LMFDB - Embedded newform 1344.2.b.e.895.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(895,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1344.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.b (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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x + 4 $ x^4 - x^3 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.3
Root			$-0.780776 + 1.17915i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.2.b.e.895.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +1.00000 q^{9} +3.02045i q^{11} -6.04090i q^{13} -1.69614i q^{15} -4.34475i q^{17} -1.12311 q^{19} +(2.56155 + 0.662153i) q^{21} +3.02045i q^{23} +2.12311 q^{25} -1.00000 q^{27} +2.00000 q^{29} -3.02045i q^{33} +(1.12311 - 4.34475i) q^{35} +7.12311 q^{37} +6.04090i q^{39} +7.73704i q^{41} -8.10887i q^{43} +1.69614i q^{45} +10.2462 q^{47} +(6.12311 + 3.39228i) q^{49} +4.34475i q^{51} +4.24621 q^{53} -5.12311 q^{55} +1.12311 q^{57} +4.00000 q^{59} +9.43318i q^{61} +(-2.56155 - 0.662153i) q^{63} +10.2462 q^{65} +2.06798i q^{67} -3.02045i q^{69} -12.4536i q^{71} -3.39228i q^{73} -2.12311 q^{75} +(2.00000 - 7.73704i) q^{77} -4.71659i q^{79} +1.00000 q^{81} +6.24621 q^{83} +7.36932 q^{85} -2.00000 q^{87} -7.73704i q^{89} +(-4.00000 + 15.4741i) q^{91} -1.90495i q^{95} -8.68951i q^{97} +3.02045i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 2 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 2 q^{7} + 4 q^{9} + 12 q^{19} + 2 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{29} - 12 q^{35} + 12 q^{37} + 8 q^{47} + 8 q^{49} - 16 q^{53} - 4 q^{55} - 12 q^{57} + 16 q^{59} - 2 q^{63} + 8 q^{65} + 8 q^{75} + 8 q^{77} + 4 q^{81} - 8 q^{83} - 20 q^{85} - 8 q^{87} - 16 q^{91}+O(q^{100}) $ 4 * q - 4 * q^3 - 2 * q^7 + 4 * q^9 + 12 * q^19 + 2 * q^21 - 8 * q^25 - 4 * q^27 + 8 * q^29 - 12 * q^35 + 12 * q^37 + 8 * q^47 + 8 * q^49 - 16 * q^53 - 4 * q^55 - 12 * q^57 + 16 * q^59 - 2 * q^63 + 8 * q^65 + 8 * q^75 + 8 * q^77 + 4 * q^81 - 8 * q^83 - 20 * q^85 - 8 * q^87 - 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			1.69614i			0.758537i	0.925287	+	0.379269i	$0.123824\pi$
$5$			1.69614i			0.758537i	−0.925287	+	0.379269i	$0.876176\pi$
$6$		0			0
$7$	−2.56155	−	0.662153i	−0.968176	−	0.250270i
$7$	−2.56155	−	0.662153i	−0.968176	−	0.250270i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.02045i			0.910699i	0.890313	+	0.455350i	$0.150486\pi$
$11$			3.02045i			0.910699i	−0.890313	+	0.455350i	$0.849514\pi$
$12$		0			0
$13$		−	6.04090i		−	1.67544i	−0.546098	−	0.837722i	$-0.683887\pi$
$13$		−	6.04090i		−	1.67544i	0.546098	−	0.837722i	$-0.316113\pi$
$14$		0			0
$15$		−	1.69614i		−	0.437942i
$16$		0			0
$17$		−	4.34475i		−	1.05376i	−0.849940	−	0.526879i	$-0.823362\pi$
$17$		−	4.34475i		−	1.05376i	0.849940	−	0.526879i	$-0.176638\pi$
$18$		0			0
$19$	−1.12311			−0.257658			−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311			−0.257658			−0.128829	+	0.991667i	$0.541122\pi$
$20$		0			0
$21$	2.56155	+	0.662153i	0.558977	+	0.144494i
$22$		0			0
$23$			3.02045i			0.629807i	0.949124	+	0.314903i	$0.101972\pi$
$23$			3.02045i			0.629807i	−0.949124	+	0.314903i	$0.898028\pi$
$24$		0			0
$25$	2.12311			0.424621
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		−	3.02045i		−	0.525792i
$34$		0			0
$35$	1.12311	−	4.34475i	0.189839	−	0.734398i
$36$		0			0
$37$	7.12311			1.17103			0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311			1.17103			0.585516	+	0.810661i	$0.300892\pi$
$38$		0			0
$39$			6.04090i			0.967317i
$40$		0			0
$41$			7.73704i			1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$41$			7.73704i			1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$42$		0			0
$43$		−	8.10887i		−	1.23659i	−0.785946	−	0.618296i	$-0.787823\pi$
$43$		−	8.10887i		−	1.23659i	0.785946	−	0.618296i	$-0.212177\pi$
$44$		0			0
$45$			1.69614i			0.252846i
$46$		0			0
$47$	10.2462			1.49456			0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462			1.49456			0.747282	+	0.664507i	$0.231359\pi$
$48$		0			0
$49$	6.12311	+	3.39228i	0.874729	+	0.484612i
$50$		0			0
$51$			4.34475i			0.608387i
$52$		0			0
$53$	4.24621			0.583262			0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621			0.583262			0.291631	+	0.956531i	$0.405802\pi$
$54$		0			0
$55$	−5.12311			−0.690799
$56$		0			0
$57$	1.12311			0.148759
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$			9.43318i			1.20779i	0.797062	+	0.603897i	$0.206386\pi$
$61$			9.43318i			1.20779i	−0.797062	+	0.603897i	$0.793614\pi$
$62$		0			0
$63$	−2.56155	−	0.662153i	−0.322725	−	0.0834235i
$64$		0			0
$65$	10.2462			1.27089
$66$		0			0
$67$			2.06798i			0.252643i	0.991989	+	0.126322i	$0.0403172\pi$
$67$			2.06798i			0.252643i	−0.991989	+	0.126322i	$0.959683\pi$
$68$		0			0
$69$		−	3.02045i		−	0.363619i
$70$		0			0
$71$		−	12.4536i		−	1.47797i	−0.673720	−	0.738987i	$-0.735305\pi$
$71$		−	12.4536i		−	1.47797i	0.673720	−	0.738987i	$-0.264695\pi$
$72$		0			0
$73$		−	3.39228i		−	0.397037i	−0.980097	−	0.198518i	$-0.936387\pi$
$73$		−	3.39228i		−	0.397037i	0.980097	−	0.198518i	$-0.0636129\pi$
$74$		0			0
$75$	−2.12311			−0.245155
$76$		0			0
$77$	2.00000	−	7.73704i	0.227921	−	0.881717i
$78$		0			0
$79$		−	4.71659i		−	0.530658i	−0.964158	−	0.265329i	$-0.914519\pi$
