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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [464,2,Mod(49,464)] 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("464.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(464, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.u (of order \(7\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3,0,0] 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: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 161.1
Root \(-1.02179 - 1.28129i\) of defining polynomial
Character \(\chi\) \(=\) 464.161
Dual form 464.2.u.h.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.02179 + 1.28129i) q^{3} +(-1.07557 - 0.517965i) q^{5} +(-1.27416 + 1.59774i) q^{7} +(0.0699247 + 0.306360i) q^{9} +(0.819415 - 3.59009i) q^{11} +(-0.479858 + 2.10239i) q^{13} +(1.76267 - 0.848856i) q^{15} -6.53517 q^{17} +(-3.31337 - 4.15484i) q^{19} +(-0.745242 - 3.26512i) q^{21} +(-5.30987 + 2.55710i) q^{23} +(-2.22890 - 2.79495i) q^{25} +(-4.89359 - 2.35663i) q^{27} +(1.75040 - 5.09275i) q^{29} +(8.10571 + 3.90350i) q^{31} +(3.76267 + 4.71824i) q^{33} +(2.19801 - 1.05851i) q^{35} +(-0.406764 - 1.78215i) q^{37} +(-2.20346 - 2.76305i) q^{39} -8.32895 q^{41} +(-3.31774 + 1.59774i) q^{43} +(0.0834753 - 0.365729i) q^{45} +(-0.220911 + 0.967876i) q^{47} +(0.628344 + 2.75296i) q^{49} +(6.67760 - 8.37344i) q^{51} +(5.10353 + 2.45773i) q^{53} +(-2.74087 + 3.43695i) q^{55} +8.70913 q^{57} -2.94918 q^{59} +(-1.12786 + 1.41429i) q^{61} +(-0.578579 - 0.278629i) q^{63} +(1.60508 - 2.01271i) q^{65} +(1.34482 + 5.89206i) q^{67} +(2.14921 - 9.41630i) q^{69} +(0.836003 - 3.66277i) q^{71} +(-11.4647 + 5.52110i) q^{73} +5.85860 q^{75} +(4.69197 + 5.88354i) q^{77} +(3.14487 + 13.7786i) q^{79} +(7.17039 - 3.45308i) q^{81} +(-1.27960 - 1.60457i) q^{83} +(7.02900 + 3.38499i) q^{85} +(4.73674 + 7.44650i) q^{87} +(14.8768 + 7.16429i) q^{89} +(-2.74767 - 3.44546i) q^{91} +(-13.2839 + 6.39717i) q^{93} +(1.41169 + 6.18501i) q^{95} +(-2.86572 - 3.59350i) q^{97} +1.15716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - q^{7} - 11 q^{9} + 2 q^{11} + q^{13} + 9 q^{15} - 12 q^{17} + 6 q^{19} - 13 q^{21} - 35 q^{23} - 6 q^{25} - 39 q^{27} - 14 q^{29} + 8 q^{31} + 33 q^{33} + 18 q^{35} + 31 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{7}\right)\)

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 0
\(3\) −1.02179 + 1.28129i −0.589933 + 0.739752i −0.983771 − 0.179428i \(-0.942575\pi\)
0.393839 + 0.919180i \(0.371147\pi\)
\(4\) 0 0
\(5\) −1.07557 − 0.517965i −0.481007 − 0.231641i 0.177636 − 0.984096i \(-0.443155\pi\)
