LMFDB - Embedded newform 1968.2.j.f.1393.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1968,2,Mod(1393,1968)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1968, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1968.1393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1968 = 2^{4} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1968.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.7145591178$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.265727878144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 984)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1393.2
Root			$0.233455i$ of defining polynomial
Character	$\chi$	$=$	1968.1393
Dual form			1968.2.j.f.1393.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +0.766545 q^{5} +4.17895i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +0.766545 q^{5} +4.17895i q^{7} -1.00000 q^{9} +1.38801i q^{11} +4.38801i q^{13} -0.766545i q^{15} -6.33351i q^{17} +4.85492i q^{19} +4.17895 q^{21} +1.44251 q^{23} -4.41241 q^{25} +1.00000i q^{27} -4.92110i q^{29} -5.26954 q^{31} +1.38801 q^{33} +3.20336i q^{35} -6.41241 q^{37} +4.38801 q^{39} +(-5.80042 + 2.71204i) q^{41} -11.8761 q^{43} -0.766545 q^{45} +12.4336i q^{47} -10.4637 q^{49} -6.33351 q^{51} +1.62146i q^{53} +1.06397i q^{55} +4.85492 q^{57} -5.63314 q^{59} +11.4580 q^{61} -4.17895i q^{63} +3.36360i q^{65} -11.2150i q^{67} -1.44251i q^{69} -0.442508i q^{71} +14.2429 q^{73} +4.41241i q^{75} -5.80042 q^{77} +12.8248i q^{79} +1.00000 q^{81} -7.54256 q^{83} -4.85492i q^{85} -4.92110 q^{87} +12.5670i q^{89} -18.3373 q^{91} +5.26954i q^{93} +3.72151i q^{95} +0.287956i q^{97} -1.38801i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q + 10 * q^5 - 8 * q^9	$ 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) $ 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1968\mathbb{Z}\right)^\times$.

$n$	$1231$	$1313$	$1441$	$1477$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	0.766545			0.342809			0.171405	−	0.985201i	$-0.445169\pi$
$5$	0.766545			0.342809			0.171405	+	0.985201i	$0.445169\pi$
$6$		0			0
$7$			4.17895i			1.57950i	0.613431	+	0.789748i	$0.289789\pi$
$7$			4.17895i			1.57950i	−0.613431	+	0.789748i	$0.710211\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			1.38801i			0.418500i	0.977862	+	0.209250i	$0.0671022\pi$
$11$			1.38801i			0.418500i	−0.977862	+	0.209250i	$0.932898\pi$
$12$		0			0
$13$			4.38801i			1.21701i	0.793548	+	0.608507i	$0.208231\pi$
$13$			4.38801i			1.21701i	−0.793548	+	0.608507i	$0.791769\pi$
$14$		0			0
$15$		−	0.766545i		−	0.197921i
$16$		0			0
$17$		−	6.33351i		−	1.53610i	−0.640389	−	0.768050i	$-0.721227\pi$
$17$		−	6.33351i		−	1.53610i	0.640389	−	0.768050i	$-0.278773\pi$
$18$		0			0
$19$			4.85492i			1.11379i	0.830581	+	0.556897i	$0.188008\pi$
$19$			4.85492i			1.11379i	−0.830581	+	0.556897i	$0.811992\pi$
$20$		0			0
$21$	4.17895			0.911922
$22$		0			0
$23$	1.44251			0.300784			0.150392	−	0.988626i	$-0.451946\pi$
$23$	1.44251			0.300784			0.150392	+	0.988626i	$0.451946\pi$
$24$		0			0
$25$	−4.41241			−0.882482
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	4.92110i		−	0.913825i	−0.889512	−	0.456912i	$-0.848955\pi$
$29$		−	4.92110i		−	0.913825i	0.889512	−	0.456912i	$-0.151045\pi$
$30$		0			0
$31$	−5.26954			−0.946437			−0.473218	−	0.880945i	$-0.656908\pi$
$31$	−5.26954			−0.946437			−0.473218	+	0.880945i	$0.656908\pi$
$32$		0			0
$33$	1.38801			0.241621
$34$		0			0
$35$			3.20336i			0.541466i
$36$		0			0
$37$	−6.41241			−1.05419			−0.527097	−	0.849805i	$-0.676719\pi$
$37$	−6.41241			−1.05419			−0.527097	+	0.849805i	$0.676719\pi$
$38$		0			0
$39$	4.38801			0.702643
$40$		0			0
$41$	−5.80042	+	2.71204i	−0.905873	+	0.423550i
$41$	−5.80042	+	2.71204i	−0.905873	+	0.423550i
$42$		0			0
$43$	−11.8761			−1.81108			−0.905541	−	0.424258i	$-0.860535\pi$
$43$	−11.8761			−1.81108			−0.905541	+	0.424258i	$0.860535\pi$
$44$		0			0
$45$	−0.766545			−0.114270
$46$		0			0
$47$			12.4336i			1.81362i	0.421539	+	0.906810i	$0.361490\pi$
$47$			12.4336i			1.81362i	−0.421539	+	0.906810i	$0.638510\pi$
$48$		0			0
$49$	−10.4637			−1.49481
$50$		0			0
$51$	−6.33351			−0.886868
$52$		0			0
$53$			1.62146i			0.222725i	0.993780	+	0.111362i	$0.0355214\pi$
$53$			1.62146i			0.222725i	−0.993780	+	0.111362i	$0.964479\pi$
$54$		0			0
$55$			1.06397i			0.143466i
$56$		0			0
$57$	4.85492			0.643049
$58$		0			0
$59$	−5.63314			−0.733372			−0.366686	−	0.930345i	$-0.619508\pi$
$59$	−5.63314			−0.733372			−0.366686	+	0.930345i	$0.619508\pi$
$60$		0			0
$61$	11.4580			1.46704			0.733521	−	0.679667i	$-0.237876\pi$
$61$	11.4580			1.46704			0.733521	+	0.679667i	$0.237876\pi$
$62$		0			0
$63$		−	4.17895i		−	0.526499i
$64$		0			0
$65$			3.36360i			0.417204i
$66$		0			0
$67$		−	11.2150i		−	1.37013i	−0.728480	−	0.685067i	$-0.759773\pi$
$67$		−	11.2150i		−	1.37013i	0.728480	−	0.685067i	$-0.240227\pi$
$68$		0			0
$69$		−	1.44251i		−	0.173658i
$70$		0			0
$71$		−	0.442508i		−	0.0525160i	−0.999655	−	0.0262580i	$-0.991641\pi$
$71$		−	0.442508i		−	0.0525160i	0.999655	−	0.0262580i	$-0.00835914\pi$
$72$		0			0
$73$	14.2429			1.66701			0.833504	−	0.552513i	$-0.186331\pi$
$73$	14.2429			1.66701			0.833504	+	0.552513i	$0.186331\pi$
