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Properties

Label 578.2.f.b.35.4
Level $578$
Weight $2$
Character 578.35
Analytic conductor $4.615$
Analytic rank $0$
Dimension $224$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [578,2,Mod(35,578)] 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("578.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.f (of order \(17\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [224] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label 35.4
Character \(\chi\) \(=\) 578.35
Dual form 578.2.f.b.545.4

$q$-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+(0.739009 + 0.673696i) q^{2} +(-0.641416 - 2.25434i) q^{3} +(0.0922684 + 0.995734i) q^{4} +(0.981061 + 1.29913i) q^{5} +(1.04473 - 2.09810i) q^{6} +(1.36765 - 0.846815i) q^{7} +(-0.602635 + 0.798017i) q^{8} +(-2.11999 + 1.31264i) q^{9} +(-0.150209 + 1.62101i) q^{10} +(0.139732 + 1.50794i) q^{11} +(2.18554 - 0.846684i) q^{12} +(3.71250 - 4.91615i) q^{13} +(1.58120 + 0.295578i) q^{14} +(2.29943 - 3.04493i) q^{15} +(-0.982973 + 0.183750i) q^{16} +(3.55688 + 2.08533i) q^{17} +(-2.45102 - 0.458175i) q^{18} +(-4.93201 + 4.49612i) q^{19} +(-1.20307 + 1.09675i) q^{20} +(-2.78625 - 2.54000i) q^{21} +(-0.912633 + 1.20852i) q^{22} +(4.80552 - 2.97545i) q^{23} +(2.18554 + 0.846684i) q^{24} +(0.643044 - 2.26007i) q^{25} +(6.05556 - 1.13198i) q^{26} +(-0.877354 - 0.799814i) q^{27} +(0.969394 + 1.28369i) q^{28} +(0.0714459 + 0.771024i) q^{29} +(3.75066 - 0.701119i) q^{30} +(5.33169 - 7.06030i) q^{31} +(-0.850217 - 0.526432i) q^{32} +(3.30980 - 1.28222i) q^{33} +(1.22369 + 3.93733i) q^{34} +(2.44188 + 0.945989i) q^{35} +(-1.50265 - 1.98984i) q^{36} +(-2.44493 - 0.947173i) q^{37} -6.67382 q^{38} +(-13.4639 - 5.21596i) q^{39} -1.62795 q^{40} +(2.65151 + 9.31909i) q^{41} +(-0.347875 - 3.75416i) q^{42} +(-1.83694 - 0.343383i) q^{43} +(-1.48862 + 0.278271i) q^{44} +(-3.78515 - 1.46637i) q^{45} +(5.55587 + 1.03857i) q^{46} +(-7.07028 - 4.37773i) q^{47} +(1.04473 + 2.09810i) q^{48} +(-1.96679 + 3.94984i) q^{49} +(1.99781 - 1.23699i) q^{50} +(2.41960 - 9.35599i) q^{51} +(5.23773 + 3.24306i) q^{52} +(11.1930 - 6.93039i) q^{53} +(-0.109541 - 1.18214i) q^{54} +(-1.82194 + 1.66092i) q^{55} +(-0.148422 + 1.60173i) q^{56} +(13.2993 + 8.23456i) q^{57} +(-0.466636 + 0.617926i) q^{58} +(-0.799194 - 1.60500i) q^{59} +(3.24411 + 2.00867i) q^{60} +(-5.28511 + 10.6139i) q^{61} +(8.69666 - 1.62569i) q^{62} +(-1.78785 + 3.59049i) q^{63} +(-0.273663 - 0.961826i) q^{64} +10.0289 q^{65} +(3.30980 + 1.28222i) q^{66} +(2.44901 - 2.23257i) q^{67} +(-1.74824 + 3.73412i) q^{68} +(-9.79002 - 8.92478i) q^{69} +(1.16726 + 