↑

Properties

Label 1421.2.b.l.1275.16
Level $1421$
Weight $2$
Character 1421.1275
Analytic conductor $11.347$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

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 1275.16
Root \(0.711468 + 0.241861i\) of defining polynomial
Character \(\chi\) \(=\) 1421.1275
Dual form 1421.2.b.l.1275.5

$q$-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)
 
\(f(q)\) \(=\) \(q+1.52717i q^{2} +3.09649i q^{3} -0.332235 q^{4} -3.78449 q^{5} -4.72885 q^{6} +2.54695i q^{8} -6.58825 q^{9} -5.77954i q^{10} -1.07045i q^{11} -1.02876i q^{12} -2.93862 q^{13} -11.7186i q^{15} -4.55409 q^{16} +5.85475i q^{17} -10.0613i q^{18} -4.24889i q^{19} +1.25734 q^{20} +1.63476 q^{22} +7.62044 q^{23} -7.88662 q^{24} +9.32236 q^{25} -4.48776i q^{26} -11.1110i q^{27} +(-5.08949 - 1.75987i) q^{29} +17.8963 q^{30} +4.39468i q^{31} -1.86094i q^{32} +3.31464 q^{33} -8.94117 q^{34} +2.18884 q^{36} +5.29253i q^{37} +6.48876 q^{38} -9.09940i q^{39} -9.63892i q^{40} -0.682869i q^{41} -1.96012i q^{43} +0.355641i q^{44} +24.9332 q^{45} +11.6377i q^{46} +5.06307i q^{47} -14.1017i q^{48} +14.2368i q^{50} -18.1292 q^{51} +0.976311 q^{52} +8.54092 q^{53} +16.9683 q^{54} +4.05111i q^{55} +13.1567 q^{57} +(2.68761 - 7.77249i) q^{58} -4.38315 q^{59} +3.89334i q^{60} -7.36859i q^{61} -6.71141 q^{62} -6.26621 q^{64} +11.1212 q^{65} +5.06200i q^{66} -11.3056 q^{67} -1.94515i q^{68} +23.5966i q^{69} -4.52848 q^{71} -16.7800i q^{72} -7.69662i q^{73} -8.08257 q^{74} +28.8666i q^{75} +1.41163i q^{76} +13.8963 q^{78} -16.8810i q^{79} +17.2349 q^{80} +14.6403 q^{81} +1.04285 q^{82} +11.6711 q^{83} -22.1572i q^{85} +2.99342 q^{86} +(5.44941 - 15.7595i) q^{87} +2.72639 q^{88} -3.79082i q^{89} +38.0771i q^{90} -2.53177 q^{92} -13.6081 q^{93} -7.73214 q^{94} +16.0799i q^{95} +5.76238 q^{96} +8.72182i q^{97} +7.05239i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} - 12 q^{9} + 36 q^{16} + 48 q^{22} + 56 q^{23} + 84 q^{25} + 12 q^{29} + 24 q^{30} - 68 q^{36} - 40 q^{51} - 32 q^{53} + 128 q^{57} - 64 q^{58} - 196 q^{64} + 104 q^{65} - 96 q^{71} - 40 q^{74}+ \cdots - 48 q^{93}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.52717i 1.07987i 0.841707 + 0.539935i \(0.181551\pi\)
−0.841707 + 0.539935i \(0.818449\pi\)
\(3\) 3.09649i 1.78776i 0.448307 + 0.893880i \(0.352027\pi\)
−0.448307 + 0.893880i \(0.647973\pi\)
\(4\) −0.332235 −0.166117
\(5\) −3.78449 −1.69248 −0.846238 − 0.532806i \(-0.821138\pi\)
−0.846238 + 0.532806i \(0.821138\pi\)
\(6\) −4.72885 −1.93055
\(7\) 0 0
\(8\) 2.54695i 0.900484i
\(9\) −6.58825 −2.19608
\(10\) − 5.77954i − 1.82765i
\(11\) − 1.07045i − 0.322753i −0.986893 − 0.161376i \(-0.948407\pi\)
0.986893 − 0.161376i \(-0.0515933\pi\)
\(12\) − 1.02876i − 0.296978i
\(13\) −2.93862 −0.815026 −0.407513 − 0.913199i \(-0.633604\pi\)
−0.407513 + 0.913199i \(0.633604\pi\)
\(14\) 0 0
\(15\) − 11.7186i − 3.02574i
\(16\) −4.55409 −1.13852
\(17\) 5.85475i 1.41999i 0.704209 + 0.709993i \(0.251302\pi\)
−0.704209 + 0.709993i \(0.748698\pi\)
\(18\) − 10.0613i − 2.37148i
\(19\) − 4.24889i − 0.974763i −0.873189 − 0.487382i \(-0.837952\pi\)
0.873189 − 0.487382i \(-0.162048\pi\)
\(20\) 1.25734 0.281149
\(21\) 0 0
\(22\) 1.63476 0.348531
\(23\) 7.62044 1.58897 0.794486 − 0.607283i \(-0.207741\pi\)
0.794486 + 0.607283i \(0.207741\pi\)
\(24\) −7.88662 −1.60985
\(25\) 9.32236 1.86447
\(26\) − 4.48776i − 0.880122i
