Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,2,Mod(5,58)] 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("58.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.463132331723\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 9.2
Root \(0.974928 - 0.222521i\) of defining polynomial
Character \(\chi\) \(=\) 58.9
Dual form 58.2.e.a.13.2

$q$-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+(0.433884 + 0.900969i) q^{2} +(-2.19064 + 1.74698i) q^{3} +(-0.623490 + 0.781831i) q^{4} +(1.63762 - 0.788637i) q^{5} +(-2.52446 - 1.21572i) q^{6} +(0.882702 + 1.10687i) q^{7} +(-0.974928 - 0.222521i) q^{8} +(1.07942 - 4.72923i) q^{9} +(1.42108 + 1.13327i) q^{10} +(3.44878 - 0.787162i) q^{11} -2.80194i q^{12} +(-1.23911 - 5.42888i) q^{13} +(-0.614269 + 1.27554i) q^{14} +(-2.20971 + 4.58851i) q^{15} +(-0.222521 - 0.974928i) q^{16} +7.46337i q^{17} +(4.72923 - 1.07942i) q^{18} +(-3.93791 - 3.14038i) q^{19} +(-0.404459 + 1.77205i) q^{20} +(-3.86737 - 0.882702i) q^{21} +(2.20558 + 2.76571i) q^{22} +(-1.61216 - 0.776374i) q^{23} +(2.52446 - 1.21572i) q^{24} +(-1.05759 + 1.32618i) q^{25} +(4.35362 - 3.47190i) q^{26} +(2.25011 + 4.67241i) q^{27} -1.41574 q^{28} +(4.21464 - 3.35213i) q^{29} -5.09287 q^{30} +(-2.07731 - 4.31359i) q^{31} +(0.781831 - 0.623490i) q^{32} +(-6.17989 + 7.74934i) q^{33} +(-6.72427 + 3.23824i) q^{34} +(2.31845 + 1.11651i) q^{35} +(3.02446 + 3.79255i) q^{36} +(1.04717 + 0.239009i) q^{37} +(1.12079 - 4.91049i) q^{38} +(12.1986 + 9.72805i) q^{39} +(-1.77205 + 0.404459i) q^{40} +1.71164i q^{41} +(-0.882702 - 3.86737i) q^{42} +(-0.881405 + 1.83026i) q^{43} +(-1.53485 + 3.18715i) q^{44} +(-1.96197 - 8.59597i) q^{45} -1.78936i q^{46} +(-0.377517 + 0.0861658i) q^{47} +(2.19064 + 1.74698i) q^{48} +(1.11164 - 4.87041i) q^{49} +(-1.65372 - 0.377450i) q^{50} +(-13.0384 - 16.3496i) q^{51} +(5.01704 + 2.41608i) q^{52} +(2.42678 - 1.16867i) q^{53} +(-3.23341 + 4.05456i) q^{54} +(5.02701 - 4.00891i) q^{55} +(-0.614269 - 1.27554i) q^{56} +14.1127 q^{57} +(4.84883 + 2.34282i) q^{58} -3.81302 q^{59} +(-2.20971 - 4.58851i) q^{60} +(-10.5585 + 8.42008i) q^{61} +(2.98510 - 3.74319i) q^{62} +(6.18747 - 2.97973i) q^{63} +(0.900969 + 0.433884i) q^{64} +(-6.31060 - 7.91325i) q^{65} +(-9.66327 - 2.20558i) q^{66} +(-0.659012 + 2.88732i) q^{67} +(-5.83510 - 4.65334i) q^{68} +(4.88797 - 1.11565i) q^{69} +2.57329i q^{70} +(1.39711 + 6.12116i) q^{71} +(-2.10471 + 4.37047i) q^{72} +(-0.416685 + 0.865255i) q^{73} +(0.239009 + 1.04717i) q^{74} -4.75277i q^{75} +(4.91049 - 1.12079i) q^{76} +(3.91554 + 3.12254i) q^{77} +(-3.47190 + 15.2114i) q^{78} +(2.71230 + 