$74$		0			0
$75$			4.41241i			0.509501i
$76$		0			0
$77$	−5.80042			−0.661019
$78$		0			0
$79$			12.8248i			1.44290i	0.692464	+	0.721452i	$0.256525\pi$
$79$			12.8248i			1.44290i	−0.692464	+	0.721452i	$0.743475\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−7.54256			−0.827903			−0.413952	−	0.910299i	$-0.635852\pi$
$83$	−7.54256			−0.827903			−0.413952	+	0.910299i	$0.635852\pi$
$84$		0			0
$85$		−	4.85492i		−	0.526590i
$86$		0			0
$87$	−4.92110			−0.527597
$88$		0			0
$89$			12.5670i			1.33210i	0.745909	+	0.666048i	$0.232015\pi$
$89$			12.5670i			1.33210i	−0.745909	+	0.666048i	$0.767985\pi$
$90$		0			0
$91$	−18.3373			−1.92227
$92$		0			0
$93$			5.26954i			0.546425i
$94$		0			0
$95$			3.72151i			0.381819i
$96$		0			0
$97$			0.287956i			0.0292375i	0.999893	+	0.0146188i	$0.00465346\pi$
$97$			0.287956i			0.0292375i	−0.999893	+	0.0146188i	$0.995347\pi$
$98$		0			0
$99$		−	1.38801i		−	0.139500i
$100$		0			0
$101$		−	6.59136i		−	0.655865i	−0.944701	−	0.327933i	$-0.893648\pi$
$101$		−	6.59136i		−	0.655865i	0.944701	−	0.327933i	$-0.106352\pi$
$102$		0			0
$103$	14.4729			1.42606			0.713028	−	0.701135i	$-0.247323\pi$
$103$	14.4729			1.42606			0.713028	+	0.701135i	$0.247323\pi$
$104$		0			0
$105$	3.20336			0.312616
$106$		0			0
$107$	18.2768			1.76688			0.883442	−	0.468540i	$-0.155220\pi$
$107$	18.2768			1.76688			0.883442	+	0.468540i	$0.155220\pi$
$108$		0			0
$109$		−	3.24513i		−	0.310827i	−0.987849	−	0.155414i	$-0.950329\pi$
$109$		−	3.24513i		−	0.310827i	0.987849	−	0.155414i	$-0.0496711\pi$
$110$		0			0
$111$			6.41241i			0.608639i
$112$		0			0
$113$	−7.64261			−0.718956			−0.359478	−	0.933154i	$-0.617045\pi$
$113$	−7.64261			−0.718956			−0.359478	+	0.933154i	$0.617045\pi$
$114$		0			0
$115$	1.10575			0.103111
$116$		0			0
$117$		−	4.38801i		−	0.405671i
$118$		0			0
$119$	26.4674			2.42626
$120$		0			0
$121$	9.07344			0.824858
$122$		0			0
$123$	2.71204	+	5.80042i	0.244537	+	0.523006i
$124$		0			0
$125$	−7.21503			−0.645332
$126$		0			0
$127$	−10.8365			−0.961584			−0.480792	−	0.876835i	$-0.659651\pi$
$127$	−10.8365			−0.961584			−0.480792	+	0.876835i	$0.659651\pi$
$128$		0			0
$129$			11.8761i			1.04563i
$130$		0			0
$131$	12.9343			1.13008			0.565039	−	0.825064i	$-0.308861\pi$
$131$	12.9343			1.13008			0.565039	+	0.825064i	$0.308861\pi$
$132$		0			0
$133$	−20.2885			−1.75923
$134$		0			0
$135$			0.766545i			0.0659737i
$136$		0			0
$137$			19.4557i			1.66222i	0.556111	+	0.831108i	$0.312293\pi$
$137$			19.4557i			1.66222i	−0.556111	+	0.831108i	$0.687707\pi$
$138$		0			0
$139$	2.37854			0.201745			0.100872	−	0.994899i	$-0.467837\pi$
$139$	2.37854			0.201745			0.100872	+	0.994899i	$0.467837\pi$
$140$		0			0
$141$	12.4336			1.04709
$142$		0			0
$143$	−6.09058			−0.509320
$144$		0			0
$145$		−	3.77224i		−	0.313268i
$146$		0			0
$147$			10.4637i			0.863028i
$148$		0			0
$149$		−	11.8910i		−	0.974148i	−0.873361	−	0.487074i	$-0.838064\pi$
$149$		−	11.8910i		−	0.974148i	0.873361	−	0.487074i	$-0.161936\pi$
$150$		0			0
$151$		−	2.17674i		−	0.177141i	−0.996070	−	0.0885704i	$-0.971770\pi$
$151$		−	2.17674i		−	0.177141i	0.996070	−	0.0885704i	$-0.0282298\pi$
$152$		0			0
$153$			6.33351i			0.512034i
$154$		0			0
$155$	−4.03934			−0.324447
$156$		0			0
$157$			12.3091i			0.982373i	0.871054	+	0.491187i	$0.163437\pi$
$157$			12.3091i			0.982373i	−0.871054	+	0.491187i	$0.836563\pi$
$158$		0			0
$159$	1.62146			0.128590
$160$		0			0
$161$			6.02817i			0.475087i
$162$		0			0
$163$	−11.7437			−0.919838			−0.459919	−	0.887961i	$-0.652122\pi$
$163$	−11.7437			−0.919838			−0.459919	+	0.887961i	$0.652122\pi$
$164$		0			0
$165$	1.06397			0.0828299
$166$		0			0
$167$		−	0.297425i		−	0.0230154i	−0.999934	−	0.0115077i	$-0.996337\pi$
$167$		−	0.297425i		−	0.0230154i	0.999934	−	0.0115077i	$-0.00366310\pi$
$168$		0			0
$169$	−6.25460			−0.481123
$170$		0			0
$171$		−	4.85492i		−	0.371265i
$172$		0			0
$173$	13.6426			1.03723			0.518614	−	0.855008i	$-0.326448\pi$
$173$	13.6426			1.03723			0.518614	+	0.855008i	$0.326448\pi$
$174$		0			0
$175$		−	18.4393i		−	1.39388i
$176$		0			0
$177$			5.63314i			0.423413i
$178$		0			0
$179$			12.2568i			0.916117i	0.888922	+	0.458059i	$0.151455\pi$
$179$			12.2568i			0.916117i	−0.888922	+	0.458059i	$0.848545\pi$
$180$		0			0
$181$			8.40915i			0.625047i	0.949910	+	0.312524i	$0.101174\pi$
$181$			8.40915i			0.625047i	−0.949910	+	0.312524i	$0.898826\pi$
$182$		0			0
$183$		−	11.4580i		−	0.846997i
$184$		0			0
$185$	−4.91540			−0.361387
$186$		0			0
$187$	8.79095			0.642858
$188$		0			0
$189$	−4.17895			−0.303974
$190$		0			0
$191$		−	1.69090i		−	0.122349i	−0.998127	−	0.0611745i	$-0.980515\pi$
$191$		−	1.69090i		−	0.122349i	0.998127	−	0.0611745i	$-0.0194846\pi$
$192$		0			0
$193$			21.4896i			1.54686i	0.633884	+	0.773428i	$0.281460\pi$
$193$			21.4896i			1.54686i	−0.633884	+	0.773428i	$0.718540\pi$
$194$		0			0
$195$	3.36360			0.240873
$196$		0			0
$197$	−15.5914			−1.11084			−0.555419	−	0.831570i	$-0.687442\pi$
$197$	−15.5914			−1.11084			−0.555419	+	0.831570i	$0.687442\pi$
$198$		0			0
$199$		−	5.29394i		−	0.375277i	−0.982238	−	0.187639i	$-0.939917\pi$
$199$		−	5.29394i		−	0.375277i	0.982238	−	0.187639i	$-0.0600834\pi$