2.34418i) q^{70} +(-2.04642 + 1.26709i) q^{71} +(0.230069 - 2.48284i) q^{72} +(0.316669 - 0.0591957i) q^{73} +(-1.16872 - 2.34711i) q^{74} -5.50742 q^{75} +(-4.93201 - 4.49612i) q^{76} +(1.46806 + 1.94402i) q^{77} +(-6.43601 - 12.9252i) q^{78} +(-10.5282 + 9.59770i) q^{79} +(-1.20307 - 1.09675i) q^{80} +(-4.57462 + 9.18707i) q^{81} +(-4.31874 + 8.67320i) q^{82} +(-3.24224 + 11.3953i) q^{83} +(2.27208 - 3.00872i) q^{84} +(0.780396 + 6.66670i) q^{85} +(-1.12618 - 1.49130i) q^{86} +(1.69233 - 0.655610i) q^{87} +(-1.28757 - 0.797231i) q^{88} +(-1.89939 - 2.51520i) q^{89} +(-1.80937 - 3.63370i) q^{90} +(0.914349 - 9.86740i) q^{91} +(3.40615 + 4.51048i) q^{92} +(-19.3362 - 7.49087i) q^{93} +(-2.27574 - 7.99840i) q^{94} +(-10.6797 - 1.99638i) q^{95} +(-0.641416 + 2.25434i) q^{96} +(-11.9798 + 7.41760i) q^{97} +(-4.11446 + 1.59395i) q^{98} +(-2.27563 - 3.01342i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q - 14 q^{2} - 2 q^{3} - 14 q^{4} + 28 q^{5} - 2 q^{6} - 4 q^{7} - 14 q^{8} - 24 q^{9} - 6 q^{10} - 18 q^{11} - 2 q^{12} - 12 q^{13} - 4 q^{14} - 22 q^{15} - 14 q^{16} - 16 q^{17} - 24 q^{18} - 12 q^{19}+ \cdots - 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{7}{17}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.739009 + 0.673696i 0.522558 + 0.476375i
\(3\) −0.641416 − 2.25434i −0.370322 − 1.30155i −0.893401 − 0.449261i \(-0.851687\pi\)
0.523079 − 0.852284i \(-0.324783\pi\)
\(4\) 0.0922684 + 0.995734i 0.0461342 + 0.497867i
\(5\) 0.981061 + 1.29913i 0.438744 + 0.580991i 0.962638 − 0.270793i \(-0.0872858\pi\)
−0.523894 + 0.851784i \(0.675521\pi\)
\(6\) 1.04473 − 2.09810i 0.426509 − 0.856545i
\(7\) 1.36765 − 0.846815i 0.516925 − 0.320066i −0.243056 − 0.970012i \(-0.578150\pi\)
0.759980 + 0.649946i \(0.225209\pi\)
\(8\) −0.602635 + 0.798017i −0.213064 + 0.282142i
\(9\) −2.11999 + 1.31264i −0.706665 + 0.437548i
\(10\) −0.150209 + 1.62101i −0.0475001 + 0.512608i
\(11\) 0.139732 + 1.50794i 0.0421307 + 0.454662i 0.990502 + 0.137500i \(0.0439066\pi\)
−0.948371 + 0.317163i \(0.897270\pi\)
\(12\) 2.18554 − 0.846684i 0.630912 − 0.244417i
\(13\) 3.71250 − 4.91615i 1.02966 − 1.36349i 0.100040 − 0.994983i \(-0.468103\pi\)
0.929623 − 0.368511i \(-0.120132\pi\)
\(14\) 1.58120 + 0.295578i 0.422595 + 0.0789966i
\(15\) 2.29943 − 3.04493i 0.593709 − 0.786198i
\(16\) −0.982973 + 0.183750i −0.245743 + 0.0459374i
\(17\) 3.55688 + 2.08533i 0.862671 + 0.505766i
\(18\) −2.45102 − 0.458175i −0.577710 − 0.107993i
\(19\) −4.93201 + 4.49612i −1.13148 + 1.03148i −0.132292 + 0.991211i \(0.542234\pi\)
−0.999189 + 0.0402701i \(0.987178\pi\)
\(20\) −1.20307 + 1.09675i −0.269015 + 0.245240i