0.619064i) q^{79} +(-1.13327 - 1.42108i) q^{80} +(0.0196143 + 0.00944576i) q^{81} +(-1.54214 + 0.742654i) q^{82} +(-5.65640 + 7.09290i) q^{83} +(3.10139 - 2.47328i) q^{84} +(5.88589 + 12.2222i) q^{85} -2.03143 q^{86} +(-3.37666 + 14.7062i) q^{87} -3.53747 q^{88} +(2.30413 + 4.78459i) q^{89} +(6.89343 - 5.49733i) q^{90} +(4.91532 - 6.16362i) q^{91} +(1.61216 - 0.776374i) q^{92} +(12.0864 + 5.82051i) q^{93} +(-0.241431 - 0.302745i) q^{94} +(-8.92543 - 2.03717i) q^{95} +(-0.623490 + 2.73169i) q^{96} +(-8.59586 - 6.85497i) q^{97} +(4.87041 - 1.11164i) q^{98} -17.1598i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{9} - 26 q^{13} - 14 q^{15} - 2 q^{16} + 2 q^{20} + 14 q^{21} + 4 q^{22} - 16 q^{23} + 12 q^{24} + 22 q^{25} + 14 q^{26} - 4 q^{28} + 18 q^{29} + 16 q^{30}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{5}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.433884 + 0.900969i 0.306802 + 0.637081i
\(3\) −2.19064 + 1.74698i −1.26477 + 1.00862i −0.265763 + 0.964039i \(0.585624\pi\)
−0.999006 + 0.0445806i \(0.985805\pi\)
\(4\) −0.623490 + 0.781831i −0.311745 + 0.390916i
\(5\) 1.63762 − 0.788637i 0.732367 − 0.352689i −0.0302478 − 0.999542i \(-0.509630\pi\)
0.762615 + 0.646853i \(0.223915\pi\)
\(6\) −2.52446 − 1.21572i −1.03061 − 0.496314i
\(7\) 0.882702 + 1.10687i 0.333630 + 0.418359i 0.920144 − 0.391580i \(-0.128071\pi\)
−0.586514 + 0.809939i \(0.699500\pi\)
\(8\) −0.974928 − 0.222521i −0.344689 − 0.0786730i
\(9\) 1.07942 − 4.72923i 0.359806 − 1.57641i
\(10\) 1.42108 + 1.13327i 0.449383 + 0.358371i
\(11\) 3.44878 − 0.787162i 1.03985 − 0.237338i 0.331686 − 0.943390i \(-0.392382\pi\)
0.708160 + 0.706052i \(0.249525\pi\)
\(12\) − 2.80194i − 0.808850i
\(13\) −1.23911 − 5.42888i −0.343666 − 1.50570i −0.791269 − 0.611468i \(-0.790579\pi\)
0.447603 − 0.894232i \(-0.352278\pi\)
\(14\) −0.614269 + 1.27554i −0.164170 + 0.340903i
\(15\) −2.20971 + 4.58851i −0.570545 + 1.18475i
\(16\) −0.222521 − 0.974928i −0.0556302 − 0.243732i
\(17\) 7.46337i 1.81013i 0.425269 + 0.905067i \(0.360180\pi\)
−0.425269 + 0.905067i \(0.639820\pi\)
\(18\) 4.72923 − 1.07942i 1.11469 − 0.254421i
\(19\) −3.93791 − 3.14038i −0.903418 − 0.720452i 0.0571884 − 0.998363i \(-0.481786\pi\)
−0.960607 + 0.277911i \(0.910358\pi\)
\(20\) −0.404459 + 1.77205i −0.0904398 + 0.396243i
\(21\) −3.86737 − 0.882702i −0.843930 − 0.192621i
\(22\) 2.20558 + 2.76571i 0.470231 + 0.589651i
\(23\) −1.61216 − 0.776374i −0.336158 − 0.161885i 0.258188 − 0.966095i \(-0.416875\pi\)
−0.594346 + 0.804210i \(0.702589\pi\)
\(24\) 2.52446 − 1.21572i 0.515303 − 0.248157i
\(25\) −1.05759 + 1.32618i −0.211518 + 0.265236i