$200$		0			0
$201$	−11.2150			−0.791048
$202$		0			0
$203$	20.5650			1.44338
$204$		0			0
$205$	−4.44628	+	2.07890i	−0.310542	+	0.145197i
$206$		0			0
$207$	−1.44251			−0.100261
$208$		0			0
$209$	−6.73866			−0.466123
$210$		0			0
$211$		−	24.6581i		−	1.69753i	−0.528770	−	0.848765i	$-0.677346\pi$
$211$		−	24.6581i		−	1.69753i	0.528770	−	0.848765i	$-0.322654\pi$
$212$		0			0
$213$	−0.442508			−0.0303201
$214$		0			0
$215$	−9.10354			−0.620856
$216$		0			0
$217$		−	22.0211i		−	1.49489i
$218$		0			0
$219$		−	14.2429i		−	0.962448i
$220$		0			0
$221$	27.7915			1.86946
$222$		0			0
$223$	1.21503			0.0813647			0.0406824	−	0.999172i	$-0.487047\pi$
$223$	1.21503			0.0813647			0.0406824	+	0.999172i	$0.487047\pi$
$224$		0			0
$225$	4.41241			0.294161
$226$		0			0
$227$			12.3940i			0.822618i	0.911496	+	0.411309i	$0.134928\pi$
$227$			12.3940i			0.822618i	−0.911496	+	0.411309i	$0.865072\pi$
$228$		0			0
$229$			6.80367i			0.449599i	0.974405	+	0.224800i	$0.0721727\pi$
$229$			6.80367i			0.449599i	−0.974405	+	0.224800i	$0.927827\pi$
$230$		0			0
$231$			5.80042i			0.381639i
$232$		0			0
$233$		−	21.1662i		−	1.38665i	−0.720627	−	0.693323i	$-0.756146\pi$
$233$		−	21.1662i		−	1.38665i	0.720627	−	0.693323i	$-0.243854\pi$
$234$		0			0
$235$			9.53088i			0.621726i
$236$		0			0
$237$	12.8248			0.833061
$238$		0			0
$239$			14.3858i			0.930540i	0.885169	+	0.465270i	$0.154043\pi$
$239$			14.3858i			0.930540i	−0.885169	+	0.465270i	$0.845957\pi$
$240$		0			0
$241$	−21.3373			−1.37445			−0.687227	−	0.726442i	$-0.741172\pi$
$241$	−21.3373			−1.37445			−0.687227	+	0.726442i	$0.741172\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−8.02086			−0.512434
$246$		0			0
$247$	−21.3034			−1.35550
$248$		0			0
$249$			7.54256i			0.477990i
$250$		0			0
$251$	5.84219			0.368756			0.184378	−	0.982855i	$-0.440973\pi$
$251$	5.84219			0.368756			0.184378	+	0.982855i	$0.440973\pi$
$252$		0			0
$253$			2.00221i			0.125878i
$254$		0			0
$255$	−4.85492			−0.304027
$256$		0			0
$257$			4.96066i			0.309438i	0.987959	+	0.154719i	$0.0494472\pi$
$257$			4.96066i			0.309438i	−0.987959	+	0.154719i	$0.950553\pi$
$258$		0			0
$259$		−	26.7972i		−	1.66509i
$260$		0			0
$261$			4.92110i			0.304608i
$262$		0			0
$263$		−	3.35570i		−	0.206921i	−0.994634	−	0.103461i	$-0.967008\pi$
$263$		−	3.35570i		−	0.206921i	0.994634	−	0.103461i	$-0.0329916\pi$
$264$		0			0
$265$			1.24292i			0.0763522i
$266$		0			0
$267$	12.5670			0.769085
$268$		0			0
$269$	17.2638			1.05259			0.526297	−	0.850301i	$-0.323580\pi$
$269$	17.2638			1.05259			0.526297	+	0.850301i	$0.323580\pi$
$270$		0			0
$271$	7.12068			0.432551			0.216275	−	0.976332i	$-0.430609\pi$
$271$	7.12068			0.432551			0.216275	+	0.976332i	$0.430609\pi$
$272$		0			0
$273$			18.3373i			1.10982i
$274$		0			0
$275$		−	6.12445i		−	0.369318i
$276$		0			0
$277$	8.70258			0.522887			0.261444	−	0.965219i	$-0.415801\pi$
$277$	8.70258			0.522887			0.261444	+	0.965219i	$0.415801\pi$
$278$		0			0
$279$	5.26954			0.315479
$280$		0			0
$281$			30.6030i			1.82562i	0.408380	+	0.912812i	$0.366094\pi$
$281$			30.6030i			1.82562i	−0.408380	+	0.912812i	$0.633906\pi$
$282$		0			0
$283$	−13.6553			−0.811725			−0.405863	−	0.913934i	$-0.633029\pi$
$283$	−13.6553			−0.811725			−0.405863	+	0.913934i	$0.633029\pi$
$284$		0			0
$285$	3.72151			0.220443
$286$		0			0
$287$	−11.3335	−	24.2397i	−0.668996	−	1.43082i
$288$		0			0
$289$	−23.1133			−1.35961
$290$		0			0
$291$	0.287956			0.0168803
$292$		0			0
$293$		−	8.27575i		−	0.483474i	−0.970342	−	0.241737i	$-0.922283\pi$
$293$		−	8.27575i		−	0.483474i	0.970342	−	0.241737i	$-0.0777172\pi$
$294$		0			0
$295$	−4.31806			−0.251407
$296$		0			0
$297$	−1.38801			−0.0805403
$298$		0			0
$299$			6.32973i			0.366058i
$300$		0			0
$301$		−	49.6295i		−	2.86060i
$302$		0			0
$303$	−6.59136			−0.378664
$304$		0			0
$305$	8.78304			0.502915
$306$		0			0
$307$	−8.58915			−0.490209			−0.245104	−	0.969497i	$-0.578822\pi$
$307$	−8.58915			−0.490209			−0.245104	+	0.969497i	$0.578822\pi$
$308$		0			0
$309$		−	14.4729i		−	0.823334i
$310$		0			0
$311$		−	13.0528i		−	0.740157i	−0.929001	−	0.370078i	$-0.879331\pi$
$311$		−	13.0528i		−	0.740157i	0.929001	−	0.370078i	$-0.120669\pi$
$312$		0			0
$313$			9.83998i			0.556189i	0.960554	+	0.278094i	$0.0897028\pi$
$313$			9.83998i			0.556189i	−0.960554	+	0.278094i	$0.910297\pi$
$314$		0			0
$315$		−	3.20336i		−	0.180489i
$316$		0			0
$317$			0.168441i			0.00946057i	0.999989	+	0.00473028i	$0.00150570\pi$
$317$			0.168441i			0.00946057i	−0.999989	+	0.00473028i	$0.998494\pi$
$318$		0			0
$319$	6.83051			0.382435
$320$		0			0
$321$		−	18.2768i		−	1.02011i
$322$		0			0
$323$	30.7486			1.71090
$324$		0			0
$325$		−	19.3617i		−	1.07399i
$326$		0			0
$327$	−3.24513			−0.179456
$328$		0			0
$329$	−51.9593			−2.86461
$330$		0			0
$331$			11.9490i			0.656776i	0.944543	+	0.328388i	$0.106505\pi$
$331$			11.9490i			0.656776i	−0.944543	+	0.328388i	$0.893495\pi$
$332$		0			0
$333$	6.41241			0.351398
$334$		0			0
$335$		−	8.59683i		−	0.469695i
$336$		0			0
$337$	10.5125			0.572650			0.286325	−	0.958133i	$-0.407566\pi$
$337$	10.5125			0.572650			0.286325	+	0.958133i	$0.407566\pi$
$338$		0			0
$339$			7.64261i			0.415090i