−0.658643 + 0.752455i \(0.728869\pi\)
\(6\) 0 0
\(7\) −1.27416 + 1.59774i −0.481585 + 0.603889i −0.961965 − 0.273171i \(-0.911927\pi\)
0.480380 + 0.877061i \(0.340499\pi\)
\(8\) 0 0
\(9\) 0.0699247 + 0.306360i 0.0233082 + 0.102120i
\(10\) 0 0
\(11\) 0.819415 − 3.59009i 0.247063 − 1.08245i −0.687368 − 0.726309i \(-0.741234\pi\)
0.934431 − 0.356144i \(-0.115909\pi\)
\(12\) 0 0
\(13\) −0.479858 + 2.10239i −0.133089 + 0.583099i 0.863769 + 0.503888i \(0.168097\pi\)
−0.996858 + 0.0792116i \(0.974760\pi\)
\(14\) 0 0
\(15\) 1.76267 − 0.848856i 0.455119 − 0.219174i
\(16\) 0 0
\(17\) −6.53517 −1.58501 −0.792506 − 0.609864i \(-0.791224\pi\)
−0.792506 + 0.609864i \(0.791224\pi\)
\(18\) 0 0
\(19\) −3.31337 − 4.15484i −0.760140 − 0.953186i 0.239704 − 0.970846i \(-0.422950\pi\)
−0.999844 + 0.0176605i \(0.994378\pi\)
\(20\) 0 0
\(21\) −0.745242 − 3.26512i −0.162625 − 0.712508i
\(22\) 0 0
\(23\) −5.30987 + 2.55710i −1.10718 + 0.533192i −0.895910 − 0.444235i \(-0.853476\pi\)
−0.211274 + 0.977427i \(0.567761\pi\)
\(24\) 0 0
\(25\) −2.22890 − 2.79495i −0.445779 − 0.558989i
\(26\) 0 0
\(27\) −4.89359 − 2.35663i −0.941771 − 0.453533i
\(28\) 0 0
\(29\) 1.75040 − 5.09275i 0.325040 − 0.945700i
\(30\) 0 0
\(31\) 8.10571 + 3.90350i 1.45583 + 0.701090i 0.983596 − 0.180385i \(-0.0577343\pi\)
0.472232 + 0.881475i \(0.343449\pi\)
\(32\) 0 0
\(33\) 3.76267 + 4.71824i 0.654996 + 0.821339i
\(34\) 0 0
\(35\) 2.19801 − 1.05851i 0.371532 − 0.178920i
\(36\) 0 0
\(37\) −0.406764 − 1.78215i −0.0668717 − 0.292984i 0.930423 − 0.366487i \(-0.119439\pi\)
−0.997295 + 0.0735026i \(0.976582\pi\)
\(38\) 0 0
\(39\) −2.20346 − 2.76305i −0.352836 − 0.442442i
\(40\) 0 0
\(41\) −8.32895 −1.30076 −0.650382 − 0.759608i \(-0.725391\pi\)
−0.650382 + 0.759608i \(0.725391\pi\)
\(42\) 0 0
\(43\) −3.31774 + 1.59774i −0.505951 + 0.243653i −0.669405 − 0.742898i \(-0.733451\pi\)
0.163454 + 0.986551i \(0.447737\pi\)
\(44\) 0 0
\(45\) 0.0834753 − 0.365729i 0.0124438 − 0.0545197i
\(46\) 0 0
\(47\) −0.220911 + 0.967876i −0.0322232 + 0.141179i −0.988481 − 0.151347i \(-0.951639\pi\)
0.956257 + 0.292526i \(0.0944960\pi\)
\(48\) 0 0
\(49\) 0.628344 + 2.75296i 0.0897635 + 0.393280i
\(50\) 0 0
\(51\) 6.67760 − 8.37344i 0.935050 − 1.17252i
\(52\) 0 0
\(53\) 5.10353 + 2.45773i 0.701023 + 0.337595i 0.750215 − 0.661194i \(-0.229950\pi\)
−0.0491913 + 0.998789i \(0.515664\pi\)
\(54\) 0 0
\(55\) −2.74087 + 3.43695i −0.369579 + 0.463438i
\(56\) 0 0
\(57\) 8.70913 1.15355
\(58\) 0 0
\(59\) −2.94918 −0.383951 −0.191975 − 0.981400i \(-0.561489\pi\)