$79$		−	4.71659i		−	0.530658i	0.964158	−	0.265329i	$-0.0854805\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	6.24621			0.685611			0.342805	−	0.939406i	$-0.388623\pi$
$83$	6.24621			0.685611			0.342805	+	0.939406i	$0.388623\pi$
$84$		0			0
$85$	7.36932			0.799315
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$		−	7.73704i		−	0.820124i	−0.912058	−	0.410062i	$-0.865507\pi$
$89$		−	7.73704i		−	0.820124i	0.912058	−	0.410062i	$-0.134493\pi$
$90$		0			0
$91$	−4.00000	+	15.4741i	−0.419314	+	1.62212i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1.90495i		−	0.195443i
$96$		0			0
$97$		−	8.68951i		−	0.882286i	−0.897437	−	0.441143i	$-0.854573\pi$
$97$		−	8.68951i		−	0.882286i	0.897437	−	0.441143i	$-0.145427\pi$
$98$		0			0
$99$			3.02045i			0.303566i
$100$		0			0
$101$		−	6.99337i		−	0.695866i	−0.937519	−	0.347933i	$-0.886884\pi$
$101$		−	6.99337i		−	0.695866i	0.937519	−	0.347933i	$-0.113116\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$	−1.12311	+	4.34475i	−0.109604	+	0.424005i
$106$		0			0
$107$		−	5.66906i		−	0.548049i	−0.961723	−	0.274024i	$-0.911645\pi$
$107$		−	5.66906i		−	0.548049i	0.961723	−	0.274024i	$-0.0883549\pi$
$108$		0			0
$109$	−8.24621			−0.789844			−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621			−0.789844			−0.394922	+	0.918715i	$0.629228\pi$
$110$		0			0
$111$	−7.12311			−0.676095
$112$		0			0
$113$	12.2462			1.15203			0.576013	−	0.817440i	$-0.304608\pi$
$113$	12.2462			1.15203			0.576013	+	0.817440i	$0.304608\pi$
$114$		0			0
$115$	−5.12311			−0.477732
$116$		0			0
$117$		−	6.04090i		−	0.558481i
$118$		0			0
$119$	−2.87689	+	11.1293i	−0.263724	+	1.02022i
$120$		0			0
$121$	1.87689			0.170627
$122$		0			0
$123$		−	7.73704i		−	0.697625i
$124$		0			0
$125$			12.0818i			1.08063i
$126$		0			0
$127$			19.4470i			1.72564i	0.505510	+	0.862821i	$0.331304\pi$
$127$			19.4470i			1.72564i	−0.505510	+	0.862821i	$0.668696\pi$
$128$		0			0
$129$			8.10887i			0.713946i
$130$		0			0
$131$	22.2462			1.94366			0.971830	−	0.235682i	$-0.0757323\pi$
$131$	22.2462			1.94366			0.971830	+	0.235682i	$0.0757323\pi$
$132$		0			0
$133$	2.87689	+	0.743668i	0.249458	+	0.0644842i
$134$		0			0
$135$		−	1.69614i		−	0.145981i
$136$		0			0
$137$	−16.2462			−1.38801			−0.694004	−	0.719971i	$-0.744155\pi$
$137$	−16.2462			−1.38801			−0.694004	+	0.719971i	$0.744155\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	−10.2462			−0.862887
$142$		0			0
$143$	18.2462			1.52582
$144$		0			0
$145$			3.39228i			0.281714i
$146$		0			0
$147$	−6.12311	−	3.39228i	−0.505025	−	0.279791i
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$		−	8.10887i		−	0.659891i	−0.944000	−	0.329945i	$-0.892970\pi$
$151$		−	8.10887i		−	0.659891i	0.944000	−	0.329945i	$-0.107030\pi$
$152$		0			0
$153$		−	4.34475i		−	0.351253i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	4.13595i		−	0.330085i	−0.986286	−	0.165042i	$-0.947224\pi$
$157$		−	4.13595i		−	0.330085i	0.986286	−	0.165042i	$-0.0527761\pi$
$158$		0			0
$159$	−4.24621			−0.336746
$160$		0			0
$161$	2.00000	−	7.73704i	0.157622	−	0.609764i
$162$		0			0
$163$		−	11.5012i		−	0.900840i	−0.892817	−	0.450420i	$-0.851274\pi$
$163$		−	11.5012i		−	0.900840i	0.892817	−	0.450420i	$-0.148726\pi$
$164$		0			0
$165$	5.12311			0.398833
$166$		0			0
$167$	2.24621			0.173817			0.0869085	−	0.996216i	$-0.472301\pi$
$167$	2.24621			0.173817			0.0869085	+	0.996216i	$0.472301\pi$
$168$		0			0
$169$	−23.4924			−1.80711
$170$		0			0
$171$	−1.12311			−0.0858860
$172$		0			0
$173$		−	8.48071i		−	0.644776i	−0.946608	−	0.322388i	$-0.895514\pi$
$173$		−	8.48071i		−	0.644776i	0.946608	−	0.322388i	$-0.104486\pi$
$174$		0			0
$175$	−5.43845	−	1.40582i	−0.411108	−	0.106270i
$176$		0			0
$177$	−4.00000			−0.300658
$178$		0			0
$179$		−	2.27678i		−	0.170175i	−0.996374	−	0.0850873i	$-0.972883\pi$
$179$		−	2.27678i		−	0.170175i	0.996374	−	0.0850873i	$-0.0271169\pi$
$180$		0			0
$181$			6.04090i			0.449016i	0.974472	+	0.224508i	$0.0720775\pi$
$181$			6.04090i			0.449016i	−0.974472	+	0.224508i	$0.927922\pi$
$182$		0			0
$183$		−	9.43318i		−	0.697321i
$184$		0			0
$185$			12.0818i			0.888271i
$186$		0			0
$187$	13.1231			0.959657
$188$		0			0
$189$	2.56155	+	0.662153i	0.186326	+	0.0481646i
$190$		0			0
$191$		−	9.06134i		−	0.655656i	−0.944737	−	0.327828i	$-0.893683\pi$
$191$		−	9.06134i		−	0.655656i	0.944737	−	0.327828i	$-0.106317\pi$
$192$		0			0
$193$	9.36932			0.674418			0.337209	−	0.941430i	$-0.390517\pi$
$193$	9.36932			0.674418			0.337209	+	0.941430i	$0.390517\pi$
$194$		0			0
$195$	−10.2462			−0.733746
$196$		0			0
$197$	−0.246211			−0.0175418			−0.00877091	−	0.999962i	$-0.502792\pi$
$197$	−0.246211			−0.0175418			−0.00877091	+	0.999962i	$0.502792\pi$
$198$		0			0
$199$	−5.12311			−0.363167			−0.181584	−	0.983375i	$-0.558122\pi$
$199$	−5.12311			−0.363167			−0.181584	+	0.983375i	$0.558122\pi$
$200$		0			0
$201$		−	2.06798i		−	0.145864i
$202$		0			0
$203$	−5.12311	−	1.32431i	−0.359572	−	0.0929481i
$204$		0			0
$205$	−13.1231			−0.916557
$206$		0			0
$207$			3.02045i			0.209936i
$208$		0			0
$209$		−	3.39228i		−	0.234649i
$210$		0			0
$211$			3.97292i			0.273507i	0.990605	+	0.136754i	$0.0436668\pi$
$211$			3.97292i			0.273507i	−0.990605	+	0.136754i	$0.956333\pi$
$212$		0			0
$213$			12.4536i			0.853308i
$214$		0			0
$215$	13.7538			0.938001
$216$		0			0
$217$		0			0
$218$		0			0
$219$			3.39228i			0.229229i