\(21\) −2.78625 − 2.54000i −0.608009 − 0.554273i
\(22\) −0.912633 + 1.20852i −0.194574 + 0.257658i
\(23\) 4.80552 − 2.97545i 1.00202 − 0.620424i 0.0756871 − 0.997132i \(-0.475885\pi\)
0.926332 + 0.376708i \(0.122944\pi\)
\(24\) 2.18554 + 0.846684i 0.446122 + 0.172829i
\(25\) 0.643044 − 2.26007i 0.128609 − 0.452013i
\(26\) 6.05556 − 1.13198i 1.18759 − 0.222000i
\(27\) −0.877354 − 0.799814i −0.168847 − 0.153924i
\(28\) 0.969394 + 1.28369i 0.183198 + 0.242594i
\(29\) 0.0714459 + 0.771024i 0.0132672 + 0.143176i 0.999791 − 0.0204418i \(-0.00650728\pi\)
−0.986524 + 0.163617i \(0.947684\pi\)
\(30\) 3.75066 − 0.701119i 0.684773 − 0.128006i
\(31\) 5.33169 − 7.06030i 0.957600 − 1.26807i −0.00579549 − 0.999983i \(-0.501845\pi\)
0.963396 − 0.268084i \(-0.0863905\pi\)
\(32\) −0.850217 − 0.526432i −0.150299 − 0.0930609i
\(33\) 3.30980 − 1.28222i 0.576162 − 0.223206i
\(34\) 1.22369 + 3.93733i 0.209861 + 0.675247i
\(35\) 2.44188 + 0.945989i 0.412753 + 0.159901i
\(36\) −1.50265 − 1.98984i −0.250442 − 0.331639i
\(37\) −2.44493 − 0.947173i −0.401945 − 0.155714i 0.151779 − 0.988414i \(-0.451500\pi\)
−0.553724 + 0.832700i \(0.686794\pi\)
\(38\) −6.67382 −1.08264
\(39\) −13.4639 − 5.21596i −2.15596 − 0.835222i
\(40\) −1.62795 −0.257402
\(41\) 2.65151 + 9.31909i 0.414096 + 1.45540i 0.835986 + 0.548751i \(0.184896\pi\)
−0.421890 + 0.906647i \(0.638633\pi\)
\(42\) −0.347875 − 3.75416i −0.0536782 − 0.579280i
\(43\) −1.83694 − 0.343383i −0.280131 − 0.0523655i 0.0418092 − 0.999126i \(-0.486688\pi\)
−0.321940 + 0.946760i \(0.604335\pi\)
\(44\) −1.48862 + 0.278271i −0.224418 + 0.0419510i
\(45\) −3.78515 − 1.46637i −0.564256 − 0.218594i
\(46\) 5.55587 + 1.03857i 0.819168 + 0.153129i
\(47\) −7.07028 − 4.37773i −1.03131 − 0.638558i −0.0970941 − 0.995275i \(-0.530955\pi\)
−0.934212 + 0.356717i \(0.883896\pi\)
\(48\) 1.04473 + 2.09810i 0.150794 + 0.302834i
\(49\) −1.96679 + 3.94984i −0.280970 + 0.564263i
\(50\) 1.99781 − 1.23699i 0.282533 − 0.174937i
\(51\) 2.41960 − 9.35599i 0.338812 − 1.31010i
\(52\) 5.23773 + 3.24306i 0.726342 + 0.449732i
\(53\) 11.1930 − 6.93039i 1.53747 − 0.951963i 0.543853 − 0.839181i \(-0.316965\pi\)
0.993620 − 0.112782i \(-0.0359762\pi\)
\(54\) −0.109541 − 1.18214i −0.0149067 − 0.160869i
\(55\) −1.82194 + 1.66092i −0.245670 + 0.223958i
\(56\) −0.148422 + 1.60173i −0.0198338 + 0.214040i
\(57\) 13.2993 + 8.23456i 1.76153 + 1.09069i
\(58\) −0.466636 + 0.617926i −0.0612724 + 0.0811377i
\(59\) −0.799194 − 1.60500i −0.104046 − 0.208953i 0.837022 − 0.547169i \(-0.184294\pi\)
−0.941069 + 0.338216i \(0.890177\pi\)
\(60\) 3.24411 + 2.00867i 0.418813 + 0.259318i
\(61\) −5.28511 + 10.6139i −0.676689 + 1.35897i 0.244420 + 0.969670i \(0.421403\pi\)
