Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [464,2,Mod(33,464)] 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("464.33"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.y (of order \(14\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0] 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: no
Analytic conductor: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 241.2
Root \(0.781831 - 0.623490i\) of defining polynomial
Character \(\chi\) \(=\) 464.241
Dual form 464.2.y.c.129.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.19064 - 1.74698i) q^{3} +(1.63762 - 0.788637i) q^{5} +(-0.882702 - 1.10687i) q^{7} +(1.07942 - 4.72923i) q^{9} +(-3.44878 + 0.787162i) q^{11} +(-1.23911 - 5.42888i) q^{13} +(2.20971 - 4.58851i) q^{15} +7.46337i q^{17} +(3.93791 + 3.14038i) q^{19} +(-3.86737 - 0.882702i) q^{21} +(1.61216 + 0.776374i) q^{23} +(-1.05759 + 1.32618i) q^{25} +(-2.25011 - 4.67241i) q^{27} +(4.21464 - 3.35213i) q^{29} +(2.07731 + 4.31359i) q^{31} +(-6.17989 + 7.74934i) q^{33} +(-2.31845 - 1.11651i) q^{35} +(1.04717 + 0.239009i) q^{37} +(-12.1986 - 9.72805i) q^{39} +1.71164i q^{41} +(0.881405 - 1.83026i) q^{43} +(-1.96197 - 8.59597i) q^{45} +(0.377517 - 0.0861658i) q^{47} +(1.11164 - 4.87041i) q^{49} +(13.0384 + 16.3496i) q^{51} +(2.42678 - 1.16867i) q^{53} +(-5.02701 + 4.00891i) q^{55} +14.1127 q^{57} +3.81302 q^{59} +(-10.5585 + 8.42008i) q^{61} +(-6.18747 + 2.97973i) q^{63} +(-6.31060 - 7.91325i) q^{65} +(0.659012 - 2.88732i) q^{67} +(4.88797 - 1.11565i) q^{69} +(-1.39711 - 6.12116i) q^{71} +(-0.416685 + 0.865255i) q^{73} +4.75277i q^{75} +(3.91554 + 3.12254i) q^{77} +(-2.71230 - 0.619064i) q^{79} +(0.0196143 + 0.00944576i) q^{81} +(5.65640 - 7.09290i) q^{83} +(5.88589 + 12.2222i) q^{85} +(3.37666 - 14.7062i) q^{87} +(2.30413 + 4.78459i) q^{89} +(-4.91532 + 6.16362i) q^{91} +(12.0864 + 5.82051i) q^{93} +(8.92543 + 2.03717i) q^{95} +(-8.59586 - 6.85497i) q^{97} +17.1598i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{5} - 4 q^{7} - 4 q^{9} - 26 q^{13} + 14 q^{15} + 14 q^{21} + 16 q^{23} + 22 q^{25} + 18 q^{29} - 28 q^{31} - 10 q^{33} - 4 q^{35} - 28 q^{37} - 28 q^{39} + 28 q^{43} - 4 q^{45} + 14 q^{47} - 8 q^{49}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{14}\right)\)

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
\(3\) 2.19064 − 1.74698i 1.26477 − 1.00862i 0.265763 − 0.964039i \(-0.414376\pi\)
0.999006 − 0.0445806i \(-0.0141952\pi\)
\(4\) 0 0
\(5\) 1.63762 − 0.788637i 0.732367 − 0.352689i −0.0302478 − 0.999542i \(-0.509630\pi\)