$220$		0			0
$221$	−26.2462			−1.76551
$222$		0			0
$223$	−18.8769			−1.26409			−0.632045	−	0.774932i	$-0.717784\pi$
$223$	−18.8769			−1.26409			−0.632045	+	0.774932i	$0.717784\pi$
$224$		0			0
$225$	2.12311			0.141540
$226$		0			0
$227$	16.4924			1.09464			0.547320	−	0.836923i	$-0.315648\pi$
$227$	16.4924			1.09464			0.547320	+	0.836923i	$0.315648\pi$
$228$		0			0
$229$		−	18.1227i		−	1.19758i	−0.800906	−	0.598790i	$-0.795648\pi$
$229$		−	18.1227i		−	1.19758i	0.800906	−	0.598790i	$-0.204352\pi$
$230$		0			0
$231$	−2.00000	+	7.73704i	−0.131590	+	0.509060i
$232$		0			0
$233$	−10.4924			−0.687381			−0.343691	−	0.939083i	$-0.611677\pi$
$233$	−10.4924			−0.687381			−0.343691	+	0.939083i	$0.611677\pi$
$234$		0			0
$235$			17.3790i			1.13368i
$236$		0			0
$237$			4.71659i			0.306375i
$238$		0			0
$239$		−	2.27678i		−	0.147273i	−0.997285	−	0.0736363i	$-0.976540\pi$
$239$		−	2.27678i		−	0.147273i	0.997285	−	0.0736363i	$-0.0234604\pi$
$240$		0			0
$241$			1.90495i			0.122708i	0.998116	+	0.0613542i	$0.0195419\pi$
$241$			1.90495i			0.122708i	−0.998116	+	0.0613542i	$0.980458\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−5.75379	+	10.3857i	−0.367596	+	0.663515i
$246$		0			0
$247$			6.78456i			0.431691i
$248$		0			0
$249$	−6.24621			−0.395838
$250$		0			0
$251$	−22.2462			−1.40417			−0.702084	−	0.712094i	$-0.747747\pi$
$251$	−22.2462			−1.40417			−0.702084	+	0.712094i	$0.747747\pi$
$252$		0			0
$253$	−9.12311			−0.573565
$254$		0			0
$255$	−7.36932			−0.461485
$256$		0			0
$257$			19.8188i			1.23626i	0.786074	+	0.618132i	$0.212110\pi$
$257$			19.8188i			1.23626i	−0.786074	+	0.618132i	$0.787890\pi$
$258$		0			0
$259$	−18.2462	−	4.71659i	−1.13376	−	0.293075i
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$			4.92539i			0.303713i	0.988403	+	0.151856i	$0.0485251\pi$
$263$			4.92539i			0.303713i	−0.988403	+	0.151856i	$0.951475\pi$
$264$		0			0
$265$			7.20217i			0.442426i
$266$		0			0
$267$			7.73704i			0.473499i
$268$		0			0
$269$			22.4674i			1.36986i	0.728607	+	0.684932i	$0.240168\pi$
$269$			22.4674i			1.36986i	−0.728607	+	0.684932i	$0.759832\pi$
$270$		0			0
$271$	−4.49242			−0.272895			−0.136448	−	0.990647i	$-0.543569\pi$
$271$	−4.49242			−0.272895			−0.136448	+	0.990647i	$0.543569\pi$
$272$		0			0
$273$	4.00000	−	15.4741i	0.242091	−	0.936534i
$274$		0			0
$275$			6.41273i			0.386702i
$276$		0			0
$277$	−3.12311			−0.187649			−0.0938246	−	0.995589i	$-0.529909\pi$
$277$	−3.12311			−0.187649			−0.0938246	+	0.995589i	$0.529909\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.246211			−0.0146877			−0.00734387	−	0.999973i	$-0.502338\pi$
$281$	−0.246211			−0.0146877			−0.00734387	+	0.999973i	$0.502338\pi$
$282$		0			0
$283$	17.1231			1.01786			0.508931	−	0.860807i	$-0.330041\pi$
$283$	17.1231			1.01786			0.508931	+	0.860807i	$0.330041\pi$
$284$		0			0
$285$			1.90495i			0.112839i
$286$		0			0
$287$	5.12311	−	19.8188i	0.302407	−	1.16987i
$288$		0			0
$289$	−1.87689			−0.110406
$290$		0			0
$291$			8.68951i			0.509388i
$292$		0			0
$293$		−	6.99337i		−	0.408557i	−0.978913	−	0.204278i	$-0.934515\pi$
$293$		−	6.99337i		−	0.408557i	0.978913	−	0.204278i	$-0.0654848\pi$
$294$		0			0
$295$			6.78456i			0.395013i
$296$		0			0
$297$		−	3.02045i		−	0.175264i
$298$		0			0
$299$	18.2462			1.05521
$300$		0			0
$301$	−5.36932	+	20.7713i	−0.309482	+	1.19724i
$302$		0			0
$303$			6.99337i			0.401759i
$304$		0			0
$305$	−16.0000			−0.916157
$306$		0			0
$307$	−21.6155			−1.23366			−0.616832	−	0.787095i	$-0.711584\pi$
$307$	−21.6155			−1.23366			−0.616832	+	0.787095i	$0.711584\pi$
$308$		0			0
$309$	−8.00000			−0.455104
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$			25.6509i			1.44988i	0.688814	+	0.724938i	$0.258132\pi$
$313$			25.6509i			1.44988i	−0.688814	+	0.724938i	$0.741868\pi$
$314$		0			0
$315$	1.12311	−	4.34475i	0.0632798	−	0.244799i
$316$		0			0
$317$	−18.4924			−1.03864			−0.519319	−	0.854580i	$-0.673814\pi$
$317$	−18.4924			−1.03864			−0.519319	+	0.854580i	$0.673814\pi$
$318$		0			0
$319$			6.04090i			0.338225i
$320$		0			0
$321$			5.66906i			0.316416i
$322$		0			0
$323$			4.87962i			0.271509i
$324$		0			0
$325$		−	12.8255i		−	0.711429i
$326$		0			0
$327$	8.24621			0.456017
$328$		0			0
$329$	−26.2462	−	6.78456i	−1.44700	−	0.374045i
$330$		0			0
$331$			5.46026i			0.300123i	0.988677	+	0.150061i	$0.0479472\pi$
$331$			5.46026i			0.300123i	−0.988677	+	0.150061i	$0.952053\pi$
$332$		0			0
$333$	7.12311			0.390344
$334$		0			0
$335$	−3.50758			−0.191639
$336$		0			0
$337$	−8.24621			−0.449200			−0.224600	−	0.974451i	$-0.572108\pi$
$337$	−8.24621			−0.449200			−0.224600	+	0.974451i	$0.572108\pi$
$338$		0			0
$339$	−12.2462			−0.665123
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−13.4384	−	12.7439i	−0.725608	−	0.688108i
$344$		0			0
$345$	5.12311			0.275819
$346$		0			0
$347$		−	34.7123i		−	1.86345i	−0.363162	−	0.931726i	$-0.618303\pi$
$347$		−	34.7123i		−	1.86345i	0.363162	−	0.931726i	$-0.381697\pi$
$348$		0			0
$349$		−	4.13595i		−	0.221392i	−0.993854	−	0.110696i	$-0.964692\pi$
$349$		−	4.13595i		−	0.221392i	0.993854	−	0.110696i	$-0.0353080\pi$
$350$		0			0
$351$			6.04090i			0.322439i
$352$		0			0
$353$		−	6.24970i		−	0.332638i	−0.986072	−	0.166319i	$-0.946812\pi$
$353$		−	6.24970i		−	0.332638i	0.986072	−	0.166319i	$-0.0531881\pi$
$354$		0			0