\(26\) 4.35362 − 3.47190i 0.853816 − 0.680895i
\(27\) 2.25011 + 4.67241i 0.433034 + 0.899205i
\(28\) −1.41574 −0.267551
\(29\) 4.21464 − 3.35213i 0.782639 − 0.622476i
\(30\) −5.09287 −0.929826
\(31\) −2.07731 − 4.31359i −0.373097 − 0.774743i 0.626894 − 0.779105i \(-0.284326\pi\)
−0.999990 + 0.00436147i \(0.998612\pi\)
\(32\) 0.781831 − 0.623490i 0.138210 − 0.110218i
\(33\) −6.17989 + 7.74934i −1.07578 + 1.34899i
\(34\) −6.72427 + 3.23824i −1.15320 + 0.555353i
\(35\) 2.31845 + 1.11651i 0.391890 + 0.188724i
\(36\) 3.02446 + 3.79255i 0.504076 + 0.632092i
\(37\) 1.04717 + 0.239009i 0.172153 + 0.0392929i 0.307729 − 0.951474i \(-0.400431\pi\)
−0.135575 + 0.990767i \(0.543288\pi\)
\(38\) 1.12079 − 4.91049i 0.181816 − 0.796587i
\(39\) 12.1986 + 9.72805i 1.95334 + 1.55773i
\(40\) −1.77205 + 0.404459i −0.280186 + 0.0639506i
\(41\) 1.71164i 0.267314i 0.991028 + 0.133657i \(0.0426720\pi\)
−0.991028 + 0.133657i \(0.957328\pi\)
\(42\) −0.882702 − 3.86737i −0.136204 − 0.596749i
\(43\) −0.881405 + 1.83026i −0.134413 + 0.279112i −0.957302 − 0.289091i \(-0.906647\pi\)
0.822888 + 0.568203i \(0.192361\pi\)
\(44\) −1.53485 + 3.18715i −0.231388 + 0.480481i
\(45\) −1.96197 − 8.59597i −0.292474 − 1.28141i
\(46\) − 1.78936i − 0.263827i
\(47\) −0.377517 + 0.0861658i −0.0550665 + 0.0125686i −0.249965 − 0.968255i \(-0.580419\pi\)
0.194899 + 0.980823i \(0.437562\pi\)
\(48\) 2.19064 + 1.74698i 0.316192 + 0.252155i
\(49\) 1.11164 − 4.87041i 0.158806 − 0.695774i
\(50\) −1.65372 − 0.377450i −0.233871 − 0.0533795i
\(51\) −13.0384 − 16.3496i −1.82574 − 2.28940i
\(52\) 5.01704 + 2.41608i 0.695738 + 0.335050i
\(53\) 2.42678 − 1.16867i 0.333343 − 0.160530i −0.259723 − 0.965683i \(-0.583631\pi\)
0.593067 + 0.805153i \(0.297917\pi\)
\(54\) −3.23341 + 4.05456i −0.440011 + 0.551756i
\(55\) 5.02701 − 4.00891i 0.677842 − 0.540561i
\(56\) −0.614269 − 1.27554i −0.0820851 − 0.170451i
\(57\) 14.1127 1.86928
\(58\) 4.84883 + 2.34282i 0.636683 + 0.307628i
\(59\) −3.81302 −0.496413 −0.248206 − 0.968707i \(-0.579841\pi\)
−0.248206 + 0.968707i \(0.579841\pi\)
\(60\) −2.20971 − 4.58851i −0.285273 − 0.592375i
\(61\) −10.5585 + 8.42008i −1.35187 + 1.07808i −0.362607 + 0.931942i \(0.618113\pi\)
−0.989264 + 0.146139i \(0.953315\pi\)
\(62\) 2.98510 − 3.74319i 0.379107 − 0.475386i
\(63\) 6.18747 − 2.97973i 0.779548 − 0.375410i
\(64\) 0.900969 + 0.433884i 0.112621 + 0.0542355i
\(65\) −6.31060 − 7.91325i −0.782734 − 0.981518i
\(66\) −9.66327 − 2.20558i −1.18947 − 0.271488i
\(67\) −0.659012 + 2.88732i −0.0805111 + 0.352742i −0.999097 − 0.0424783i \(-0.986475\pi\)
0.918586 + 0.395221i \(0.129332\pi\)
