f(q)\) \(=\) \(q-1.87376i q^{2} +0.306676i q^{3} -1.51098 q^{4} +1.29728 q^{5} +0.574637 q^{6} -0.916302i q^{8} +2.90595 q^{9} -2.43078i q^{10} -2.73767i q^{11} -0.463382i q^{12} -5.66364 q^{13} +0.397843i q^{15} -4.73890 q^{16} -4.76779i q^{17} -5.44506i q^{18} +3.70274i q^{19} -1.96016 q^{20} -5.12974 q^{22} +5.83875 q^{23} +0.281008 q^{24} -3.31708 q^{25} +10.6123i q^{26} +1.81121i q^{27} +(5.10945 - 1.70104i) q^{29} +0.745463 q^{30} -1.89877i q^{31} +7.04696i q^{32} +0.839576 q^{33} -8.93370 q^{34} -4.39084 q^{36} -9.65656i q^{37} +6.93806 q^{38} -1.73690i q^{39} -1.18870i q^{40} +4.48626i q^{41} -7.66427i q^{43} +4.13657i q^{44} +3.76982 q^{45} -10.9404i q^{46} -13.0933i q^{47} -1.45330i q^{48} +6.21541i q^{50} +1.46216 q^{51} +8.55766 q^{52} -1.55621 q^{53} +3.39378 q^{54} -3.55151i q^{55} -1.13554 q^{57} +(-3.18734 - 9.57389i) q^{58} +9.50155 q^{59} -0.601134i q^{60} +8.63943i q^{61} -3.55785 q^{62} +3.72653 q^{64} -7.34730 q^{65} -1.57317i q^{66} -5.05841 q^{67} +7.20404i q^{68} +1.79060i q^{69} -14.3552 q^{71} -2.66273i q^{72} +2.17483i q^{73} -18.0941 q^{74} -1.01727i q^{75} -5.59478i q^{76} -3.25454 q^{78} +6.08527i q^{79} -6.14765 q^{80} +8.16240 q^{81} +8.40619 q^{82} -1.57828 q^{83} -6.18513i q^{85} -14.3610 q^{86} +(0.521667 + 1.56694i) q^{87} -2.50853 q^{88} -9.55399i q^{89} -7.06374i q^{90} -8.82225 q^{92} +0.582308 q^{93} -24.5337 q^{94} +4.80348i q^{95} -2.16113 q^{96} +2.96953i q^{97} -7.95552i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} - 12 q^{9} + 36 q^{16} + 48 q^{22} + 56 q^{23} + 84 q^{25} + 12 q^{29} + 24 q^{30} - 68 q^{36} - 40 q^{51} - 32 q^{53} + 128 q^{57} - 64 q^{58} - 196 q^{64} + 104 q^{65} - 96 q^{71} - 40 q^{74}+ \cdots - 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1421\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1277\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.87376i − 1.32495i −0.749084 − 0.662475i \(-0.769506\pi\)
0.749084 − 0.662475i \(-0.230494\pi\)
\(3\) 0.306676i 0.177059i 0.996074 + 0.0885297i \(0.0282168\pi\)
−0.996074 + 0.0885297i \(0.971783\pi\)
\(4\) −1.51098 −0.755491
\(5\) 1.29728 0.580159 0.290080 − 0.957003i \(-0.406318\pi\)
0.290080 + 0.957003i \(0.406318\pi\)
\(6\) 0.574637 0.234595
\(7\) 0 0
\(8\) − 0.916302i − 0.323962i
\(9\) 2.90595 0.968650
\(10\) − 2.43078i − 0.768682i
\(11\) − 2.73767i − 0.825438i −0.910858 − 0.412719i \(-0.864579\pi\)
0.910858 − 0.412719i \(-0.135421\pi\)
\(12\) − 0.463382i − 0.133767i
\(13\) −5.66364 −1.57081 −0.785405 − 0.618982i \(-0.787545\pi\)
−0.785405 + 0.618982i \(0.787545\pi\)
\(14\) 0 0
\(15\) 0.397843i 0.102723i
\(16\) −4.73890 −1.18472
\(17\) − 4.76779i − 1.15636i −0.815910 − 0.578179i \(-0.803764\pi\)
0.815910 − 0.578179i \(-0.196236\pi\)
\(18\) − 5.44506i − 1.28341i
\(19\) 3.70274i 0.849468i 0.905318 + 0.424734i \(0.139632\pi\)
−0.905318 + 0.424734i \(0.860368\pi\)
\(20\) −1.96016 −0.438305
\(21\) 0 0
\(22\) −5.12974 −1.09366
\(23\) 5.83875 1.21746 0.608732 − 0.793376i \(-0.291678\pi\)
0.608732 + 0.793376i \(0.291678\pi\)
\(24\) 0.281008 0.0573605
\(25\) −3.31708 −0.663415
\(26\) 10.6123i 2.08124i
\(27\) 1.81121i 0.348568i
\(28\) 0 0
\(29\) 5.10945 − 1.70104i 0.948801 − 0.315875i
\(30\) 0.745463 0.136102
\(31\) − 1.89877i − 0.341030i −0.985355 − 0.170515i \(-0.945457\pi\)