$355$	21.1231			1.12110
$356$		0			0
$357$	2.87689	−	11.1293i	0.152261	−	0.589026i
$358$		0			0
$359$			1.11550i			0.0588740i	0.999567	+	0.0294370i	$0.00937144\pi$
$359$			1.11550i			0.0588740i	−0.999567	+	0.0294370i	$0.990629\pi$
$360$		0			0
$361$	−17.7386			−0.933612
$362$		0			0
$363$	−1.87689			−0.0985114
$364$		0			0
$365$	5.75379			0.301167
$366$		0			0
$367$	8.63068			0.450518			0.225259	−	0.974299i	$-0.427677\pi$
$367$	8.63068			0.450518			0.225259	+	0.974299i	$0.427677\pi$
$368$		0			0
$369$			7.73704i			0.402774i
$370$		0			0
$371$	−10.8769	−	2.81164i	−0.564700	−	0.145973i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$		−	12.0818i		−	0.623901i
$376$		0			0
$377$		−	12.0818i		−	0.622244i
$378$		0			0
$379$		−	18.7033i		−	0.960725i	−0.877070	−	0.480363i	$-0.840505\pi$
$379$		−	18.7033i		−	0.960725i	0.877070	−	0.480363i	$-0.159495\pi$
$380$		0			0
$381$		−	19.4470i		−	0.996300i
$382$		0			0
$383$	26.2462			1.34112			0.670559	−	0.741856i	$-0.266054\pi$
$383$	26.2462			1.34112			0.670559	+	0.741856i	$0.266054\pi$
$384$		0			0
$385$	13.1231	+	3.39228i	0.668815	+	0.172887i
$386$		0			0
$387$		−	8.10887i		−	0.412197i
$388$		0			0
$389$	−0.246211			−0.0124834			−0.00624170	−	0.999981i	$-0.501987\pi$
$389$	−0.246211			−0.0124834			−0.00624170	+	0.999981i	$0.501987\pi$
$390$		0			0
$391$	13.1231			0.663664
$392$		0			0
$393$	−22.2462			−1.12217
$394$		0			0
$395$	8.00000			0.402524
$396$		0			0
$397$			16.2177i			0.813945i	0.913440	+	0.406973i	$0.133416\pi$
$397$			16.2177i			0.813945i	−0.913440	+	0.406973i	$0.866584\pi$
$398$		0			0
$399$	−2.87689	−	0.743668i	−0.144025	−	0.0372300i
$400$		0			0
$401$	−8.24621			−0.411796			−0.205898	−	0.978573i	$-0.566012\pi$
$401$	−8.24621			−0.411796			−0.205898	+	0.978573i	$0.566012\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$			1.69614i			0.0842819i
$406$		0			0
$407$			21.5150i			1.06646i
$408$		0			0
$409$		−	27.5559i		−	1.36255i	−0.732028	−	0.681275i	$-0.761426\pi$
$409$		−	27.5559i		−	1.36255i	0.732028	−	0.681275i	$-0.238574\pi$
$410$		0			0
$411$	16.2462			0.801367
$412$		0			0
$413$	−10.2462	−	2.64861i	−0.504183	−	0.130330i
$414$		0			0
$415$			10.5945i			0.520061i
$416$		0			0
$417$	12.0000			0.587643
$418$		0			0
$419$	16.4924			0.805708			0.402854	−	0.915264i	$-0.368018\pi$
$419$	16.4924			0.805708			0.402854	+	0.915264i	$0.368018\pi$
$420$		0			0
$421$	−19.1231			−0.932003			−0.466002	−	0.884784i	$-0.654306\pi$
$421$	−19.1231			−0.932003			−0.466002	+	0.884784i	$0.654306\pi$
$422$		0			0
$423$	10.2462			0.498188
$424$		0			0
$425$		−	9.22437i		−	0.447448i
$426$		0			0
$427$	6.24621	−	24.1636i	0.302275	−	1.16936i
$428$		0			0
$429$	−18.2462			−0.880935
$430$		0			0
$431$			15.1022i			0.727449i	0.931507	+	0.363725i	$0.118495\pi$
$431$			15.1022i			0.727449i	−0.931507	+	0.363725i	$0.881505\pi$
$432$		0			0
$433$			6.78456i			0.326045i	0.986622	+	0.163023i	$0.0521244\pi$
$433$			6.78456i			0.326045i	−0.986622	+	0.163023i	$0.947876\pi$
$434$		0			0
$435$		−	3.39228i		−	0.162647i
$436$		0			0
$437$		−	3.39228i		−	0.162275i
$438$		0			0
$439$	31.3693			1.49718			0.748588	−	0.663036i	$-0.230732\pi$
$439$	31.3693			1.49718			0.748588	+	0.663036i	$0.230732\pi$
$440$		0			0
$441$	6.12311	+	3.39228i	0.291576	+	0.161537i
$442$		0			0
$443$			35.8735i			1.70440i	0.523213	+	0.852202i	$0.324733\pi$
$443$			35.8735i			1.70440i	−0.523213	+	0.852202i	$0.675267\pi$
$444$		0			0
$445$	13.1231			0.622095
$446$		0			0
$447$	−10.0000			−0.472984
$448$		0			0
$449$	28.2462			1.33302			0.666511	−	0.745496i	$-0.267787\pi$
$449$	28.2462			1.33302			0.666511	+	0.745496i	$0.267787\pi$
$450$		0			0
$451$	−23.3693			−1.10042
$452$		0			0
$453$			8.10887i			0.380988i
$454$		0			0
$455$	−26.2462	−	6.78456i	−1.23044	−	0.318065i
$456$		0			0
$457$	−16.2462			−0.759966			−0.379983	−	0.924994i	$-0.624070\pi$
$457$	−16.2462			−0.759966			−0.379983	+	0.924994i	$0.624070\pi$
$458$		0			0
$459$			4.34475i			0.202796i
$460$		0			0
$461$		−	17.1702i		−	0.799697i	−0.916581	−	0.399848i	$-0.869063\pi$
$461$		−	17.1702i		−	0.799697i	0.916581	−	0.399848i	$-0.130937\pi$
$462$		0			0
$463$		−	39.4746i		−	1.83454i	−0.398264	−	0.917271i	$-0.630387\pi$
$463$		−	39.4746i		−	1.83454i	0.398264	−	0.917271i	$-0.369613\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.7538			0.821547			0.410774	−	0.911737i	$-0.365259\pi$
$467$	17.7538			0.821547			0.410774	+	0.911737i	$0.365259\pi$
$468$		0			0
$469$	1.36932	−	5.29723i	0.0632292	−	0.244603i
$470$		0			0
$471$			4.13595i			0.190575i
$472$		0			0
$473$	24.4924			1.12616
$474$		0			0
$475$	−2.38447			−0.109407
$476$		0			0
$477$	4.24621			0.194421
$478$		0			0
$479$	20.4924			0.936323			0.468161	−	0.883643i	$-0.344917\pi$
$479$	20.4924			0.936323			0.468161	+	0.883643i	$0.344917\pi$
$480$		0			0
$481$		−	43.0299i		−	1.96200i
$482$		0			0
$483$	−2.00000	+	7.73704i	−0.0910032	+	0.352047i
$484$		0			0
$485$	14.7386			0.669247
$486$		0			0
$487$		−	32.2725i		−	1.46240i	−0.682161	−	0.731202i	$-0.738960\pi$
$487$		−	32.2725i		−	1.46240i	0.682161	−	0.731202i	$-0.261040\pi$
$488$		0			0
$489$			11.5012i			0.520100i
$490$		0			0
$491$			11.7100i			0.528463i	0.964459	+	0.264231i	$0.0851183\pi$
$491$			11.7100i			0.528463i	−0.964459	+	0.264231i	$0.914882\pi$
$492$		0			0
$493$		−	8.68951i		−	0.391356i
$494$		0			0
