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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 129.2
Root \(0.781831 + 0.623490i\) of defining polynomial
Character \(\chi\) \(=\) 464.129
Dual form 464.2.y.c.241.2

$q$-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 \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{9}{14}\right)\)

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 0.999006 + 0.0445806i \(0.0141952\pi\)
0.265763 + 0.964039i \(0.414376\pi\)
\(4\) 0 0
\(5\) 1.63762 + 0.788637i 0.732367 + 0.352689i 0.762615 − 0.646853i \(-0.223915\pi\)
−0.0302478 + 0.999542i \(0.509630\pi\)
\(6\) 0 0
\(7\) −0.882702 + 1.10687i −0.333630 + 0.418359i −0.920144 − 0.391580i \(-0.871929\pi\)
0.586514 + 0.809939i \(0.300500\pi\)
\(8\) 0 0
\(9\) 1.07942 + 4.72923i 0.359806 + 1.57641i
\(10\) 0 0
\(11\) −3.44878 − 0.787162i −1.03985 − 0.237338i −0.331686 − 0.943390i \(-0.607618\pi\)
−0.708160 + 0.706052i \(0.750475\pi\)
\(12\) 0 0
\(13\) −1.23911 + 5.42888i −0.343666 + 1.50570i 0.447603 + 0.894232i \(0.352278\pi\)
−0.791269 + 0.611468i \(0.790579\pi\)
\(14\) 0 0
\(15\) 2.20971 + 4.58851i 0.570545 + 1.18475i
\(16\) 0 0
\(17\) − 7.46337i − 1.81013i −0.425269 − 0.905067i \(-0.639820\pi\)
0.425269 − 0.905067i \(-0.360180\pi\)
\(18\) 0 0
\(19\) 3.93791 − 3.14038i 0.903418 − 0.720452i −0.0571884 − 0.998363i \(-0.518214\pi\)
0.960607 + 0.277911i \(0.0896421\pi\)
\(20\) 0 0
\(21\) −3.86737 + 0.882702i −0.843930 + 0.192621i
\(22\) 0 0
\(23\) 1.61216 − 0.776374i 0.336158 − 0.161885i −0.258188 − 0.966095i \(-0.583125\pi\)
0.594346 + 0.804210i \(0.297411\pi\)
\(24\) 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 −0.626894 − 0.779105i \(-0.715674\pi\)
0.999990 + 0.00436147i \(0.00138830\pi\)
\(32\) 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 −0.135575 − 0.990767i \(-0.543288\pi\)
0.307729 + 0.951474i \(0.400431\pi\)
\(38\) 0 0
\(39\) −12.1986 + 9.72805i −1.95334 + 1.55773i
\(40\) 0 0
\(41\) − 1.71164i − 0.267314i −0.991028 − 0.133657i \(-0.957328\pi\)
0.991028 − 0.133657i \(-0.0426720\pi\)
\(42\) 0 0
\(43\) 0.881405 + 1.83026i 0.134413 + 0.279112i 0.957302 − 0.289091i \(-0.0933531\pi\)
−0.822888 + 0.568203i \(0.807639\pi\)
\(44\) 0 0
\(45\) −1.96197 + 8.59597i −0.292474 + 1.28141i
\(46\) 0 0
\(47\) 0.377517 + 0.0861658i 0.0550665 + 0.0125686i 0.249965 − 0.968255i \(-0.419581\pi\)
−0.194899 + 0.980823i \(0.562438\pi\)
\(48\) 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 0.593067 − 0.805153i \(-0.297917\pi\)
−0.259723 + 0.965683i \(0.583631\pi\)
\(54\) 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 −0.989264 − 0.146139i \(-0.953315\pi\)
−0.362607 − 0.931942i \(-0.618113\pi\)
\(62\) 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 0.999097 − 0.0424783i \(-0.0135253\pi\)
−0.918586 + 0.395221i \(0.870668\pi\)
\(68\) 0 0
\(69\) 4.88797 + 1.11565i 0.588442 + 0.134308i
\(70\) 0 0
\(71\) −1.39711 + 6.12116i −0.165807 + 0.726448i 0.821836 + 0.569725i \(0.192950\pi\)
−0.987643 + 0.156723i \(0.949907\pi\)
\(72\) 0 0
\(73\) −0.416685 − 0.865255i −0.0487693 − 0.101270i 0.875160 − 0.483833i \(-0.160756\pi\)
−0.923930 + 0.382563i \(0.875042\pi\)
\(74\) 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 −0.372357 − 0.928089i \(-0.621450\pi\)
0.0672004 + 0.997739i \(0.478593\pi\)
\(80\) 0 0
\(81\) 0.0196143 − 0.00944576i 0.00217937 − 0.00104953i
\(82\) 0 0
\(83\) 5.65640 + 7.09290i 0.620870 + 0.778547i 0.988467 − 0.151439i \(-0.0483906\pi\)
−0.367596 + 0.929985i \(0.619819\pi\)
\(84\) 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 −0.742428 − 0.669926i \(-0.766326\pi\)
0.986665 + 0.162761i \(0.0520400\pi\)
\(90\) 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 −0.953718 − 0.300702i \(-0.902779\pi\)
0.0809406 + 0.996719i \(0.474208\pi\)
\(98\) 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.129.2 12
4.3 odd 2 58.2.e.a.13.2 yes 12
12.11 even 2 522.2.n.a.361.1 12
29.9 even 14 inner 464.2.y.c.241.2 12
116.3 even 28 1682.2.a.s.1.2 6
116.7 odd 14 1682.2.b.j.1681.7 12
116.51 odd 14 1682.2.b.j.1681.5 12
116.55 even 28 1682.2.a.r.1.6 6
116.67 odd 14 58.2.e.a.9.2 ✓ 12
348.299 even 14 522.2.n.a.415.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.9.2 ✓ 12 116.67 odd 14
58.2.e.a.13.2 yes 12 4.3 odd 2
464.2.y.c.129.2 12 1.1 even 1 trivial
464.2.y.c.241.2 12 29.9 even 14 inner
522.2.n.a.361.1 12 12.11 even 2
522.2.n.a.415.1 12 348.299 even 14
1682.2.a.r.1.6 6 116.55 even 28
1682.2.a.s.1.2 6 116.3 even 28
1682.2.b.j.1681.5 12 116.51 odd 14
1682.2.b.j.1681.7 12 116.7 odd 14