$340$		0			0
$341$		−	7.31415i		−	0.396083i
$342$		0			0
$343$		−	14.4745i		−	0.781547i
$344$		0			0
$345$		−	1.10575i		−	0.0595314i
$346$		0			0
$347$		−	7.49974i		−	0.402607i	−0.979529	−	0.201303i	$-0.935482\pi$
$347$		−	7.49974i		−	0.402607i	0.979529	−	0.201303i	$-0.0645177\pi$
$348$		0			0
$349$	−2.14287			−0.114705			−0.0573527	−	0.998354i	$-0.518266\pi$
$349$	−2.14287			−0.114705			−0.0573527	+	0.998354i	$0.518266\pi$
$350$		0			0
$351$	−4.38801			−0.234214
$352$		0			0
$353$	−11.2473			−0.598636			−0.299318	−	0.954153i	$-0.596759\pi$
$353$	−11.2473			−0.598636			−0.299318	+	0.954153i	$0.596759\pi$
$354$		0			0
$355$		−	0.339202i		−	0.0180030i
$356$		0			0
$357$		−	26.4674i		−	1.40080i
$358$		0			0
$359$	−25.7592			−1.35952			−0.679758	−	0.733436i	$-0.737915\pi$
$359$	−25.7592			−1.35952			−0.679758	+	0.733436i	$0.737915\pi$
$360$		0			0
$361$	−4.57022			−0.240538
$362$		0			0
$363$		−	9.07344i		−	0.476232i
$364$		0			0
$365$	10.9178			0.571466
$366$		0			0
$367$	25.4769			1.32988			0.664942	−	0.746895i	$-0.268456\pi$
$367$	25.4769			1.32988			0.664942	+	0.746895i	$0.268456\pi$
$368$		0			0
$369$	5.80042	−	2.71204i	0.301958	−	0.141183i
$370$		0			0
$371$	−6.77601			−0.351793
$372$		0			0
$373$	−10.0339			−0.519534			−0.259767	−	0.965671i	$-0.583646\pi$
$373$	−10.0339			−0.519534			−0.259767	+	0.965671i	$0.583646\pi$
$374$		0			0
$375$			7.21503i			0.372583i
$376$		0			0
$377$	21.5938			1.11214
$378$		0			0
$379$	−22.3796			−1.14956			−0.574781	−	0.818308i	$-0.694913\pi$
$379$	−22.3796			−1.14956			−0.574781	+	0.818308i	$0.694913\pi$
$380$		0			0
$381$			10.8365i			0.555171i
$382$		0			0
$383$		−	27.4186i		−	1.40103i	−0.713640	−	0.700513i	$-0.752955\pi$
$383$		−	27.4186i		−	1.40103i	0.713640	−	0.700513i	$-0.247045\pi$
$384$		0			0
$385$	−4.44628			−0.226603
$386$		0			0
$387$	11.8761			0.603694
$388$		0			0
$389$	15.2757			0.774511			0.387256	−	0.921972i	$-0.373423\pi$
$389$	15.2757			0.774511			0.387256	+	0.921972i	$0.373423\pi$
$390$		0			0
$391$		−	9.13613i		−	0.462034i
$392$		0			0
$393$		−	12.9343i		−	0.652451i
$394$		0			0
$395$			9.83080i			0.494641i
$396$		0			0
$397$		−	6.77578i		−	0.340067i	−0.985438	−	0.170033i	$-0.945612\pi$
$397$		−	6.77578i		−	0.340067i	0.985438	−	0.170033i	$-0.0543876\pi$
$398$		0			0
$399$			20.2885i			1.01569i
$400$		0			0
$401$	35.2250			1.75905			0.879527	−	0.475849i	$-0.157859\pi$
$401$	35.2250			1.75905			0.879527	+	0.475849i	$0.157859\pi$
$402$		0			0
$403$		−	23.1228i		−	1.15183i
$404$		0			0
$405$	0.766545			0.0380899
$406$		0			0
$407$		−	8.90047i		−	0.441180i
$408$		0			0
$409$	13.0722			0.646377			0.323188	−	0.946335i	$-0.395245\pi$
$409$	13.0722			0.646377			0.323188	+	0.946335i	$0.395245\pi$
$410$		0			0
$411$	19.4557			0.959681
$412$		0			0
$413$		−	23.5406i		−	1.15836i
$414$		0			0
$415$	−5.78171			−0.283813
$416$		0			0
$417$		−	2.37854i		−	0.116478i
$418$		0			0
$419$	−15.8582			−0.774722			−0.387361	−	0.921928i	$-0.626613\pi$
$419$	−15.8582			−0.774722			−0.387361	+	0.921928i	$0.626613\pi$
$420$		0			0
$421$		−	18.6770i		−	0.910261i	−0.890425	−	0.455131i	$-0.849593\pi$
$421$		−	18.6770i		−	0.910261i	0.890425	−	0.455131i	$-0.150407\pi$
$422$		0			0
$423$		−	12.4336i		−	0.604540i
$424$		0			0
$425$			27.9460i			1.35558i
$426$		0			0
$427$			47.8823i			2.31719i
$428$		0			0
$429$			6.09058i			0.294056i
$430$		0			0
$431$	40.6252			1.95685			0.978424	−	0.206606i	$-0.0662417\pi$
$431$	40.6252			1.95685			0.978424	+	0.206606i	$0.0662417\pi$
$432$		0			0
$433$	4.36116			0.209584			0.104792	−	0.994494i	$-0.466582\pi$
$433$	4.36116			0.209584			0.104792	+	0.994494i	$0.466582\pi$
$434$		0			0
$435$	−3.77224			−0.180865
$436$		0			0
$437$			7.00326i			0.335011i
$438$		0			0
$439$			6.33246i			0.302232i	0.988516	+	0.151116i	$0.0482867\pi$
$439$			6.33246i			0.302232i	−0.988516	+	0.151116i	$0.951713\pi$
$440$		0			0
$441$	10.4637			0.498269
$442$		0			0
$443$	−8.92929			−0.424243			−0.212122	−	0.977243i	$-0.568037\pi$
$443$	−8.92929			−0.424243			−0.212122	+	0.977243i	$0.568037\pi$
$444$		0			0
$445$			9.63314i			0.456655i
$446$		0			0
$447$	−11.8910			−0.562425
$448$		0			0
$449$	39.8731			1.88173			0.940864	−	0.338785i	$-0.110016\pi$
$449$	39.8731			1.88173			0.940864	+	0.338785i	$0.110016\pi$
$450$		0			0
$451$	−3.76433	−	8.05101i	−0.177256	−	0.379107i
$452$		0			0
$453$	−2.17674			−0.102272
$454$		0			0
$455$	−14.0563			−0.658972
$456$		0			0
$457$			12.2066i			0.571001i	0.958379	+	0.285501i	$0.0921599\pi$
$457$			12.2066i			0.571001i	−0.958379	+	0.285501i	$0.907840\pi$
$458$		0			0
$459$	6.33351			0.295623
$460$		0			0
$461$	0.696879			0.0324569			0.0162284	−	0.999868i	$-0.494834\pi$
$461$	0.696879			0.0324569			0.0162284	+	0.999868i	$0.494834\pi$
$462$		0			0
$463$			29.4940i			1.37070i	0.728212	+	0.685352i	$0.240352\pi$
$463$			29.4940i			1.37070i	−0.728212	+	0.685352i	$0.759648\pi$
$464$		0			0
$465$			4.03934i			0.187320i
$466$		0			0
$467$	21.9866			1.01742			0.508710	−	0.860938i	$-0.330123\pi$
$467$	21.9866			1.01742			0.508710	+	0.860938i	$0.330123\pi$
$468$		0			0
$469$	46.8671			2.16412
$470$		0			0
$471$	12.3091			0.567174
$472$		0			0
$473$		−	16.4841i		−	0.757938i
$474$		0			0