$495$	−5.12311			−0.230266
$496$		0			0
$497$	−8.24621	+	31.9006i	−0.369893	+	1.43094i
$498$		0			0
$499$			17.5420i			0.785290i	0.919690	+	0.392645i	$0.128440\pi$
$499$			17.5420i			0.785290i	−0.919690	+	0.392645i	$0.871560\pi$
$500$		0			0
$501$	−2.24621			−0.100353
$502$		0			0
$503$	−22.7386			−1.01387			−0.506933	−	0.861986i	$-0.669221\pi$
$503$	−22.7386			−1.01387			−0.506933	+	0.861986i	$0.669221\pi$
$504$		0			0
$505$	11.8617			0.527840
$506$		0			0
$507$	23.4924			1.04334
$508$		0			0
$509$		−	1.69614i		−	0.0751801i	−0.999293	−	0.0375901i	$-0.988032\pi$
$509$		−	1.69614i		−	0.0751801i	0.999293	−	0.0375901i	$-0.0119681\pi$
$510$		0			0
$511$	−2.24621	+	8.68951i	−0.0993665	+	0.384401i
$512$		0			0
$513$	1.12311			0.0495863
$514$		0			0
$515$			13.5691i			0.597927i
$516$		0			0
$517$			30.9481i			1.36110i
$518$		0			0
$519$			8.48071i			0.372262i
$520$		0			0
$521$			33.8056i			1.48105i	0.672030	+	0.740524i	$0.265423\pi$
$521$			33.8056i			1.48105i	−0.672030	+	0.740524i	$0.734577\pi$
$522$		0			0
$523$	−0.492423			−0.0215321			−0.0107661	−	0.999942i	$-0.503427\pi$
$523$	−0.492423			−0.0215321			−0.0107661	+	0.999942i	$0.503427\pi$
$524$		0			0
$525$	5.43845	+	1.40582i	0.237353	+	0.0613551i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.8769			0.603343
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$	46.7386			2.02447
$534$		0			0
$535$	9.61553			0.415716
$536$		0			0
$537$			2.27678i			0.0982503i
$538$		0			0
$539$	−10.2462	+	18.4945i	−0.441336	+	0.796615i
$540$		0			0
$541$	−11.1231			−0.478220			−0.239110	−	0.970993i	$-0.576856\pi$
$541$	−11.1231			−0.478220			−0.239110	+	0.970993i	$0.576856\pi$
$542$		0			0
$543$		−	6.04090i		−	0.259240i
$544$		0			0
$545$		−	13.9867i		−	0.599126i
$546$		0			0
$547$			33.0161i			1.41167i	0.708378	+	0.705834i	$0.249427\pi$
$547$			33.0161i			1.41167i	−0.708378	+	0.705834i	$0.750573\pi$
$548$		0			0
$549$			9.43318i			0.402598i
$550$		0			0
$551$	−2.24621			−0.0956918
$552$		0			0
$553$	−3.12311	+	12.0818i	−0.132808	+	0.513770i
$554$		0			0
$555$		−	12.0818i		−	0.512843i
$556$		0			0
$557$	−14.0000			−0.593199			−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000			−0.593199			−0.296600	+	0.955002i	$0.595853\pi$
$558$		0			0
$559$	−48.9848			−2.07184
$560$		0			0
$561$	−13.1231			−0.554058
$562$		0			0
$563$	−14.2462			−0.600406			−0.300203	−	0.953875i	$-0.597054\pi$
$563$	−14.2462			−0.600406			−0.300203	+	0.953875i	$0.597054\pi$
$564$		0			0
$565$			20.7713i			0.873855i
$566$		0			0
$567$	−2.56155	−	0.662153i	−0.107575	−	0.0278078i
$568$		0			0
$569$	34.9848			1.46664			0.733320	−	0.679883i	$-0.237969\pi$
$569$	34.9848			1.46664			0.733320	+	0.679883i	$0.237969\pi$
$570$		0			0
$571$		−	40.9620i		−	1.71420i	−0.515146	−	0.857102i	$-0.672262\pi$
$571$		−	40.9620i		−	1.71420i	0.515146	−	0.857102i	$-0.327738\pi$
$572$		0			0
$573$			9.06134i			0.378543i
$574$		0			0
$575$			6.41273i			0.267429i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$	−9.36932			−0.389376
$580$		0			0
$581$	−16.0000	−	4.13595i	−0.663792	−	0.171588i
$582$		0			0
$583$			12.8255i			0.531176i
$584$		0			0
$585$	10.2462			0.423629
$586$		0			0
$587$	−38.2462			−1.57859			−0.789295	−	0.614014i	$-0.789554\pi$
$587$	−38.2462			−1.57859			−0.789295	+	0.614014i	$0.789554\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	0.246211			0.0101278
$592$		0			0
$593$		−	21.7238i		−	0.892088i	−0.895011	−	0.446044i	$-0.852832\pi$
$593$		−	21.7238i		−	0.892088i	0.895011	−	0.446044i	$-0.147168\pi$
$594$		0			0
$595$	−18.8769	−	4.87962i	−0.773877	−	0.200045i
$596$		0			0
$597$	5.12311			0.209675
$598$		0			0
$599$		−	19.2382i		−	0.786051i	−0.919528	−	0.393026i	$-0.871428\pi$
$599$		−	19.2382i		−	0.786051i	0.919528	−	0.393026i	$-0.128572\pi$
$600$		0			0
$601$		−	5.29723i		−	0.216078i	−0.994147	−	0.108039i	$-0.965543\pi$
$601$		−	5.29723i		−	0.216078i	0.994147	−	0.108039i	$-0.0344572\pi$
$602$		0			0
$603$			2.06798i			0.0842145i
$604$		0			0
$605$			3.18348i			0.129427i
$606$		0			0
$607$	33.6155			1.36441			0.682206	−	0.731160i	$-0.261021\pi$
$607$	33.6155			1.36441			0.682206	+	0.731160i	$0.261021\pi$
$608$		0			0
$609$	5.12311	+	1.32431i	0.207599	+	0.0536636i
$610$		0			0
$611$		−	61.8963i		−	2.50406i
$612$		0			0
$613$	40.7386			1.64542			0.822709	−	0.568463i	$-0.192462\pi$
$613$	40.7386			1.64542			0.822709	+	0.568463i	$0.192462\pi$
$614$		0			0
$615$	13.1231			0.529175
$616$		0			0
$617$	15.7538			0.634224			0.317112	−	0.948388i	$-0.397287\pi$
$617$	15.7538			0.634224			0.317112	+	0.948388i	$0.397287\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$		−	3.02045i		−	0.121206i
$622$		0			0
$623$	−5.12311	+	19.8188i	−0.205253	+	0.794025i
$624$		0			0
$625$	−9.87689			−0.395076
$626$		0			0
$627$			3.39228i			0.135475i
$628$		0			0
$629$		−	30.9481i		−	1.23398i
$630$		0			0
$631$			17.9597i			0.714963i	0.933920	+	0.357481i	$0.116364\pi$
$631$			17.9597i			0.714963i	−0.933920	+	0.357481i	$0.883636\pi$
$632$		0			0
$633$		−	3.97292i		−	0.157909i
$634$		0			0
$635$	−32.9848			−1.30896
$636$		0			0
$637$	20.4924	−	36.9890i	0.811939	−	1.46556i
$638$		0			0
$639$		−	12.4536i		−	0.492658i
$640$		0			0
$641$	−9.50758			−0.375527			−0.187763	−	0.982214i	$-0.560124\pi$
$641$	−9.50758			−0.375527			−0.187763	+	0.982214i	$0.560124\pi$
$642$		0			0