0.985355 − 0.170515i \(-0.0545431\pi\)
\(32\) 7.04696i 1.24574i
\(33\) 0.839576 0.146151
\(34\) −8.93370 −1.53212
\(35\) 0 0
\(36\) −4.39084 −0.731807
\(37\) − 9.65656i − 1.58753i −0.608225 − 0.793765i \(-0.708118\pi\)
0.608225 − 0.793765i \(-0.291882\pi\)
\(38\) 6.93806 1.12550
\(39\) − 1.73690i − 0.278127i
\(40\) − 1.18870i − 0.187949i
\(41\) 4.48626i 0.700637i 0.936631 + 0.350318i \(0.113927\pi\)
−0.936631 + 0.350318i \(0.886073\pi\)
\(42\) 0 0
\(43\) − 7.66427i − 1.16879i −0.811470 − 0.584395i \(-0.801332\pi\)
0.811470 − 0.584395i \(-0.198668\pi\)
\(44\) 4.13657i 0.623611i
\(45\) 3.76982 0.561971
\(46\) − 10.9404i − 1.61308i
\(47\) − 13.0933i − 1.90985i −0.296840 − 0.954927i \(-0.595933\pi\)
0.296840 − 0.954927i \(-0.404067\pi\)
\(48\) − 1.45330i − 0.209766i
\(49\) 0 0
\(50\) 6.21541i 0.878992i
\(51\) 1.46216 0.204744
\(52\) 8.55766 1.18673
\(53\) −1.55621 −0.213763 −0.106881 − 0.994272i \(-0.534086\pi\)
−0.106881 + 0.994272i \(0.534086\pi\)
\(54\) 3.39378 0.461835
\(55\) − 3.55151i − 0.478885i
\(56\) 0 0
\(57\) −1.13554 −0.150406
\(58\) −3.18734 − 9.57389i −0.418518 − 1.25711i
\(59\) 9.50155 1.23700 0.618498 − 0.785786i \(-0.287741\pi\)
0.618498 + 0.785786i \(0.287741\pi\)
\(60\) − 0.601134i − 0.0776060i
\(61\) 8.63943i 1.10617i 0.833126 + 0.553083i \(0.186549\pi\)
−0.833126 + 0.553083i \(0.813451\pi\)
\(62\) −3.55785 −0.451847
\(63\) 0 0
\(64\) 3.72653 0.465816
\(65\) −7.34730 −0.911320
\(66\) − 1.57317i − 0.193643i
\(67\) −5.05841 −0.617983 −0.308992 − 0.951065i \(-0.599992\pi\)
−0.308992 + 0.951065i \(0.599992\pi\)
\(68\) 7.20404i 0.873618i
\(69\) 1.79060i 0.215563i
\(70\) 0 0
\(71\) −14.3552 −1.70365 −0.851824 − 0.523827i \(-0.824504\pi\)
−0.851824 + 0.523827i \(0.824504\pi\)
\(72\) − 2.66273i − 0.313806i
\(73\) 2.17483i 0.254544i 0.991868 + 0.127272i \(0.0406221\pi\)
−0.991868 + 0.127272i \(0.959378\pi\)
\(74\) −18.0941 −2.10340
\(75\) − 1.01727i − 0.117464i
\(76\) − 5.59478i − 0.641766i
\(77\) 0 0
\(78\) −3.25454 −0.368504
\(79\) 6.08527i 0.684646i 0.939582 + 0.342323i \(0.111214\pi\)
−0.939582 + 0.342323i \(0.888786\pi\)
\(80\) −6.14765 −0.687329
\(81\) 8.16240 0.906933
\(82\) 8.40619 0.928308
\(83\) −1.57828 −0.173239 −0.0866195 − 0.996241i \(-0.527606\pi\)
−0.0866195 + 0.996241i \(0.527606\pi\)
\(84\) 0 0
\(85\) − 6.18513i − 0.670872i
\(86\) −14.3610 −1.54859
\(87\) 0.521667 + 1.56694i 0.0559286 + 0.167994i
\(88\) −2.50853 −0.267410
\(89\) − 9.55399i − 1.01272i −0.862322 − 0.506360i \(-0.830991\pi\)
0.862322 − 0.506360i \(-0.169009\pi\)
\(90\) − 7.06374i − 0.744583i
\(91\) 0 0
\(92\) −8.82225 −0.919783
\(93\) 0.582308 0.0603825
\(94\) −24.5337 −2.53046
\(95\) 4.80348i 0.492827i
\(96\) −2.16113 −0.220570
\(97\) 2.96953i 0.301510i 0.988571 + 0.150755i \(0.0481705\pi\)
−0.988571 + 0.150755i \(0.951829\pi\)
\(98\) 0 0
\(99\) − 7.95552i − 0.799560i
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 1421.2.b.l.1275.4 yes 20
7.6 odd 2 inner 1421.2.b.l.1275.3 ✓ 20
29.28 even 2 inner 1421.2.b.l.1275.17 yes 20
203.202 odd 2 inner 1421.2.b.l.1275.18 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1421.2.b.l.1275.3 ✓ 20 7.6 odd 2 inner
1421.2.b.l.1275.4 yes 20 1.1 even 1 trivial
1421.2.b.l.1275.17 yes 20 29.28 even 2 inner
1421.2.b.l.1275.18 yes 20 203.202 odd 2 inner