\(68\) −5.83510 − 4.65334i −0.707610 − 0.564300i
\(69\) 4.88797 − 1.11565i 0.588442 − 0.134308i
\(70\) 2.57329i 0.307567i
\(71\) 1.39711 + 6.12116i 0.165807 + 0.726448i 0.987643 + 0.156723i \(0.0500930\pi\)
−0.821836 + 0.569725i \(0.807050\pi\)
\(72\) −2.10471 + 4.37047i −0.248042 + 0.515065i
\(73\) −0.416685 + 0.865255i −0.0487693 + 0.101270i −0.923930 − 0.382563i \(-0.875042\pi\)
0.875160 + 0.483833i \(0.160756\pi\)
\(74\) 0.239009 + 1.04717i 0.0277843 + 0.121731i
\(75\) − 4.75277i − 0.548803i
\(76\) 4.91049 − 1.12079i 0.563272 − 0.128563i
\(77\) 3.91554 + 3.12254i 0.446217 + 0.355846i
\(78\) −3.47190 + 15.2114i −0.393115 + 1.72235i
\(79\) 2.71230 + 0.619064i 0.305157 + 0.0696501i 0.372357 − 0.928089i \(-0.378550\pi\)
−0.0672004 + 0.997739i \(0.521407\pi\)
\(80\) −1.13327 − 1.42108i −0.126703 − 0.158881i
\(81\) 0.0196143 + 0.00944576i 0.00217937 + 0.00104953i
\(82\) −1.54214 + 0.742654i −0.170300 + 0.0820124i
\(83\) −5.65640 + 7.09290i −0.620870 + 0.778547i −0.988467 − 0.151439i \(-0.951609\pi\)
0.367596 + 0.929985i \(0.380181\pi\)
\(84\) 3.10139 − 2.47328i 0.338390 − 0.269857i
\(85\) 5.88589 + 12.2222i 0.638415 + 1.32568i
\(86\) −2.03143 −0.219055
\(87\) −3.37666 + 14.7062i −0.362016 + 1.57667i
\(88\) −3.53747 −0.377096
\(89\) 2.30413 + 4.78459i 0.244238 + 0.507165i 0.986665 − 0.162761i \(-0.0520400\pi\)
−0.742428 + 0.669926i \(0.766326\pi\)
\(90\) 6.89343 − 5.49733i 0.726631 − 0.579469i
\(91\) 4.91532 − 6.16362i 0.515266 − 0.646123i
\(92\) 1.61216 − 0.776374i 0.168079 − 0.0809426i
\(93\) 12.0864 + 5.82051i 1.25330 + 0.603558i
\(94\) −0.241431 − 0.302745i −0.0249017 − 0.0312258i
\(95\) −8.92543 − 2.03717i −0.915729 − 0.209009i
\(96\) −0.623490 + 2.73169i −0.0636347 + 0.278802i
\(97\) −8.59586 − 6.85497i −0.872778 − 0.696017i 0.0809406 − 0.996719i \(-0.474208\pi\)
−0.953718 + 0.300702i \(0.902779\pi\)
\(98\) 4.87041 − 1.11164i 0.491986 − 0.112293i
\(99\) − 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 58.2.e.a.9.2 ✓ 12
3.2 odd 2 522.2.n.a.415.1 12
4.3 odd 2 464.2.y.c.241.2 12
29.4 even 14 1682.2.b.j.1681.7 12
29.10 odd 28 1682.2.a.s.1.2 6
29.13 even 14 inner 58.2.e.a.13.2 yes 12
29.19 odd 28 1682.2.a.r.1.6 6
29.25 even 7 1682.2.b.j.1681.5 12
87.71 odd 14 522.2.n.a.361.1 12
116.71 odd 14 464.2.y.c.129.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.9.2 ✓ 12 1.1 even 1 trivial
58.2.e.a.13.2 yes 12 29.13 even 14 inner
464.2.y.c.129.2 12 116.71 odd 14
464.2.y.c.241.2 12 4.3 odd 2
522.2.n.a.361.1 12 87.71 odd 14
522.2.n.a.415.1 12 3.2 odd 2
1682.2.a.r.1.6 6 29.19 odd 28
1682.2.a.s.1.2 6 29.10 odd 28
1682.2.b.j.1681.5 12 29.25 even 7
1682.2.b.j.1681.7 12 29.4 even 14