$643$	29.6155			1.16792			0.583961	−	0.811782i	$-0.301502\pi$
$643$	29.6155			1.16792			0.583961	+	0.811782i	$0.301502\pi$
$644$		0			0
$645$	−13.7538			−0.541555
$646$		0			0
$647$	−32.9848			−1.29677			−0.648384	−	0.761313i	$-0.724555\pi$
$647$	−32.9848			−1.29677			−0.648384	+	0.761313i	$0.724555\pi$
$648$		0			0
$649$			12.0818i			0.474252i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	32.7386			1.28116			0.640581	−	0.767891i	$-0.278694\pi$
$653$	32.7386			1.28116			0.640581	+	0.767891i	$0.278694\pi$
$654$		0			0
$655$			37.7327i			1.47434i
$656$		0			0
$657$		−	3.39228i		−	0.132346i
$658$		0			0
$659$		−	38.1045i		−	1.48434i	−0.670210	−	0.742171i	$-0.733796\pi$
$659$		−	38.1045i		−	1.48434i	0.670210	−	0.742171i	$-0.266204\pi$
$660$		0			0
$661$		−	2.64861i		−	0.103019i	−0.998673	−	0.0515096i	$-0.983597\pi$
$661$		−	2.64861i		−	0.103019i	0.998673	−	0.0515096i	$-0.0164033\pi$
$662$		0			0
$663$	26.2462			1.01932
$664$		0			0
$665$	−1.26137	+	4.87962i	−0.0489137	+	0.189223i
$666$		0			0
$667$			6.04090i			0.233904i
$668$		0			0
$669$	18.8769			0.729823
$670$		0			0
$671$	−28.4924			−1.09994
$672$		0			0
$673$	29.8617			1.15109			0.575543	−	0.817772i	$-0.304791\pi$
$673$	29.8617			1.15109			0.575543	+	0.817772i	$0.304791\pi$
$674$		0			0
$675$	−2.12311			−0.0817184
$676$		0			0
$677$			32.6443i			1.25462i	0.778769	+	0.627311i	$0.215844\pi$
$677$			32.6443i			1.25462i	−0.778769	+	0.627311i	$0.784156\pi$
$678$		0			0
$679$	−5.75379	+	22.2586i	−0.220810	+	0.854208i
$680$		0			0
$681$	−16.4924			−0.631991
$682$		0			0
$683$			29.0890i			1.11306i	0.830828	+	0.556529i	$0.187867\pi$
$683$			29.0890i			1.11306i	−0.830828	+	0.556529i	$0.812133\pi$
$684$		0			0
$685$		−	27.5559i		−	1.05286i
$686$		0			0
$687$			18.1227i			0.691424i
$688$		0			0
$689$		−	25.6509i		−	0.977222i
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$	2.00000	−	7.73704i	0.0759737	−	0.293906i
$694$		0			0
$695$		−	20.3537i		−	0.772060i
$696$		0			0
$697$	33.6155			1.27328
$698$		0			0
$699$	10.4924			0.396860
$700$		0			0
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$		−	17.3790i		−	0.654532i
$706$		0			0
$707$	−4.63068	+	17.9139i	−0.174155	+	0.673721i
$708$		0			0
$709$	−6.00000			−0.225335			−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$		−	4.71659i		−	0.176886i
$712$		0			0
$713$		0			0
$714$		0			0
$715$			30.9481i			1.15740i
$716$		0			0
$717$			2.27678i			0.0850279i
$718$		0			0
$719$	4.49242			0.167539			0.0837695	−	0.996485i	$-0.473304\pi$
$719$	4.49242			0.167539			0.0837695	+	0.996485i	$0.473304\pi$
$720$		0			0
$721$	−20.4924	−	5.29723i	−0.763178	−	0.197279i
$722$		0			0
$723$		−	1.90495i		−	0.0708457i
$724$		0			0
$725$	4.24621			0.157700
$726$		0			0
$727$	32.9848			1.22334			0.611670	−	0.791113i	$-0.290498\pi$
$727$	32.9848			1.22334			0.611670	+	0.791113i	$0.290498\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−35.2311			−1.30307
$732$		0			0
$733$		−	16.6354i		−	0.614441i	−0.951638	−	0.307220i	$-0.900601\pi$
$733$		−	16.6354i		−	0.614441i	0.951638	−	0.307220i	$-0.0993989\pi$
$734$		0			0
$735$	5.75379	−	10.3857i	0.212232	−	0.383080i
$736$		0			0
$737$	−6.24621			−0.230082
$738$		0			0
$739$			5.87787i			0.216221i	0.994139	+	0.108110i	$0.0344800\pi$
$739$			5.87787i			0.216221i	−0.994139	+	0.108110i	$0.965520\pi$
$740$		0			0
$741$		−	6.78456i		−	0.249237i
$742$		0			0
$743$			13.6149i			0.499482i	0.968313	+	0.249741i	$0.0803455\pi$
$743$			13.6149i			0.499482i	−0.968313	+	0.249741i	$0.919654\pi$
$744$		0			0
$745$			16.9614i			0.621418i
$746$		0			0
$747$	6.24621			0.228537
$748$		0			0
$749$	−3.75379	+	14.5216i	−0.137160	+	0.530608i
$750$		0			0
$751$		−	30.7851i		−	1.12336i	−0.827353	−	0.561682i	$-0.810154\pi$
$751$		−	30.7851i		−	1.12336i	0.827353	−	0.561682i	$-0.189846\pi$
$752$		0			0
$753$	22.2462			0.810697
$754$		0			0
$755$	13.7538			0.500552
$756$		0			0
$757$	−30.9848			−1.12616			−0.563082	−	0.826401i	$-0.690384\pi$
$757$	−30.9848			−1.12616			−0.563082	+	0.826401i	$0.690384\pi$
$758$		0			0
$759$	9.12311			0.331148
$760$		0			0
$761$		−	33.8056i		−	1.22545i	−0.790296	−	0.612725i	$-0.790073\pi$
$761$		−	33.8056i		−	1.22545i	0.790296	−	0.612725i	$-0.209927\pi$
$762$		0			0
$763$	21.1231	+	5.46026i	0.764708	+	0.197675i
$764$		0			0
$765$	7.36932			0.266438
$766$		0			0
$767$		−	24.1636i		−	0.872496i
$768$		0			0
$769$			44.5173i			1.60533i	0.596427	+	0.802667i	$0.296586\pi$
$769$			44.5173i			1.60533i	−0.596427	+	0.802667i	$0.703414\pi$
$770$		0			0
$771$		−	19.8188i		−	0.713758i
$772$		0			0
$773$			1.69614i			0.0610060i	0.999535	+	0.0305030i	$0.00971091\pi$
$773$			1.69614i			0.0610060i	−0.999535	+	0.0305030i	$0.990289\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	18.2462	+	4.71659i	0.654579	+	0.169207i
$778$		0			0
$779$		−	8.68951i		−	0.311334i
$780$		0			0
$781$	37.6155			1.34599
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$	7.01515			0.250382
$786$		0			0
$787$	20.9848			0.748029			0.374014	−	0.927423i	$-0.377981\pi$
$787$	20.9848			0.748029			0.374014	+	0.927423i	$0.377981\pi$
$788$		0			0
$789$		−	4.92539i		−	0.175349i
$790$		0			0
$791$	−31.3693	−	8.10887i	−1.11536	−	0.288318i
$792$		0			0
$793$	56.9848			2.02359
$794$		0			0
$795$		−	7.20217i		−	0.255435i
$796$		0			0
$797$			34.1316i			1.20900i	0.796604	+	0.604502i	$0.206628\pi$