−0.191975 + 0.981400i \(0.561489\pi\)
\(60\) 0 0
\(61\) −1.12786 + 1.41429i −0.144407 + 0.181081i −0.848775 − 0.528754i \(-0.822659\pi\)
0.704368 + 0.709835i \(0.251231\pi\)
\(62\) 0 0
\(63\) −0.578579 − 0.278629i −0.0728941 − 0.0351040i
\(64\) 0 0
\(65\) 1.60508 − 2.01271i 0.199086 − 0.249646i
\(66\) 0 0
\(67\) 1.34482 + 5.89206i 0.164296 + 0.719830i 0.988209 + 0.153113i \(0.0489297\pi\)
−0.823912 + 0.566717i \(0.808213\pi\)
\(68\) 0 0
\(69\) 2.14921 − 9.41630i 0.258734 − 1.13359i
\(70\) 0 0
\(71\) 0.836003 − 3.66277i 0.0992153 − 0.434691i −0.900785 − 0.434266i \(-0.857008\pi\)
1.00000 0.000424562i \(-0.000135142\pi\)
\(72\) 0 0
\(73\) −11.4647 + 5.52110i −1.34184 + 0.646196i −0.960511 − 0.278243i \(-0.910248\pi\)
−0.381329 + 0.924439i \(0.624534\pi\)
\(74\) 0 0
\(75\) 5.85860 0.676493
\(76\) 0 0
\(77\) 4.69197 + 5.88354i 0.534700 + 0.670492i
\(78\) 0 0
\(79\) 3.14487 + 13.7786i 0.353826 + 1.55021i 0.768264 + 0.640133i \(0.221121\pi\)
−0.414438 + 0.910077i \(0.636022\pi\)
\(80\) 0 0
\(81\) 7.17039 − 3.45308i 0.796710 − 0.383676i
\(82\) 0 0
\(83\) −1.27960 − 1.60457i −0.140455 − 0.176125i 0.706629 − 0.707584i \(-0.250215\pi\)
−0.847083 + 0.531460i \(0.821644\pi\)
\(84\) 0 0
\(85\) 7.02900 + 3.38499i 0.762403 + 0.367154i
\(86\) 0 0
\(87\) 4.73674 + 7.44650i 0.507832 + 0.798349i
\(88\) 0 0
\(89\) 14.8768 + 7.16429i 1.57694 + 0.759413i 0.998416 − 0.0562548i \(-0.0179159\pi\)
0.578521 + 0.815668i \(0.303630\pi\)
\(90\) 0 0
\(91\) −2.74767 − 3.44546i −0.288034 − 0.361183i
\(92\) 0 0
\(93\) −13.2839 + 6.39717i −1.37747 + 0.663356i
\(94\) 0 0
\(95\) 1.41169 + 6.18501i 0.144836 + 0.634569i
\(96\) 0 0
\(97\) −2.86572 − 3.59350i −0.290970 − 0.364864i 0.614764 − 0.788711i \(-0.289251\pi\)
−0.905734 + 0.423846i \(0.860680\pi\)
\(98\) 0 0
\(99\) 1.15716 0.116299
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 464.2.u.h.161.1 12
4.3 odd 2 58.2.d.b.45.2 ✓ 12
12.11 even 2 522.2.k.h.451.2 12
29.20 even 7 inner 464.2.u.h.49.1 12
116.3 even 28 1682.2.b.i.1681.3 12
116.7 odd 14 1682.2.a.t.1.4 6
116.51 odd 14 1682.2.a.q.1.3 6
116.55 even 28 1682.2.b.i.1681.10 12
116.107 odd 14 58.2.d.b.49.2 yes 12
348.107 even 14 522.2.k.h.397.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.2 ✓ 12 4.3 odd 2
58.2.d.b.49.2 yes 12 116.107 odd 14
464.2.u.h.49.1 12 29.20 even 7 inner
464.2.u.h.161.1 12 1.1 even 1 trivial
522.2.k.h.397.2 12 348.107 even 14
522.2.k.h.451.2 12 12.11 even 2
1682.2.a.q.1.3 6 116.51 odd 14
1682.2.a.t.1.4 6 116.7 odd 14
1682.2.b.i.1681.3 12 116.3 even 28
1682.2.b.i.1681.10 12 116.55 even 28