$475$		−	21.4219i		−	0.982903i
$476$		0			0
$477$		−	1.62146i		−	0.0742416i
$478$		0			0
$479$			33.0889i			1.51187i	0.654647	+	0.755935i	$0.272817\pi$
$479$			33.0889i			1.51187i	−0.654647	+	0.755935i	$0.727183\pi$
$480$		0			0
$481$		−	28.1377i		−	1.28297i
$482$		0			0
$483$	6.02817			0.274291
$484$		0			0
$485$			0.220731i			0.0100229i
$486$		0			0
$487$	27.1561			1.23056			0.615280	−	0.788308i	$-0.289043\pi$
$487$	27.1561			1.23056			0.615280	+	0.788308i	$0.289043\pi$
$488$		0			0
$489$			11.7437i			0.531069i
$490$		0			0
$491$	−7.09458			−0.320174			−0.160087	−	0.987103i	$-0.551177\pi$
$491$	−7.09458			−0.320174			−0.160087	+	0.987103i	$0.551177\pi$
$492$		0			0
$493$	−31.1678			−1.40373
$494$		0			0
$495$		−	1.06397i		−	0.0478219i
$496$		0			0
$497$	1.84922			0.0829488
$498$		0			0
$499$		−	5.40317i		−	0.241879i	−0.992660	−	0.120940i	$-0.961409\pi$
$499$		−	5.40317i		−	0.241879i	0.992660	−	0.120940i	$-0.0385907\pi$
$500$		0			0
$501$	−0.297425			−0.0132879
$502$		0			0
$503$		−	23.1421i		−	1.03186i	−0.856632	−	0.515928i	$-0.827447\pi$
$503$		−	23.1421i		−	1.03186i	0.856632	−	0.515928i	$-0.172553\pi$
$504$		0			0
$505$		−	5.05258i		−	0.224837i
$506$		0			0
$507$			6.25460i			0.277777i
$508$		0			0
$509$		−	34.8565i		−	1.54499i	−0.635023	−	0.772493i	$-0.719009\pi$
$509$		−	34.8565i		−	1.54499i	0.635023	−	0.772493i	$-0.280991\pi$
$510$		0			0
$511$			59.5205i			2.63303i
$512$		0			0
$513$	−4.85492			−0.214350
$514$		0			0
$515$	11.0941			0.488865
$516$		0			0
$517$	−17.2579			−0.759000
$518$		0			0
$519$		−	13.6426i		−	0.598844i
$520$		0			0
$521$		−	39.9137i		−	1.74865i	−0.485340	−	0.874326i	$-0.661304\pi$
$521$		−	39.9137i		−	1.74865i	0.485340	−	0.874326i	$-0.338696\pi$
$522$		0			0
$523$	−39.6290			−1.73286			−0.866428	−	0.499303i	$-0.833590\pi$
$523$	−39.6290			−1.73286			−0.866428	+	0.499303i	$0.833590\pi$
$524$		0			0
$525$	−18.4393			−0.804755
$526$		0			0
$527$			33.3746i			1.45382i
$528$		0			0
$529$	−20.9192			−0.909529
$530$		0			0
$531$	5.63314			0.244457
$532$		0			0
$533$	−11.9005	−	25.4523i	−0.515466	−	1.10246i
$534$		0			0
$535$	14.0100			0.605704
$536$		0			0
$537$	12.2568			0.528920
$538$		0			0
$539$		−	14.5236i		−	0.625577i
$540$		0			0
$541$	29.5829			1.27187			0.635935	−	0.771743i	$-0.280615\pi$
$541$	29.5829			1.27187			0.635935	+	0.771743i	$0.280615\pi$
$542$		0			0
$543$	8.40915			0.360871
$544$		0			0
$545$		−	2.48754i		−	0.106555i
$546$		0			0
$547$			22.2101i			0.949635i	0.880084	+	0.474817i	$0.157486\pi$
$547$			22.2101i			0.949635i	−0.880084	+	0.474817i	$0.842514\pi$
$548$		0			0
$549$	−11.4580			−0.489014
$550$		0			0
$551$	23.8915			1.01781
$552$		0			0
$553$	−53.5943			−2.27906
$554$		0			0
$555$			4.91540i			0.208647i
$556$		0			0
$557$			25.5090i			1.08085i	0.841392	+	0.540425i	$0.181737\pi$
$557$			25.5090i			1.08085i	−0.841392	+	0.540425i	$0.818263\pi$
$558$		0			0
$559$		−	52.1122i		−	2.20411i
$560$		0			0
$561$		−	8.79095i		−	0.371154i
$562$		0			0
$563$		−	36.5501i		−	1.54040i	−0.637801	−	0.770202i	$-0.720156\pi$
$563$		−	36.5501i		−	1.54040i	0.637801	−	0.770202i	$-0.279844\pi$
$564$		0			0
$565$	−5.85840			−0.246465
$566$		0			0
$567$			4.17895i			0.175500i
$568$		0			0
$569$	44.0683			1.84744			0.923719	−	0.383071i	$-0.125134\pi$
$569$	44.0683			1.84744			0.923719	+	0.383071i	$0.125134\pi$
$570$		0			0
$571$			13.0428i			0.545825i	0.962039	+	0.272913i	$0.0879870\pi$
$571$			13.0428i			0.545825i	−0.962039	+	0.272913i	$0.912013\pi$
$572$		0			0
$573$	−1.69090			−0.0706382
$574$		0			0
$575$	−6.36493			−0.265436
$576$		0			0
$577$			34.3699i			1.43084i	0.698696	+	0.715418i	$0.253764\pi$
$577$			34.3699i			1.43084i	−0.698696	+	0.715418i	$0.746236\pi$
$578$		0			0
$579$	21.4896			0.893078
$580$		0			0
$581$		−	31.5200i		−	1.30767i
$582$		0			0
$583$	−2.25060			−0.0932103
$584$		0			0
$585$		−	3.36360i		−	0.139068i
$586$		0			0
$587$		−	0.0935525i		−	0.00386132i	−0.999998	−	0.00193066i	$-0.999385\pi$
$587$		−	0.0935525i		−	0.00386132i	0.999998	−	0.00193066i	$-0.000614549\pi$
$588$		0			0
$589$		−	25.5832i		−	1.05414i
$590$		0			0
$591$			15.5914i			0.641343i
$592$		0			0
$593$			3.96420i			0.162790i	0.996682	+	0.0813952i	$0.0259376\pi$
$593$			3.96420i			0.162790i	−0.996682	+	0.0813952i	$0.974062\pi$
$594$		0			0
$595$	20.2885			0.831746
$596$		0			0
$597$	−5.29394			−0.216666
$598$		0			0
$599$	−41.4202			−1.69238			−0.846191	−	0.532879i	$-0.821110\pi$
$599$	−41.4202			−1.69238			−0.846191	+	0.532879i	$0.821110\pi$
$600$		0			0
$601$		−	6.61123i		−	0.269678i	−0.990868	−	0.134839i	$-0.956948\pi$
$601$		−	6.61123i		−	0.269678i	0.990868	−	0.134839i	$-0.0430517\pi$
$602$		0			0
$603$			11.2150i			0.456712i
$604$		0			0
$605$	6.95520			0.282769
$606$		0			0
$607$	−0.348670			−0.0141521			−0.00707605	−	0.999975i	$-0.502252\pi$
$607$	−0.348670			−0.0141521			−0.00707605	+	0.999975i	$0.502252\pi$
$608$		0			0
$609$		−	20.5650i		−	0.833337i
$610$		0			0
$611$	−54.5585			−2.20720
$612$		0			0
$613$	−6.94794			−0.280625			−0.140312	−	0.990107i	$-0.544811\pi$
$613$	−6.94794			−0.280625			−0.140312	+	0.990107i	$0.544811\pi$
$614$		0			0
$615$	2.07890	+	4.44628i	0.0838295	+	0.179291i
$616$		0			0
