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 5.1
Root \(-0.433884 + 0.900969i\) of defining polynomial
Character \(\chi\) \(=\) 58.5
Dual form 58.2.e.a.35.1

$q$-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+(-0.781831 + 0.623490i) q^{2} +(0.240787 - 0.0549581i) q^{3} +(0.222521 - 0.974928i) q^{4} +(2.13946 + 2.68280i) q^{5} +(-0.153989 + 0.193096i) q^{6} +(-0.349501 - 1.53126i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-2.64795 + 1.27518i) q^{9} +(-3.34540 - 0.763565i) q^{10} +(1.89794 - 3.94111i) q^{11} -0.246980i q^{12} +(-4.09493 - 1.97201i) q^{13} +(1.22798 + 0.979280i) q^{14} +(0.662597 + 0.528403i) q^{15} +(-0.900969 - 0.433884i) q^{16} -2.21432i q^{17} +(1.27518 - 2.64795i) q^{18} +(-0.412686 - 0.0941928i) q^{19} +(3.09161 - 1.48884i) q^{20} +(-0.168311 - 0.349501i) q^{21} +(0.973375 + 4.26463i) q^{22} +(-3.83783 + 4.81248i) q^{23} +(0.153989 + 0.193096i) q^{24} +(-1.50752 + 6.60486i) q^{25} +(4.43107 - 1.01136i) q^{26} +(-1.14680 + 0.914542i) q^{27} -1.57064 q^{28} +(4.45880 + 3.01978i) q^{29} -0.847493 q^{30} +(5.83914 - 4.65656i) q^{31} +(0.974928 - 0.222521i) q^{32} +(0.240404 - 1.05328i) q^{33} +(1.38061 + 1.73123i) q^{34} +(3.36033 - 4.21372i) q^{35} +(0.653989 + 2.86531i) q^{36} +(0.689279 + 1.43130i) q^{37} +(0.381379 - 0.183662i) q^{38} +(-1.09438 - 0.249786i) q^{39} +(-1.48884 + 3.09161i) q^{40} +4.56642i q^{41} +(0.349501 + 0.168311i) q^{42} +(-7.24319 - 5.77625i) q^{43} +(-3.41997 - 2.72733i) q^{44} +(-9.08625 - 4.37571i) q^{45} -6.15540i q^{46} +(-1.87017 + 3.88344i) q^{47} +(-0.240787 - 0.0549581i) q^{48} +(4.08416 - 1.96683i) q^{49} +(-2.93944 - 6.10381i) q^{50} +(-0.121695 - 0.533180i) q^{51} +(-2.83378 + 3.55344i) q^{52} +(4.07593 + 5.11105i) q^{53} +(0.326396 - 1.43004i) q^{54} +(14.6338 - 3.34007i) q^{55} +(1.22798 - 0.979280i) q^{56} -0.104546 q^{57} +(-5.36883 + 0.419060i) q^{58} -14.5556 q^{59} +(0.662597 - 0.528403i) q^{60} +(6.00919 - 1.37156i) q^{61} +(-1.66191 + 7.28128i) q^{62} +(2.87811 + 3.60903i) q^{63} +(-0.623490 + 0.781831i) q^{64} +(-3.47042 - 15.2049i) q^{65} +(0.468752 + 0.973375i) q^{66} +(-7.42071 + 3.57363i) q^{67} +(-2.15880 - 0.492733i) q^{68} +(-0.659615 + 1.36970i) q^{69} +5.38955i q^{70} +(11.4165 + 5.49788i) q^{71} +(-2.29780 - 1.83244i) q^{72} +(6.75568 + 5.38748i) q^{73} +(-1.43130 - 0.689279i) q^{74} +1.67322i q^{75} +(-0.183662 + 0.381379i) q^{76} +(-6.69822 - 1.52882i) q^{77} +(1.01136 - 0.487047i) q^{78} +(2.79634 + 5.80666i) q^{79} +(-0.763565 - 3.34540i) q^{80} +(5.27144 - 6.61017i) q^{81} +(-2.84712 - 