0.762615 + 0.646853i \(0.223915\pi\)
\(6\) 0 0
\(7\) −0.882702 − 1.10687i −0.333630 − 0.418359i 0.586514 − 0.809939i \(-0.300500\pi\)
−0.920144 + 0.391580i \(0.871929\pi\)
\(8\) 0 0
\(9\) 1.07942 − 4.72923i 0.359806 − 1.57641i
\(10\) 0 0
\(11\) −3.44878 + 0.787162i −1.03985 + 0.237338i −0.708160 − 0.706052i \(-0.750475\pi\)
−0.331686 + 0.943390i \(0.607618\pi\)
\(12\) 0 0
\(13\) −1.23911 − 5.42888i −0.343666 − 1.50570i −0.791269 − 0.611468i \(-0.790579\pi\)
0.447603 − 0.894232i \(-0.352278\pi\)
\(14\) 0 0
\(15\) 2.20971 − 4.58851i 0.570545 − 1.18475i
\(16\) 0 0
\(17\) 7.46337i 1.81013i 0.425269 + 0.905067i \(0.360180\pi\)
−0.425269 + 0.905067i \(0.639820\pi\)
\(18\) 0 0
\(19\) 3.93791 + 3.14038i 0.903418 + 0.720452i 0.960607 − 0.277911i \(-0.0896421\pi\)
−0.0571884 + 0.998363i \(0.518214\pi\)
\(20\) 0 0
\(21\) −3.86737 − 0.882702i −0.843930 − 0.192621i
\(22\) 0 0
\(23\) 1.61216 + 0.776374i 0.336158 + 0.161885i 0.594346 − 0.804210i \(-0.297411\pi\)
−0.258188 + 0.966095i \(0.583125\pi\)
\(24\) 0 0
\(25\) −1.05759 + 1.32618i −0.211518 + 0.265236i
\(26\) 0 0
\(27\) −2.25011 − 4.67241i −0.433034 − 0.899205i
\(28\) 0 0
\(29\) 4.21464 − 3.35213i 0.782639 − 0.622476i
\(30\) 0 0
\(31\) 2.07731 + 4.31359i 0.373097 + 0.774743i 0.999990 − 0.00436147i \(-0.00138830\pi\)
−0.626894 + 0.779105i \(0.715674\pi\)
\(32\) 0 0
\(33\) −6.17989 + 7.74934i −1.07578 + 1.34899i
\(34\) 0 0
\(35\) −2.31845 − 1.11651i −0.391890 − 0.188724i
\(36\) 0 0
\(37\) 1.04717 + 0.239009i 0.172153 + 0.0392929i 0.307729 − 0.951474i \(-0.400431\pi\)
−0.135575 + 0.990767i \(0.543288\pi\)
\(38\) 0 0
\(39\) −12.1986 − 9.72805i −1.95334 − 1.55773i
\(40\) 0 0
\(41\) 1.71164i 0.267314i 0.991028 + 0.133657i \(0.0426720\pi\)
−0.991028 + 0.133657i \(0.957328\pi\)
\(42\) 0 0
\(43\) 0.881405 − 1.83026i 0.134413 − 0.279112i −0.822888 − 0.568203i \(-0.807639\pi\)
0.957302 + 0.289091i \(0.0933531\pi\)
\(44\) 0 0
\(45\) −1.96197 − 8.59597i −0.292474 − 1.28141i
\(46\) 0 0
\(47\) 0.377517 − 0.0861658i 0.0550665 − 0.0125686i −0.194899 − 0.980823i \(-0.562438\pi\)
0.249965 + 0.968255i \(0.419581\pi\)
\(48\) 0 0
\(49\) 1.11164 − 4.87041i 0.158806 − 0.695774i
\(50\) 0 0
\(51\) 13.0384 + 16.3496i 1.82574 + 2.28940i
\(52\) 0 0
\(53\) 2.42678 − 1.16867i 0.333343 − 0.160530i −0.259723 − 0.965683i \(-0.583631\pi\)
0.593067 + 0.805153i \(0.297917\pi\)
\(54\) 0 0
\(55\) −5.02701 + 4.00891i −0.677842 + 0.540561i
\(56\) 0 0
\(57\) 14.1127 1.86928
\(58\) 0 0
\(59\) 3.81302 0.496413 0.248206 − 0.968707i \(-0.420159\pi\)
0.248206 + 0.968707i \(0.420159\pi\)