$797$			34.1316i			1.20900i	−0.796604	+	0.604502i	$0.793372\pi$
$798$		0			0
$799$		−	44.5173i		−	1.57491i
$800$		0			0
$801$		−	7.73704i		−	0.273375i
$802$		0			0
$803$	10.2462			0.361581
$804$		0			0
$805$	13.1231	+	3.39228i	0.462529	+	0.119562i
$806$		0			0
$807$		−	22.4674i		−	0.790891i
$808$		0			0
$809$	−10.4924			−0.368894			−0.184447	−	0.982842i	$-0.559049\pi$
$809$	−10.4924			−0.368894			−0.184447	+	0.982842i	$0.559049\pi$
$810$		0			0
$811$	−32.4924			−1.14096			−0.570482	−	0.821310i	$-0.693243\pi$
$811$	−32.4924			−1.14096			−0.570482	+	0.821310i	$0.693243\pi$
$812$		0			0
$813$	4.49242			0.157556
$814$		0			0
$815$	19.5076			0.683321
$816$		0			0
$817$			9.10712i			0.318618i
$818$		0			0
$819$	−4.00000	+	15.4741i	−0.139771	+	0.540708i
$820$		0			0
$821$	−25.2311			−0.880570			−0.440285	−	0.897858i	$-0.645123\pi$
$821$	−25.2311			−0.880570			−0.440285	+	0.897858i	$0.645123\pi$
$822$		0			0
$823$			45.0979i			1.57201i	0.618217	+	0.786007i	$0.287855\pi$
$823$			45.0979i			1.57201i	−0.618217	+	0.786007i	$0.712145\pi$
$824$		0			0
$825$		−	6.41273i		−	0.223263i
$826$		0			0
$827$			30.9939i			1.07776i	0.842381	+	0.538882i	$0.181153\pi$
$827$			30.9939i			1.07776i	−0.842381	+	0.538882i	$0.818847\pi$
$828$		0			0
$829$			31.6918i			1.10070i	0.834933	+	0.550351i	$0.185506\pi$
$829$			31.6918i			1.10070i	−0.834933	+	0.550351i	$0.814494\pi$
$830$		0			0
$831$	3.12311			0.108339
$832$		0			0
$833$	14.7386	−	26.6034i	0.510663	−	0.921753i
$834$		0			0
$835$			3.80989i			0.131847i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6.73863			−0.232643			−0.116322	−	0.993212i	$-0.537110\pi$
$839$	−6.73863			−0.232643			−0.116322	+	0.993212i	$0.537110\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	0.246211			0.00847997
$844$		0			0
$845$		−	39.8465i		−	1.37076i
$846$		0			0
$847$	−4.80776	−	1.24279i	−0.165197	−	0.0427028i
$848$		0			0
$849$	−17.1231			−0.587663
$850$		0			0
$851$			21.5150i			0.737524i
$852$		0			0
$853$		−	37.4067i		−	1.28078i	−0.768050	−	0.640390i	$-0.778773\pi$
$853$		−	37.4067i		−	1.28078i	0.768050	−	0.640390i	$-0.221227\pi$
$854$		0			0
$855$		−	1.90495i		−	0.0651478i
$856$		0			0
$857$		−	20.2364i		−	0.691264i	−0.938370	−	0.345632i	$-0.887665\pi$
$857$		−	20.2364i		−	0.691264i	0.938370	−	0.345632i	$-0.112335\pi$
$858$		0			0
$859$	31.8617			1.08711			0.543554	−	0.839374i	$-0.317078\pi$
$859$	31.8617			1.08711			0.543554	+	0.839374i	$0.317078\pi$
$860$		0			0
$861$	−5.12311	+	19.8188i	−0.174595	+	0.675424i
$862$		0			0
$863$			21.8868i			0.745035i	0.928025	+	0.372518i	$0.121505\pi$
$863$			21.8868i			0.745035i	−0.928025	+	0.372518i	$0.878495\pi$
$864$		0			0
$865$	14.3845			0.489087
$866$		0			0
$867$	1.87689			0.0637427
$868$		0			0
$869$	14.2462			0.483270
$870$		0			0
$871$	12.4924			0.423290
$872$		0			0
$873$		−	8.68951i		−	0.294095i
$874$		0			0
$875$	8.00000	−	30.9481i	0.270449	−	1.04624i
$876$		0			0
$877$	−19.7538			−0.667038			−0.333519	−	0.942743i	$-0.608236\pi$
$877$	−19.7538			−0.667038			−0.333519	+	0.942743i	$0.608236\pi$
$878$		0			0
$879$			6.99337i			0.235880i
$880$		0			0
$881$			39.1028i			1.31741i	0.752403	+	0.658703i	$0.228895\pi$
$881$			39.1028i			1.31741i	−0.752403	+	0.658703i	$0.771105\pi$
$882$		0			0
$883$		−	26.9752i		−	0.907789i	−0.891056	−	0.453894i	$-0.850034\pi$
$883$		−	26.9752i		−	0.907789i	0.891056	−	0.453894i	$-0.149966\pi$
$884$		0			0
$885$		−	6.78456i		−	0.228061i
$886$		0			0
$887$	−22.7386			−0.763489			−0.381744	−	0.924268i	$-0.624677\pi$
$887$	−22.7386			−0.763489			−0.381744	+	0.924268i	$0.624677\pi$
$888$		0			0
$889$	12.8769	−	49.8145i	0.431877	−	1.67072i
$890$		0			0
$891$			3.02045i			0.101189i
$892$		0			0
$893$	−11.5076			−0.385086
$894$		0			0
$895$	3.86174			0.129084
$896$		0			0
$897$	−18.2462			−0.609223
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	18.4487i		−	0.614617i
$902$		0			0
$903$	5.36932	−	20.7713i	0.178680	−	0.691226i
$904$		0			0
$905$	−10.2462			−0.340596
$906$		0			0
$907$		−	11.9188i		−	0.395756i	−0.980227	−	0.197878i	$-0.936595\pi$
$907$		−	11.9188i		−	0.395756i	0.980227	−	0.197878i	$-0.0634050\pi$
$908$		0			0
$909$		−	6.99337i		−	0.231955i
$910$		0			0
$911$			6.41273i			0.212463i	0.994341	+	0.106232i	$0.0338785\pi$
$911$			6.41273i			0.212463i	−0.994341	+	0.106232i	$0.966122\pi$
$912$		0			0
$913$			18.8664i			0.624385i
$914$		0			0
$915$	16.0000			0.528944
$916$		0			0
$917$	−56.9848	−	14.7304i	−1.88181	−	0.486441i
$918$		0			0
$919$		−	8.10887i		−	0.267487i	−0.991016	−	0.133743i	$-0.957300\pi$
$919$		−	8.10887i		−	0.267487i	0.991016	−	0.133743i	$-0.0426998\pi$
$920$		0			0
$921$	21.6155			0.712256
$922$		0			0
$923$	−75.2311			−2.47626
$924$		0			0
$925$	15.1231			0.497245
$926$		0			0
$927$	8.00000			0.262754
$928$		0			0
$929$			50.7670i			1.66561i	0.553566	+	0.832805i	$0.313267\pi$
$929$			50.7670i			1.66561i	−0.553566	+	0.832805i	$0.686733\pi$
$930$		0			0
$931$	−6.87689	−	3.80989i	−0.225381	−	0.124864i
$932$		0			0
$933$	8.00000			0.261908
$934$		0			0
$935$			22.2586i			0.727935i
$936$		0			0
$937$		−	56.5991i		−	1.84901i	−0.381169	−	0.924505i	$-0.624478\pi$
$937$		−	56.5991i		−	1.84901i	0.381169	−	0.924505i	$-0.375522\pi$
$938$		0			0
$939$		−	25.6509i		−	0.837086i
$940$		0			0
$941$		−	13.3603i		−	0.435534i	−0.976001	−	0.217767i	$-0.930123\pi$