−0.921109 + 0.389305i \(0.872715\pi\)
\(62\) 8.69666 − 1.62569i 1.10448 − 0.206463i
\(63\) −1.78785 + 3.59049i −0.225248 + 0.452359i
\(64\) −0.273663 − 0.961826i −0.0342079 − 0.120228i
\(65\) 10.0289 1.24394
\(66\) 3.30980 + 1.28222i 0.407408 + 0.157831i
\(67\) 2.44901 − 2.23257i 0.299194 − 0.272752i −0.510151 − 0.860085i \(-0.670410\pi\)
0.809345 + 0.587333i \(0.199822\pi\)
\(68\) −1.74824 + 3.73412i −0.212006 + 0.452828i
\(69\) −9.79002 − 8.92478i −1.17858 − 1.07442i
\(70\) 1.16726 + 2.34418i 0.139515 + 0.280183i
\(71\) −2.04642 + 1.26709i −0.242866 + 0.150376i −0.642468 − 0.766312i \(-0.722089\pi\)
0.399602 + 0.916689i \(0.369148\pi\)
\(72\) 0.230069 − 2.48284i 0.0271139 − 0.292605i
\(73\) 0.316669 − 0.0591957i 0.0370633 − 0.00692834i −0.165185 − 0.986263i \(-0.552822\pi\)
0.202249 + 0.979334i \(0.435175\pi\)
\(74\) −1.16872 − 2.34711i −0.135861 − 0.272846i
\(75\) −5.50742 −0.635942
\(76\) −4.93201 − 4.49612i −0.565740 − 0.515740i
\(77\) 1.46806 + 1.94402i 0.167300 + 0.221542i
\(78\) −6.43601 − 12.9252i −0.728734 − 1.46350i
\(79\) −10.5282 + 9.59770i −1.18451 + 1.07983i −0.189317 + 0.981916i \(0.560627\pi\)
−0.995195 + 0.0979093i \(0.968784\pi\)
\(80\) −1.20307 − 1.09675i −0.134508 − 0.122620i
\(81\) −4.57462 + 9.18707i −0.508291 + 1.02079i
\(82\) −4.31874 + 8.67320i −0.476925 + 0.957795i
\(83\) −3.24224 + 11.3953i −0.355882 + 1.25080i 0.552803 + 0.833312i \(0.313558\pi\)
−0.908685 + 0.417483i \(0.862912\pi\)
\(84\) 2.27208 − 3.00872i 0.247904 − 0.328279i
\(85\) 0.780396 + 6.66670i 0.0846459 + 0.723106i
\(86\) −1.12618 − 1.49130i −0.121439 − 0.160811i
\(87\) 1.69233 − 0.655610i 0.181436 − 0.0702888i
\(88\) −1.28757 − 0.797231i −0.137256 − 0.0849851i
\(89\) −1.89939 − 2.51520i −0.201335 − 0.266611i 0.686137 − 0.727472i \(-0.259305\pi\)
−0.887472 + 0.460862i \(0.847540\pi\)
\(90\) −1.80937 − 3.63370i −0.190724 − 0.383026i
\(91\) 0.914349 − 9.86740i 0.0958498 − 1.03438i
\(92\) 3.40615 + 4.51048i 0.355116 + 0.470250i
\(93\) −19.3362 − 7.49087i −2.00507 − 0.776767i
\(94\) −2.27574 − 7.99840i −0.234725 − 0.824972i
\(95\) −10.6797 − 1.99638i −1.09571 − 0.204824i
\(96\) −0.641416 + 2.25434i −0.0654642 + 0.230083i
\(97\) −11.9798 + 7.41760i −1.21637 + 0.753143i −0.976049 − 0.217551i \(-0.930193\pi\)
−0.240319 + 0.970694i \(0.577252\pi\)
\(98\) −4.11446 + 1.59395i −0.415624 + 0.161013i
\(99\) −2.27563 − 3.01342i −0.228709 − 0.302860i
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 578.2.f.b.35.4 ✓ 224
289.256 even 17 inner 578.2.f.b.545.4 yes 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
578.2.f.b.35.4 ✓ 224 1.1 even 1 trivial
578.2.f.b.545.4 yes 224 289.256 even 17 inner