3.57017i) q^{82} +(-0.137003 + 0.600250i) q^{83} +(-0.378191 + 0.0863196i) q^{84} +(5.94058 - 4.73746i) q^{85} +9.26439 q^{86} +(1.23958 + 0.482077i) q^{87} +4.37431 q^{88} +(-3.96126 + 3.15900i) q^{89} +(9.83213 - 2.24412i) q^{90} +(-1.58849 + 6.95964i) q^{91} +(3.83783 + 4.81248i) q^{92} +(1.15007 - 1.44215i) q^{93} +(-0.959131 - 4.20223i) q^{94} +(-0.630225 - 1.30868i) q^{95} +(0.222521 - 0.107160i) q^{96} +(-7.93496 - 1.81110i) q^{97} +(-1.96683 + 4.08416i) q^{98} +12.8561i 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 \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{11}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.781831 + 0.623490i −0.552838 + 0.440874i
\(3\) 0.240787 − 0.0549581i 0.139019 − 0.0317301i −0.152445 − 0.988312i \(-0.548715\pi\)
0.291464 + 0.956582i \(0.405858\pi\)
\(4\) 0.222521 − 0.974928i 0.111260 − 0.487464i
\(5\) 2.13946 + 2.68280i 0.956796 + 1.19978i 0.979786 + 0.200049i \(0.0641102\pi\)
−0.0229894 + 0.999736i \(0.507318\pi\)
\(6\) −0.153989 + 0.193096i −0.0628659 + 0.0788313i
\(7\) −0.349501 − 1.53126i −0.132099 − 0.578764i −0.997040 − 0.0768893i \(-0.975501\pi\)
0.864941 − 0.501874i \(-0.167356\pi\)
\(8\) 0.433884 + 0.900969i 0.153401 + 0.318541i
\(9\) −2.64795 + 1.27518i −0.882649 + 0.425062i
\(10\) −3.34540 − 0.763565i −1.05791 − 0.241460i
\(11\) 1.89794 − 3.94111i 0.572250 − 1.18829i −0.391176 − 0.920316i \(-0.627931\pi\)
0.963426 − 0.267974i \(-0.0863542\pi\)
\(12\) − 0.246980i − 0.0712969i
\(13\) −4.09493 − 1.97201i −1.13573 − 0.546938i −0.231011 − 0.972951i \(-0.574203\pi\)
−0.904717 + 0.426013i \(0.859918\pi\)
\(14\) 1.22798 + 0.979280i 0.328191 + 0.261724i
\(15\) 0.662597 + 0.528403i 0.171082 + 0.136433i
\(16\) −0.900969 − 0.433884i −0.225242 − 0.108471i
\(17\) − 2.21432i − 0.537052i −0.963272 − 0.268526i \(-0.913464\pi\)
0.963272 − 0.268526i \(-0.0865365\pi\)
\(18\) 1.27518 − 2.64795i 0.300564 − 0.624127i
\(19\) −0.412686 − 0.0941928i −0.0946766 − 0.0216093i 0.174920 − 0.984583i \(-0.444033\pi\)
−0.269597 + 0.962973i \(0.586890\pi\)
\(20\) 3.09161 − 1.48884i 0.691305 − 0.332915i
\(21\) −0.168311 − 0.349501i −0.0367284 − 0.0762674i
\(22\) 0.973375 + 4.26463i 0.207524 + 0.909223i
\(23\) −3.83783 + 4.81248i −0.800242 + 1.00347i 0.199480 + 0.979902i \(0.436075\pi\)
−0.999722 + 0.0235701i \(0.992497\pi\)
\(24\) 0.153989 + 0.193096i 0.0314329 + 0.0394156i
\(25\) −1.50752 + 6.60486i −0.301503 + 1.32097i
\(26\) 4.43107 − 1.01136i 0.869005 − 0.198345i
\(27\) −1.14680 + 0.914542i −0.220702 + 0.176004i
\(28\) −1.57064 −0.296824
\(29\) 4.45880 + 3.01978i 0.827979 + 0.560759i