\(27\) − 11.1110i − 2.13831i
\(28\) 0 0
\(29\) −5.08949 − 1.75987i −0.945094 − 0.326799i
\(30\) 17.8963 3.26740
\(31\) 4.39468i 0.789308i 0.918830 + 0.394654i \(0.129136\pi\)
−0.918830 + 0.394654i \(0.870864\pi\)
\(32\) − 1.86094i − 0.328971i
\(33\) 3.31464 0.577005
\(34\) −8.94117 −1.53340
\(35\) 0 0
\(36\) 2.18884 0.364807
\(37\) 5.29253i 0.870086i 0.900410 + 0.435043i \(0.143267\pi\)
−0.900410 + 0.435043i \(0.856733\pi\)
\(38\) 6.48876 1.05262
\(39\) − 9.09940i − 1.45707i
\(40\) − 9.63892i − 1.52405i
\(41\) − 0.682869i − 0.106646i −0.998577 − 0.0533231i \(-0.983019\pi\)
0.998577 − 0.0533231i \(-0.0169813\pi\)
\(42\) 0 0
\(43\) − 1.96012i − 0.298915i −0.988768 − 0.149458i \(-0.952247\pi\)
0.988768 − 0.149458i \(-0.0477527\pi\)
\(44\) 0.355641i 0.0536149i
\(45\) 24.9332 3.71682
\(46\) 11.6377i 1.71588i
\(47\) 5.06307i 0.738524i 0.929325 + 0.369262i \(0.120390\pi\)
−0.929325 + 0.369262i \(0.879610\pi\)
\(48\) − 14.1017i − 2.03540i
\(49\) 0 0
\(50\) 14.2368i 2.01339i
\(51\) −18.1292 −2.53859
\(52\) 0.976311 0.135390
\(53\) 8.54092 1.17319 0.586593 − 0.809882i \(-0.300469\pi\)
0.586593 + 0.809882i \(0.300469\pi\)
\(54\) 16.9683 2.30909
\(55\) 4.05111i 0.546251i
\(56\) 0 0
\(57\) 13.1567 1.74264
\(58\) 2.68761 − 7.77249i 0.352900 − 1.02058i
\(59\) −4.38315 −0.570638 −0.285319 − 0.958433i \(-0.592100\pi\)
−0.285319 + 0.958433i \(0.592100\pi\)
\(60\) 3.89334i 0.502628i
\(61\) − 7.36859i − 0.943452i −0.881745 − 0.471726i \(-0.843631\pi\)
0.881745 − 0.471726i \(-0.156369\pi\)
\(62\) −6.71141 −0.852350
\(63\) 0 0
\(64\) −6.26621 −0.783277
\(65\) 11.1212 1.37941
\(66\) 5.06200i 0.623089i
\(67\) −11.3056 −1.38120 −0.690598 − 0.723239i \(-0.742652\pi\)
−0.690598 + 0.723239i \(0.742652\pi\)
\(68\) − 1.94515i − 0.235884i
\(69\) 23.5966i 2.84070i
\(70\) 0 0
\(71\) −4.52848 −0.537431 −0.268716 − 0.963220i \(-0.586599\pi\)
−0.268716 + 0.963220i \(0.586599\pi\)
\(72\) − 16.7800i − 1.97754i
\(73\) − 7.69662i − 0.900821i −0.892822 − 0.450411i \(-0.851278\pi\)
0.892822 − 0.450411i \(-0.148722\pi\)
\(74\) −8.08257 −0.939579
\(75\) 28.8666i 3.33323i
\(76\) 1.41163i 0.161925i
\(77\) 0 0
\(78\) 13.8963 1.57345
\(79\) − 16.8810i − 1.89927i −0.313365 − 0.949633i \(-0.601456\pi\)
0.313365 − 0.949633i \(-0.398544\pi\)
\(80\) 17.2349 1.92692
\(81\) 14.6403 1.62670
\(82\) 1.04285 0.115164
\(83\) 11.6711 1.28107 0.640535 − 0.767929i \(-0.278713\pi\)
0.640535 + 0.767929i \(0.278713\pi\)
\(84\) 0 0
\(85\) − 22.1572i − 2.40329i
\(86\) 2.99342 0.322789
\(87\) 5.44941 − 15.7595i 0.584238 − 1.68960i
\(88\) 2.72639 0.290634
\(89\) − 3.79082i − 0.401826i −0.979609 − 0.200913i \(-0.935609\pi\)
0.979609 − 0.200913i \(-0.0643909\pi\)
\(90\) 38.0771i 4.01368i
\(91\) 0 0
\(92\) −2.53177 −0.263956
\(93\) −13.6081 −1.41109
\(94\) −7.73214 −0.797510
\(95\) 16.0799i 1.64976i
\(96\) 5.76238 0.588121
\(97\) 8.72182i 0.885567i 0.896629 + 0.442783i \(0.146009\pi\)
−0.896629 + 0.442783i \(0.853991\pi\)
\(98\) 0 0
\(99\) 7.05239i 0.708792i
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.1275.16 yes 20
7.6 odd 2 inner 1421.2.b.l.1275.15 yes 20
29.28 even 2 inner 1421.2.b.l.1275.5 ✓ 20
203.202 odd 2 inner 1421.2.b.l.1275.6 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1421.2.b.l.1275.5 ✓ 20 29.28 even 2 inner
1421.2.b.l.1275.6 yes 20 203.202 odd 2 inner
1421.2.b.l.1275.15 yes 20 7.6 odd 2 inner
1421.2.b.l.1275.16 yes 20 1.1 even 1 trivial