\(60\) 0 0
\(61\) −10.5585 + 8.42008i −1.35187 + 1.07808i −0.362607 + 0.931942i \(0.618113\pi\)
−0.989264 + 0.146139i \(0.953315\pi\)
\(62\) 0 0
\(63\) −6.18747 + 2.97973i −0.779548 + 0.375410i
\(64\) 0 0
\(65\) −6.31060 − 7.91325i −0.782734 − 0.981518i
\(66\) 0 0
\(67\) 0.659012 − 2.88732i 0.0805111 − 0.352742i −0.918586 − 0.395221i \(-0.870668\pi\)
0.999097 + 0.0424783i \(0.0135253\pi\)
\(68\) 0 0
\(69\) 4.88797 − 1.11565i 0.588442 − 0.134308i
\(70\) 0 0
\(71\) −1.39711 − 6.12116i −0.165807 − 0.726448i −0.987643 − 0.156723i \(-0.949907\pi\)
0.821836 − 0.569725i \(-0.192950\pi\)
\(72\) 0 0
\(73\) −0.416685 + 0.865255i −0.0487693 + 0.101270i −0.923930 − 0.382563i \(-0.875042\pi\)
0.875160 + 0.483833i \(0.160756\pi\)
\(74\) 0 0
\(75\) 4.75277i 0.548803i
\(76\) 0 0
\(77\) 3.91554 + 3.12254i 0.446217 + 0.355846i
\(78\) 0 0
\(79\) −2.71230 − 0.619064i −0.305157 − 0.0696501i 0.0672004 − 0.997739i \(-0.478593\pi\)
−0.372357 + 0.928089i \(0.621450\pi\)
\(80\) 0 0
\(81\) 0.0196143 + 0.00944576i 0.00217937 + 0.00104953i
\(82\) 0 0
\(83\) 5.65640 − 7.09290i 0.620870 − 0.778547i −0.367596 − 0.929985i \(-0.619819\pi\)
0.988467 + 0.151439i \(0.0483906\pi\)
\(84\) 0 0
\(85\) 5.88589 + 12.2222i 0.638415 + 1.32568i
\(86\) 0 0
\(87\) 3.37666 − 14.7062i 0.362016 − 1.57667i
\(88\) 0 0
\(89\) 2.30413 + 4.78459i 0.244238 + 0.507165i 0.986665 − 0.162761i \(-0.0520400\pi\)
−0.742428 + 0.669926i \(0.766326\pi\)
\(90\) 0 0
\(91\) −4.91532 + 6.16362i −0.515266 + 0.646123i
\(92\) 0 0
\(93\) 12.0864 + 5.82051i 1.25330 + 0.603558i
\(94\) 0 0
\(95\) 8.92543 + 2.03717i 0.915729 + 0.209009i
\(96\) 0 0
\(97\) −8.59586 − 6.85497i −0.872778 − 0.696017i 0.0809406 − 0.996719i \(-0.474208\pi\)
−0.953718 + 0.300702i \(0.902779\pi\)
\(98\) 0 0
\(99\) 17.1598i 1.72462i
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.y.c.241.2 12
4.3 odd 2 58.2.e.a.9.2 ✓ 12
12.11 even 2 522.2.n.a.415.1 12
29.13 even 14 inner 464.2.y.c.129.2 12
116.19 even 28 1682.2.a.r.1.6 6
116.39 even 28 1682.2.a.s.1.2 6
116.71 odd 14 58.2.e.a.13.2 yes 12
116.83 odd 14 1682.2.b.j.1681.5 12
116.91 odd 14 1682.2.b.j.1681.7 12
348.71 even 14 522.2.n.a.361.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.9.2 ✓ 12 4.3 odd 2
58.2.e.a.13.2 yes 12 116.71 odd 14
464.2.y.c.129.2 12 29.13 even 14 inner
464.2.y.c.241.2 12 1.1 even 1 trivial
522.2.n.a.361.1 12 348.71 even 14
522.2.n.a.415.1 12 12.11 even 2
1682.2.a.r.1.6 6 116.19 even 28
1682.2.a.s.1.2 6 116.39 even 28
1682.2.b.j.1681.5 12 116.83 odd 14
1682.2.b.j.1681.7 12 116.91 odd 14