$617$	−2.06071			−0.0829612			−0.0414806	−	0.999139i	$-0.513207\pi$
$617$	−2.06071			−0.0829612			−0.0414806	+	0.999139i	$0.513207\pi$
$618$		0			0
$619$	16.1222			0.648008			0.324004	−	0.946056i	$-0.394971\pi$
$619$	16.1222			0.648008			0.324004	+	0.946056i	$0.394971\pi$
$620$		0			0
$621$			1.44251i			0.0578858i
$622$		0			0
$623$	−52.5167			−2.10404
$624$		0			0
$625$	16.5314			0.661256
$626$		0			0
$627$			6.73866i			0.269116i
$628$		0			0
$629$			40.6130i			1.61935i
$630$		0			0
$631$	−15.5140			−0.617603			−0.308802	−	0.951126i	$-0.599928\pi$
$631$	−15.5140			−0.617603			−0.308802	+	0.951126i	$0.599928\pi$
$632$		0			0
$633$	−24.6581			−0.980070
$634$		0			0
$635$	−8.30666			−0.329640
$636$		0			0
$637$		−	45.9146i		−	1.81920i
$638$		0			0
$639$			0.442508i			0.0175053i
$640$		0			0
$641$		−	17.3348i		−	0.684683i	−0.939576	−	0.342341i	$-0.888780\pi$
$641$		−	17.3348i		−	0.684683i	0.939576	−	0.342341i	$-0.111220\pi$
$642$		0			0
$643$			2.56696i			0.101231i	0.998718	+	0.0506155i	$0.0161183\pi$
$643$			2.56696i			0.101231i	−0.998718	+	0.0506155i	$0.983882\pi$
$644$		0			0
$645$			9.10354i			0.358451i
$646$		0			0
$647$	−1.99297			−0.0783519			−0.0391759	−	0.999232i	$-0.512473\pi$
$647$	−1.99297			−0.0783519			−0.0391759	+	0.999232i	$0.512473\pi$
$648$		0			0
$649$		−	7.81884i		−	0.306916i
$650$		0			0
$651$	−22.0211			−0.863077
$652$		0			0
$653$			6.03608i			0.236210i	0.993001	+	0.118105i	$0.0376820\pi$
$653$			6.03608i			0.236210i	−0.993001	+	0.118105i	$0.962318\pi$
$654$		0			0
$655$	9.91475			0.387401
$656$		0			0
$657$	−14.2429			−0.555669
$658$		0			0
$659$		−	6.24891i		−	0.243423i	−0.992566	−	0.121711i	$-0.961162\pi$
$659$		−	6.24891i		−	0.243423i	0.992566	−	0.121711i	$-0.0388382\pi$
$660$		0			0
$661$	36.6209			1.42439			0.712195	−	0.701982i	$-0.247701\pi$
$661$	36.6209			1.42439			0.712195	+	0.701982i	$0.247701\pi$
$662$		0			0
$663$		−	27.7915i		−	1.07933i
$664$		0			0
$665$	−15.5520			−0.603082
$666$		0			0
$667$		−	7.09872i		−	0.274864i
$668$		0			0
$669$		−	1.21503i		−	0.0469760i
$670$		0			0
$671$			15.9037i			0.613956i
$672$		0			0
$673$			27.2852i			1.05177i	0.850556	+	0.525884i	$0.176265\pi$
$673$			27.2852i			1.05177i	−0.850556	+	0.525884i	$0.823735\pi$
$674$		0			0
$675$		−	4.41241i		−	0.169834i
$676$		0			0
$677$	25.3734			0.975177			0.487589	−	0.873073i	$-0.337877\pi$
$677$	25.3734			0.975177			0.487589	+	0.873073i	$0.337877\pi$
$678$		0			0
$679$	−1.20336			−0.0461806
$680$		0			0
$681$	12.3940			0.474939
$682$		0			0
$683$		−	19.1432i		−	0.732493i	−0.930518	−	0.366246i	$-0.880643\pi$
$683$		−	19.1432i		−	0.732493i	0.930518	−	0.366246i	$-0.119357\pi$
$684$		0			0
$685$			14.9137i			0.569823i
$686$		0			0
$687$	6.80367			0.259576
$688$		0			0
$689$	−7.11498			−0.271059
$690$		0			0
$691$		−	13.8888i		−	0.528354i	−0.964474	−	0.264177i	$-0.914900\pi$
$691$		−	13.8888i		−	0.528354i	0.964474	−	0.264177i	$-0.0851004\pi$
$692$		0			0
$693$	5.80042			0.220340
$694$		0			0
$695$	1.82326			0.0691601
$696$		0			0
$697$	17.1767	+	36.7370i	0.650616	+	1.39151i
$698$		0			0
$699$	−21.1662			−0.800580
$700$		0			0
$701$	−3.04282			−0.114926			−0.0574629	−	0.998348i	$-0.518301\pi$
$701$	−3.04282			−0.114926			−0.0574629	+	0.998348i	$0.518301\pi$
$702$		0			0
$703$		−	31.1317i		−	1.17415i
$704$		0			0
$705$	9.53088			0.358954
$706$		0			0
$707$	27.5450			1.03594
$708$		0			0
$709$		−	40.9737i		−	1.53880i	−0.638768	−	0.769399i	$-0.720556\pi$
$709$		−	40.9737i		−	1.53880i	0.638768	−	0.769399i	$-0.279444\pi$
$710$		0			0
$711$		−	12.8248i		−	0.480968i
$712$		0			0
$713$	−7.60135			−0.284673
$714$		0			0
$715$	−4.66871			−0.174600
$716$		0			0
$717$	14.3858			0.537247
$718$		0			0
$719$			31.9017i			1.18973i	0.803824	+	0.594867i	$0.202795\pi$
$719$			31.9017i			1.18973i	−0.803824	+	0.594867i	$0.797205\pi$
$720$		0			0
$721$			60.4815i			2.25245i
$722$		0			0
$723$			21.3373i			0.793542i
$724$		0			0
$725$			21.7139i			0.806434i
$726$		0			0
$727$			0.341182i			0.0126537i	0.999980	+	0.00632686i	$0.00201392\pi$
$727$			0.341182i			0.0126537i	−0.999980	+	0.00632686i	$0.997986\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			75.2171i			2.78201i
$732$		0			0
$733$	24.3619			0.899828			0.449914	−	0.893072i	$-0.351455\pi$
$733$	24.3619			0.899828			0.449914	+	0.893072i	$0.351455\pi$
$734$		0			0
$735$			8.02086i			0.295854i
$736$		0			0
$737$	15.5665			0.573401
$738$		0			0
$739$	5.79693			0.213243			0.106622	−	0.994300i	$-0.465997\pi$
$739$	5.79693			0.213243			0.106622	+	0.994300i	$0.465997\pi$
$740$		0			0
$741$			21.3034i			0.782600i
$742$		0			0
$743$	6.41113			0.235202			0.117601	−	0.993061i	$-0.462480\pi$
$743$	6.41113			0.235202			0.117601	+	0.993061i	$0.462480\pi$
$744$		0			0
$745$		−	9.11498i		−	0.333947i
$746$		0			0
$747$	7.54256			0.275968
$748$		0			0
$749$			76.3779i			2.79079i
$750$		0			0
$751$		−	5.58841i		−	0.203924i	−0.994788	−	0.101962i	$-0.967488\pi$
$751$		−	5.58841i		−	0.203924i	0.994788	−	0.101962i	$-0.0325120\pi$
$752$		0			0
$753$		−	5.84219i		−	0.212901i
$754$		0			0
$755$		−	1.66857i		−	0.0607255i
$756$		0			0
$757$		−	26.0466i		−	0.946680i	−0.880880	−	0.473340i	$-0.843048\pi$
$757$		−	26.0466i		−	0.946680i	0.880880	−	0.473340i	$-0.156952\pi$