\(30\) −0.847493 −0.154730
\(31\) 5.83914 − 4.65656i 1.04874 − 0.836342i 0.0619065 − 0.998082i \(-0.480282\pi\)
0.986833 + 0.161740i \(0.0517105\pi\)
\(32\) 0.974928 − 0.222521i 0.172345 − 0.0393365i
\(33\) 0.240404 − 1.05328i 0.0418489 − 0.183352i
\(34\) 1.38061 + 1.73123i 0.236772 + 0.296903i
\(35\) 3.36033 − 4.21372i 0.568000 − 0.712249i
\(36\) 0.653989 + 2.86531i 0.108998 + 0.477552i
\(37\) 0.689279 + 1.43130i 0.113317 + 0.235305i 0.949912 − 0.312516i \(-0.101172\pi\)
−0.836596 + 0.547821i \(0.815458\pi\)
\(38\) 0.381379 − 0.183662i 0.0618678 − 0.0297940i
\(39\) −1.09438 − 0.249786i −0.175242 − 0.0399978i
\(40\) −1.48884 + 3.09161i −0.235407 + 0.488827i
\(41\) 4.56642i 0.713155i 0.934266 + 0.356577i \(0.116056\pi\)
−0.934266 + 0.356577i \(0.883944\pi\)
\(42\) 0.349501 + 0.168311i 0.0539292 + 0.0259709i
\(43\) −7.24319 − 5.77625i −1.10458 − 0.880870i −0.110976 − 0.993823i \(-0.535398\pi\)
−0.993600 + 0.112953i \(0.963969\pi\)
\(44\) −3.41997 − 2.72733i −0.515580 − 0.411161i
\(45\) −9.08625 − 4.37571i −1.35450 − 0.652292i
\(46\) − 6.15540i − 0.907564i
\(47\) −1.87017 + 3.88344i −0.272792 + 0.566458i −0.991689 − 0.128658i \(-0.958933\pi\)
0.718897 + 0.695117i \(0.244647\pi\)
\(48\) −0.240787 − 0.0549581i −0.0347547 − 0.00793252i
\(49\) 4.08416 − 1.96683i 0.583452 − 0.280976i
\(50\) −2.93944 − 6.10381i −0.415699 − 0.863209i
\(51\) −0.121695 − 0.533180i −0.0170407 − 0.0746602i
\(52\) −2.83378 + 3.55344i −0.392974 + 0.492774i
\(53\) 4.07593 + 5.11105i 0.559872 + 0.702057i 0.978534 − 0.206085i \(-0.0660724\pi\)
−0.418663 + 0.908142i \(0.637501\pi\)
\(54\) 0.326396 − 1.43004i 0.0444169 − 0.194603i
\(55\) 14.6338 − 3.34007i 1.97322 − 0.450375i
\(56\) 1.22798 − 0.979280i 0.164096 − 0.130862i
\(57\) −0.104546 −0.0138475
\(58\) −5.36883 + 0.419060i −0.704963 + 0.0550253i
\(59\) −14.5556 −1.89498 −0.947490 − 0.319787i \(-0.896389\pi\)
−0.947490 + 0.319787i \(0.896389\pi\)
\(60\) 0.662597 − 0.528403i 0.0855409 − 0.0682166i
\(61\) 6.00919 − 1.37156i 0.769398 − 0.175610i 0.180234 − 0.983624i \(-0.442314\pi\)
0.589164 + 0.808014i \(0.299457\pi\)
\(62\) −1.66191 + 7.28128i −0.211062 + 0.924724i
\(63\) 2.87811 + 3.60903i 0.362607 + 0.454695i
\(64\) −0.623490 + 0.781831i −0.0779362 + 0.0977289i
\(65\) −3.47042 − 15.2049i −0.430453 − 1.88594i
\(66\) 0.468752 + 0.973375i 0.0576994 + 0.119814i
\(67\) −7.42071 + 3.57363i −0.906584 + 0.436588i −0.828262 − 0.560340i \(-0.810670\pi\)
−0.0783214 + 0.996928i \(0.524956\pi\)
\(68\) −2.15880 − 0.492733i −0.261793 − 0.0597526i
\(69\) −0.659615 + 1.36970i −0.0794083 + 0.164893i