$941$		−	13.3603i		−	0.435534i	0.976001	−	0.217767i	$-0.0698773\pi$
$942$		0			0
$943$	−23.3693			−0.761010
$944$		0			0
$945$	−1.12311	+	4.34475i	−0.0365346	+	0.141335i
$946$		0			0
$947$			42.2405i			1.37263i	0.727304	+	0.686316i	$0.240773\pi$
$947$			42.2405i			1.37263i	−0.727304	+	0.686316i	$0.759227\pi$
$948$		0			0
$949$	−20.4924			−0.665212
$950$		0			0
$951$	18.4924			0.599658
$952$		0			0
$953$	61.2311			1.98347			0.991734	−	0.128309i	$-0.0409549\pi$
$953$	61.2311			1.98347			0.991734	+	0.128309i	$0.0409549\pi$
$954$		0			0
$955$	15.3693			0.497339
$956$		0			0
$957$		−	6.04090i		−	0.195274i
$958$		0			0
$959$	41.6155	+	10.7575i	1.34384	+	0.347377i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		−	5.66906i		−	0.182683i
$964$		0			0
$965$			15.8917i			0.511571i
$966$		0			0
$967$		−	40.9620i		−	1.31725i	−0.752472	−	0.658624i	$-0.771139\pi$
$967$		−	40.9620i		−	1.31725i	0.752472	−	0.658624i	$-0.228861\pi$
$968$		0			0
$969$		−	4.87962i		−	0.156756i
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$	30.7386	+	7.94584i	0.985435	+	0.254732i
$974$		0			0
$975$			12.8255i			0.410743i
$976$		0			0
$977$	−60.7386			−1.94320			−0.971601	−	0.236627i	$-0.923958\pi$
$977$	−60.7386			−1.94320			−0.971601	+	0.236627i	$0.923958\pi$
$978$		0			0
$979$	23.3693			0.746887
$980$		0			0
$981$	−8.24621			−0.263281
$982$		0			0
$983$	−54.7386			−1.74589			−0.872946	−	0.487818i	$-0.837793\pi$
$983$	−54.7386			−1.74589			−0.872946	+	0.487818i	$0.837793\pi$
$984$		0			0
$985$		−	0.417609i		−	0.0133061i
$986$		0			0
$987$	26.2462	+	6.78456i	0.835426	+	0.215955i
$988$		0			0
$989$	24.4924			0.778814
$990$		0			0
$991$			10.7575i			0.341723i	0.985295	+	0.170861i	$0.0546550\pi$
$991$			10.7575i			0.341723i	−0.985295	+	0.170861i	$0.945345\pi$
$992$		0			0
$993$		−	5.46026i		−	0.173276i
$994$		0			0
$995$		−	8.68951i		−	0.275476i
$996$		0			0
$997$			38.8940i			1.23178i	0.787830	+	0.615892i	$0.211204\pi$
$997$			38.8940i			1.23178i	−0.787830	+	0.615892i	$0.788796\pi$
$998$		0			0
$999$	−7.12311			−0.225365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.b.e.895.3		4
3.2	odd	2		4032.2.b.j.3583.2		4
4.3	odd	2		1344.2.b.f.895.3		4
7.6	odd	2		1344.2.b.f.895.2		4
8.3	odd	2		84.2.b.a.55.3	✓	4
8.5	even	2		84.2.b.b.55.4	yes	4
12.11	even	2		4032.2.b.n.3583.2		4
21.20	even	2		4032.2.b.n.3583.3		4
24.5	odd	2		252.2.b.d.55.1		4
24.11	even	2		252.2.b.e.55.2		4
28.27	even	2	inner	1344.2.b.e.895.2		4
56.3	even	6		588.2.o.a.19.3		8
56.5	odd	6		588.2.o.c.31.3		8
56.11	odd	6		588.2.o.c.19.3		8
56.13	odd	2		84.2.b.a.55.4	yes	4
56.19	even	6		588.2.o.a.31.1		8
56.27	even	2		84.2.b.b.55.3	yes	4
56.37	even	6		588.2.o.a.31.3		8
56.45	odd	6		588.2.o.c.19.1		8
56.51	odd	6		588.2.o.c.31.1		8
56.53	even	6		588.2.o.a.19.1		8
84.83	odd	2		4032.2.b.j.3583.3		4
168.83	odd	2		252.2.b.d.55.2		4
168.125	even	2		252.2.b.e.55.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.b.a.55.3	✓	4	8.3	odd	2
84.2.b.a.55.4	yes	4	56.13	odd	2
84.2.b.b.55.3	yes	4	56.27	even	2
84.2.b.b.55.4	yes	4	8.5	even	2
252.2.b.d.55.1		4	24.5	odd	2
252.2.b.d.55.2		4	168.83	odd	2
252.2.b.e.55.1		4	168.125	even	2
252.2.b.e.55.2		4	24.11	even	2
588.2.o.a.19.1		8	56.53	even	6
588.2.o.a.19.3		8	56.3	even	6
588.2.o.a.31.1		8	56.19	even	6
588.2.o.a.31.3		8	56.37	even	6
588.2.o.c.19.1		8	56.45	odd	6
588.2.o.c.19.3		8	56.11	odd	6
588.2.o.c.31.1		8	56.51	odd	6
588.2.o.c.31.3		8	56.5	odd	6
1344.2.b.e.895.2		4	28.27	even	2	inner
1344.2.b.e.895.3		4	1.1	even	1	trivial
1344.2.b.f.895.2		4	7.6	odd	2
1344.2.b.f.895.3		4	4.3	odd	2
4032.2.b.j.3583.2		4	3.2	odd	2
4032.2.b.j.3583.3		4	84.83	odd	2
4032.2.b.n.3583.2		4	12.11	even	2
4032.2.b.n.3583.3		4	21.20	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +1.00000 q^{9} +3.02045i q^{11} -6.04090i q^{13} -1.69614i q^{15} -4.34475i q^{17} -1.12311 q^{19} +(2.56155 + 0.662153i) q^{21} +3.02045i q^{23} +2.12311 q^{25} -1.00000 q^{27} +2.00000 q^{29} -3.02045i q^{33} +(1.12311 - 4.34475i) q^{35} +7.12311 q^{37} +6.04090i q^{39} +7.73704i q^{41} -8.10887i q^{43} +1.69614i q^{45} +10.2462 q^{47} +(6.12311 + 3.39228i) q^{49} +4.34475i q^{51} +4.24621 q^{53} -5.12311 q^{55} +1.12311 q^{57} +4.00000 q^{59} +9.43318i q^{61} +(-2.56155 - 0.662153i) q^{63} +10.2462 q^{65} +2.06798i q^{67} -3.02045i q^{69} -12.4536i q^{71} -3.39228i q^{73} -2.12311 q^{75} +(2.00000 - 7.73704i) q^{77} -4.71659i q^{79} +1.00000 q^{81} +6.24621 q^{83} +7.36932 q^{85} -2.00000 q^{87} -7.73704i q^{89} +(-4.00000 + 15.4741i) q^{91} -1.90495i q^{95} -8.68951i q^{97} +3.02045i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 2 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 2 q^{7} + 4 q^{9} + 12 q^{19} + 2 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{29} - 12 q^{35} + 12 q^{37} + 8 q^{47} + 8 q^{49} - 16 q^{53} - 4 q^{55} - 12 q^{57} + 16 q^{59} - 2 q^{63} + 8 q^{65} + 8 q^{75} + 8 q^{77} + 4 q^{81} - 8 q^{83} - 20 q^{85} - 8 q^{87} - 16 q^{91}+O(q^{100}) \) 4 * q - 4 * q^3 - 2 * q^7 + 4 * q^9 + 12 * q^19 + 2 * q^21 - 8 * q^25 - 4 * q^27 + 8 * q^29 - 12 * q^35 + 12 * q^37 + 8 * q^47 + 8 * q^49 - 16 * q^53 - 4 * q^55 - 12 * q^57 + 16 * q^59 - 2 * q^63 + 8 * q^65 + 8 * q^75 + 8 * q^77 + 4 * q^81 - 8 * q^83 - 20 * q^85 - 8 * q^87 - 16 * q^91

Introduction

L-functions

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