$758$		0			0
$759$	2.00221			0.0726756
$760$		0			0
$761$	−2.73372			−0.0990973			−0.0495486	−	0.998772i	$-0.515778\pi$
$761$	−2.73372			−0.0990973			−0.0495486	+	0.998772i	$0.515778\pi$
$762$		0			0
$763$	13.5613			0.490951
$764$		0			0
$765$			4.85492i			0.175530i
$766$		0			0
$767$		−	24.7183i		−	0.892525i
$768$		0			0
$769$	−4.54505			−0.163899			−0.0819494	−	0.996636i	$-0.526115\pi$
$769$	−4.54505			−0.163899			−0.0819494	+	0.996636i	$0.526115\pi$
$770$		0			0
$771$	4.96066			0.178654
$772$		0			0
$773$			7.54139i			0.271245i	0.990761	+	0.135623i	$0.0433034\pi$
$773$			7.54139i			0.271245i	−0.990761	+	0.135623i	$0.956697\pi$
$774$		0			0
$775$	23.2513			0.835213
$776$		0			0
$777$	−26.7972			−0.961343
$778$		0			0
$779$	−13.1667	−	28.1605i	−0.471748	−	1.00896i
$780$		0			0
$781$	0.614204			0.0219779
$782$		0			0
$783$	4.92110			0.175866
$784$		0			0
$785$			9.43548i			0.336767i
$786$		0			0
$787$	−20.1364			−0.717784			−0.358892	−	0.933379i	$-0.616845\pi$
$787$	−20.1364			−0.717784			−0.358892	+	0.933379i	$0.616845\pi$
$788$		0			0
$789$	−3.35570			−0.119466
$790$		0			0
$791$		−	31.9381i		−	1.13559i
$792$		0			0
$793$			50.2776i			1.78541i
$794$		0			0
$795$	1.24292			0.0440819
$796$		0			0
$797$	21.2106			0.751318			0.375659	−	0.926758i	$-0.377416\pi$
$797$	21.2106			0.751318			0.375659	+	0.926758i	$0.377416\pi$
$798$		0			0
$799$	78.7480			2.78590
$800$		0			0
$801$		−	12.5670i		−	0.444032i
$802$		0			0
$803$			19.7693i			0.697642i
$804$		0			0
$805$			4.62087i			0.162864i
$806$		0			0
$807$		−	17.2638i		−	0.607716i
$808$		0			0
$809$		−	28.3858i		−	0.997991i	−0.866604	−	0.498996i	$-0.833702\pi$
$809$		−	28.3858i		−	0.997991i	0.866604	−	0.498996i	$-0.166298\pi$
$810$		0			0
$811$	17.6598			0.620118			0.310059	−	0.950717i	$-0.399651\pi$
$811$	17.6598			0.620118			0.310059	+	0.950717i	$0.399651\pi$
$812$		0			0
$813$		−	7.12068i		−	0.249733i
$814$		0			0
$815$	−9.00208			−0.315329
$816$		0			0
$817$		−	57.6573i		−	2.01717i
$818$		0			0
$819$	18.3373			0.640756
$820$		0			0
$821$	40.7865			1.42346			0.711730	−	0.702454i	$-0.247912\pi$
$821$	40.7865			1.42346			0.711730	+	0.702454i	$0.247912\pi$
$822$		0			0
$823$			36.7880i			1.28235i	0.767395	+	0.641174i	$0.221552\pi$
$823$			36.7880i			1.28235i	−0.767395	+	0.641174i	$0.778448\pi$
$824$		0			0
$825$	−6.12445			−0.213226
$826$		0			0
$827$		−	16.4401i		−	0.571677i	−0.958278	−	0.285839i	$-0.907728\pi$
$827$		−	16.4401i		−	0.571677i	0.958278	−	0.285839i	$-0.0922721\pi$
$828$		0			0
$829$	−44.4460			−1.54367			−0.771837	−	0.635820i	$-0.780662\pi$
$829$	−44.4460			−1.54367			−0.771837	+	0.635820i	$0.780662\pi$
$830$		0			0
$831$		−	8.70258i		−	0.301889i
$832$		0			0
$833$			66.2716i			2.29618i
$834$		0			0
$835$		−	0.227989i		−	0.00788989i
$836$		0			0
$837$		−	5.26954i		−	0.182142i
$838$		0			0
$839$		−	3.54307i		−	0.122321i	−0.998128	−	0.0611603i	$-0.980520\pi$
$839$		−	3.54307i		−	0.122321i	0.998128	−	0.0611603i	$-0.0194801\pi$
$840$		0			0
$841$	4.78281			0.164924
$842$		0			0
$843$	30.6030			1.05402
$844$		0			0
$845$	−4.79443			−0.164934
$846$		0			0
$847$			37.9175i			1.30286i
$848$		0			0
$849$			13.6553i			0.468650i
$850$		0			0
$851$	−9.24995			−0.317084
$852$		0			0
$853$	−17.2387			−0.590241			−0.295121	−	0.955460i	$-0.595360\pi$
$853$	−17.2387			−0.590241			−0.295121	+	0.955460i	$0.595360\pi$
$854$		0			0
$855$		−	3.72151i		−	0.127273i
$856$		0			0
$857$	−16.6605			−0.569112			−0.284556	−	0.958659i	$-0.591846\pi$
$857$	−16.6605			−0.569112			−0.284556	+	0.958659i	$0.591846\pi$
$858$		0			0
$859$	−48.5729			−1.65729			−0.828644	−	0.559776i	$-0.810887\pi$
$859$	−48.5729			−1.65729			−0.828644	+	0.559776i	$0.810887\pi$
$860$		0			0
$861$	−24.2397	+	11.3335i	−0.826086	+	0.386245i
$862$		0			0
$863$	−0.873106			−0.0297209			−0.0148604	−	0.999890i	$-0.504730\pi$
$863$	−0.873106			−0.0297209			−0.0148604	+	0.999890i	$0.504730\pi$
$864$		0			0
$865$	10.4577			0.355572
$866$		0			0
$867$			23.1133i			0.784968i
$868$		0			0
$869$	−17.8009			−0.603855
$870$		0			0
$871$	49.2116			1.66747
$872$		0			0
$873$		−	0.287956i		−	0.00974584i
$874$		0			0
$875$		−	30.1513i		−	1.01930i
$876$		0			0
$877$	−7.46935			−0.252222			−0.126111	−	0.992016i	$-0.540250\pi$
$877$	−7.46935			−0.252222			−0.126111	+	0.992016i	$0.540250\pi$
$878$		0			0
$879$	−8.27575			−0.279134
$880$		0			0
$881$	−44.6257			−1.50348			−0.751740	−	0.659460i	$-0.770785\pi$
$881$	−44.6257			−1.50348			−0.751740	+	0.659460i	$0.770785\pi$
$882$		0			0
$883$		−	31.2627i		−	1.05207i	−0.850462	−	0.526036i	$-0.823678\pi$
$883$		−	31.2627i		−	1.05207i	0.850462	−	0.526036i	$-0.176322\pi$
$884$		0			0
$885$			4.31806i			0.145150i
$886$		0			0
$887$			5.31613i			0.178498i	0.996009	+	0.0892491i	$0.0284467\pi$
$887$			5.31613i			0.178498i	−0.996009	+	0.0892491i	$0.971553\pi$
$888$		0			0
$889$		−	45.2852i		−	1.51882i
$890$		0			0
$891$			1.38801i			0.0465000i
$892$		0			0
$893$	−60.3639			−2.02000
$894$		0			0
$895$			9.39540i			0.314054i
$896$		0			0
$897$	6.32973			0.211344
$898$		0			0
$899$			25.9319i			0.864877i
$900$		0			0
$901$	10.2695			0.342128
$902$		0			0
$903$	−49.6295			−1.65157
$904$		0			0