\(70\) 5.38955i 0.644175i
\(71\) 11.4165 + 5.49788i 1.35489 + 0.652478i 0.963490 − 0.267744i \(-0.0862783\pi\)
0.391395 + 0.920223i \(0.371993\pi\)
\(72\) −2.29780 − 1.83244i −0.270799 − 0.215955i
\(73\) 6.75568 + 5.38748i 0.790693 + 0.630557i 0.933249 − 0.359230i \(-0.116961\pi\)
−0.142556 + 0.989787i \(0.545532\pi\)
\(74\) −1.43130 − 0.689279i −0.166386 − 0.0801271i
\(75\) 1.67322i 0.193206i
\(76\) −0.183662 + 0.381379i −0.0210675 + 0.0437472i
\(77\) −6.69822 − 1.52882i −0.763333 − 0.174226i
\(78\) 1.01136 − 0.487047i 0.114514 − 0.0551472i
\(79\) 2.79634 + 5.80666i 0.314613 + 0.653300i 0.996976 − 0.0777134i \(-0.0247619\pi\)
−0.682363 + 0.731014i \(0.739048\pi\)
\(80\) −0.763565 − 3.34540i −0.0853692 − 0.374027i
\(81\) 5.27144 − 6.61017i 0.585715 − 0.734464i
\(82\) −2.84712 − 3.57017i −0.314411 − 0.394259i
\(83\) −0.137003 + 0.600250i −0.0150381 + 0.0658860i −0.981890 − 0.189451i \(-0.939329\pi\)
0.966852 + 0.255337i \(0.0821864\pi\)
\(84\) −0.378191 + 0.0863196i −0.0412640 + 0.00941825i
\(85\) 5.94058 − 4.73746i 0.644346 − 0.513849i
\(86\) 9.26439 0.999005
\(87\) 1.23958 + 0.482077i 0.132897 + 0.0516841i
\(88\) 4.37431 0.466303
\(89\) −3.96126 + 3.15900i −0.419893 + 0.334853i −0.810536 − 0.585689i \(-0.800824\pi\)
0.390643 + 0.920542i \(0.372253\pi\)
\(90\) 9.83213 − 2.24412i 1.03640 − 0.236551i
\(91\) −1.58849 + 6.95964i −0.166519 + 0.729568i
\(92\) 3.83783 + 4.81248i 0.400121 + 0.501736i
\(93\) 1.15007 − 1.44215i 0.119257 − 0.149544i
\(94\) −0.959131 − 4.20223i −0.0989268 − 0.433427i
\(95\) −0.630225 − 1.30868i −0.0646597 − 0.134267i
\(96\) 0.222521 − 0.107160i 0.0227109 − 0.0109370i
\(97\) −7.93496 − 1.81110i −0.805673 − 0.183890i −0.200199 − 0.979755i \(-0.564159\pi\)
−0.605473 + 0.795866i \(0.707016\pi\)
\(98\) −1.96683 + 4.08416i −0.198680 + 0.412563i
\(99\) 12.8561i 1.29209i
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.5.1 ✓ 12
3.2 odd 2 522.2.n.a.469.2 12
4.3 odd 2 464.2.y.c.353.1 12
29.6 even 14 inner 58.2.e.a.35.1 yes 12
29.8 odd 28 1682.2.a.r.1.1 6
29.9 even 14 1682.2.b.j.1681.2 12
29.20 even 7 1682.2.b.j.1681.12 12
29.21 odd 28 1682.2.a.s.1.5 6
87.35 odd 14 522.2.n.a.325.2 12
116.35 odd 14 464.2.y.c.209.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.5.1 ✓ 12 1.1 even 1 trivial
58.2.e.a.35.1 yes 12 29.6 even 14 inner
464.2.y.c.209.1 12 116.35 odd 14
464.2.y.c.353.1 12 4.3 odd 2
522.2.n.a.325.2 12 87.35 odd 14
522.2.n.a.469.2 12 3.2 odd 2
1682.2.a.r.1.1 6 29.8 odd 28
1682.2.a.s.1.5 6 29.21 odd 28
1682.2.b.j.1681.2 12 29.9 even 14
1682.2.b.j.1681.12 12 29.20 even 7