$905$			6.44599i			0.214272i
$906$		0			0
$907$	18.8112			0.624614			0.312307	−	0.949981i	$-0.398898\pi$
$907$	18.8112			0.624614			0.312307	+	0.949981i	$0.398898\pi$
$908$		0			0
$909$			6.59136i			0.218622i
$910$		0			0
$911$	−45.5803			−1.51014			−0.755072	−	0.655642i	$-0.772398\pi$
$911$	−45.5803			−1.51014			−0.755072	+	0.655642i	$0.772398\pi$
$912$		0			0
$913$		−	10.4691i		−	0.346477i
$914$		0			0
$915$		−	8.78304i		−	0.290358i
$916$		0			0
$917$			54.0520i			1.78495i
$918$		0			0
$919$		−	4.56719i		−	0.150658i	−0.997159	−	0.0753289i	$-0.975999\pi$
$919$		−	4.56719i		−	0.150658i	0.997159	−	0.0753289i	$-0.0240007\pi$
$920$		0			0
$921$			8.58915i			0.283022i
$922$		0			0
$923$	1.94173			0.0639127
$924$		0			0
$925$	28.2942			0.930307
$926$		0			0
$927$	−14.4729			−0.475352
$928$		0			0
$929$		−	1.10028i		−	0.0360991i	−0.999837	−	0.0180495i	$-0.994254\pi$
$929$		−	1.10028i		−	0.0360991i	0.999837	−	0.0180495i	$-0.00574566\pi$
$930$		0			0
$931$		−	50.8002i		−	1.66491i
$932$		0			0
$933$	−13.0528			−0.427330
$934$		0			0
$935$	6.73866			0.220378
$936$		0			0
$937$			23.7820i			0.776924i	0.921465	+	0.388462i	$0.126994\pi$
$937$			23.7820i			0.776924i	−0.921465	+	0.388462i	$0.873006\pi$
$938$		0			0
$939$	9.83998			0.321116
$940$		0			0
$941$	−50.0967			−1.63310			−0.816552	−	0.577272i	$-0.804117\pi$
$941$	−50.0967			−1.63310			−0.816552	+	0.577272i	$0.804117\pi$
$942$		0			0
$943$	−8.36714	+	3.91214i	−0.272472	+	0.127397i
$944$		0			0
$945$	−3.20336			−0.104205
$946$		0			0
$947$	57.6640			1.87383			0.936915	−	0.349558i	$-0.113668\pi$
$947$	57.6640			1.87383			0.936915	+	0.349558i	$0.113668\pi$
$948$		0			0
$949$			62.4980i			2.02877i
$950$		0			0
$951$	0.168441			0.00546206
$952$		0			0
$953$	14.7456			0.477658			0.238829	−	0.971062i	$-0.423236\pi$
$953$	14.7456			0.477658			0.238829	+	0.971062i	$0.423236\pi$
$954$		0			0
$955$		−	1.29615i		−	0.0419424i
$956$		0			0
$957$		−	6.83051i		−	0.220799i
$958$		0			0
$959$	−81.3047			−2.62546
$960$		0			0
$961$	−3.23199			−0.104258
$962$		0			0
$963$	−18.2768			−0.588961
$964$		0			0
$965$			16.4728i			0.530277i
$966$		0			0
$967$			40.8795i			1.31460i	0.753630	+	0.657299i	$0.228301\pi$
$967$			40.8795i			1.31460i	−0.753630	+	0.657299i	$0.771699\pi$
$968$		0			0
$969$		−	30.7486i		−	0.987789i
$970$		0			0
$971$		−	29.2887i		−	0.939919i	−0.882688	−	0.469960i	$-0.844268\pi$
$971$		−	29.2887i		−	0.939919i	0.882688	−	0.469960i	$-0.155732\pi$
$972$		0			0
$973$			9.93980i			0.318655i
$974$		0			0
$975$	−19.3617			−0.620070
$976$		0			0
$977$			26.0135i			0.832245i	0.909309	+	0.416122i	$0.136611\pi$
$977$			26.0135i			0.832245i	−0.909309	+	0.416122i	$0.863389\pi$
$978$		0			0
$979$	−17.4430			−0.557481
$980$		0			0
$981$			3.24513i			0.103609i
$982$		0			0
$983$	52.7288			1.68179			0.840893	−	0.541201i	$-0.182030\pi$
$983$	52.7288			1.68179			0.840893	+	0.541201i	$0.182030\pi$
$984$		0			0
$985$	−11.9515			−0.380806
$986$		0			0
$987$			51.9593i			1.65388i
$988$		0			0
$989$	−17.1313			−0.544744
$990$		0			0
$991$		−	2.14584i		−	0.0681650i	−0.999419	−	0.0340825i	$-0.989149\pi$
$991$		−	2.14584i		−	0.0681650i	0.999419	−	0.0340825i	$-0.0108509\pi$
$992$		0			0
$993$	11.9490			0.379190
$994$		0			0
$995$		−	4.05804i		−	0.128649i
$996$		0			0
$997$			37.6434i			1.19218i	0.802919	+	0.596089i	$0.203279\pi$
$997$			37.6434i			1.19218i	−0.802919	+	0.596089i	$0.796721\pi$
$998$		0			0
$999$		−	6.41241i		−	0.202880i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1968.2.j.f.1393.2		8
4.3	odd	2		984.2.j.b.409.6	yes	8
12.11	even	2		2952.2.j.d.2377.5		8
41.40	even	2	inner	1968.2.j.f.1393.6		8
164.163	odd	2		984.2.j.b.409.2	✓	8
492.491	even	2		2952.2.j.d.2377.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.j.b.409.2	✓	8	164.163	odd	2
984.2.j.b.409.6	yes	8	4.3	odd	2
1968.2.j.f.1393.2		8	1.1	even	1	trivial
1968.2.j.f.1393.6		8	41.40	even	2	inner
2952.2.j.d.2377.5		8	12.11	even	2
2952.2.j.d.2377.6		8	492.491	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1968.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +0.766545 q^{5} +4.17895i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +0.766545 q^{5} +4.17895i q^{7} -1.00000 q^{9} +1.38801i q^{11} +4.38801i q^{13} -0.766545i q^{15} -6.33351i q^{17} +4.85492i q^{19} +4.17895 q^{21} +1.44251 q^{23} -4.41241 q^{25} +1.00000i q^{27} -4.92110i q^{29} -5.26954 q^{31} +1.38801 q^{33} +3.20336i q^{35} -6.41241 q^{37} +4.38801 q^{39} +(-5.80042 + 2.71204i) q^{41} -11.8761 q^{43} -0.766545 q^{45} +12.4336i q^{47} -10.4637 q^{49} -6.33351 q^{51} +1.62146i q^{53} +1.06397i q^{55} +4.85492 q^{57} -5.63314 q^{59} +11.4580 q^{61} -4.17895i q^{63} +3.36360i q^{65} -11.2150i q^{67} -1.44251i q^{69} -0.442508i q^{71} +14.2429 q^{73} +4.41241i q^{75} -5.80042 q^{77} +12.8248i q^{79} +1.00000 q^{81} -7.54256 q^{83} -4.85492i q^{85} -4.92110 q^{87} +12.5670i q^{89} -18.3373 q^{91} +5.26954i q^{93} +3.72151i q^{95} +0.287956i q^{97} -1.38801i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q + 10 * q^5 - 8 * q^9	\( 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) \) 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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