LMFDB - Embedded newform 912.2.d.b.191.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [912,2,Mod(191,912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("912.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.3
Character	$\chi$	$=$	912.191
Dual form			912.2.d.b.191.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.57777 - 0.714595i) q^{3} -1.85004i q^{5} -0.707624i q^{7} +(1.97871 + 2.25493i) q^{9} -4.63214 q^{11} -1.78223 q^{13} +(-1.32203 + 2.91894i) q^{15} -1.47660i q^{17} -1.00000i q^{19} +(-0.505665 + 1.11647i) q^{21} -2.68346 q^{23} +1.57733 q^{25} +(-1.51058 - 4.97174i) q^{27} +2.44439i q^{29} +6.63667i q^{31} +(7.30844 + 3.31010i) q^{33} -1.30914 q^{35} -3.04433 q^{37} +(2.81195 + 1.27357i) q^{39} +4.95675i q^{41} +6.80404i q^{43} +(4.17173 - 3.66070i) q^{45} -8.19420 q^{47} +6.49927 q^{49} +(-1.05517 + 2.32973i) q^{51} +3.89976i q^{53} +8.56966i q^{55} +(-0.714595 + 1.57777i) q^{57} -8.14537 q^{59} -11.9620 q^{61} +(1.59564 - 1.40018i) q^{63} +3.29720i q^{65} +4.84424i q^{67} +(4.23388 + 1.91759i) q^{69} +6.09961 q^{71} -15.0559 q^{73} +(-2.48867 - 1.12716i) q^{75} +3.27781i q^{77} -2.21403i q^{79} +(-1.16944 + 8.92370i) q^{81} +2.65489 q^{83} -2.73177 q^{85} +(1.74675 - 3.85669i) q^{87} +10.9884i q^{89} +1.26115i q^{91} +(4.74253 - 10.4711i) q^{93} -1.85004 q^{95} -9.45643 q^{97} +(-9.16564 - 10.4452i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) $ 24 * q + 4 * q^9 - 12 * q^21 - 64 * q^25 + 12 * q^33 + 64 * q^37 - 16 * q^45 - 8 * q^49 - 24 * q^61 - 8 * q^69 - 16 * q^73 - 4 * q^81 - 8 * q^85 + 32 * q^93 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−1.57777	−	0.714595i	−0.910925	−	0.412572i
$3$	−1.57777	−	0.714595i	−0.910925	−	0.412572i
$4$		0			0
$5$		−	1.85004i		−	0.827365i	−0.910421	−	0.413683i	$-0.864242\pi$
$5$		−	1.85004i		−	0.827365i	0.910421	−	0.413683i	$-0.135758\pi$
$6$		0			0
$7$		−	0.707624i		−	0.267457i	−0.991018	−	0.133728i	$-0.957305\pi$
$7$		−	0.707624i		−	0.267457i	0.991018	−	0.133728i	$-0.0426949\pi$
$8$		0			0
$9$	1.97871	+	2.25493i	0.659569	+	0.751644i
$10$		0			0
$11$	−4.63214			−1.39664			−0.698321	−	0.715785i	$-0.746069\pi$
$11$	−4.63214			−1.39664			−0.698321	+	0.715785i	$0.746069\pi$
$12$		0			0
$13$	−1.78223			−0.494302			−0.247151	−	0.968977i	$-0.579494\pi$
$13$	−1.78223			−0.494302			−0.247151	+	0.968977i	$0.579494\pi$
$14$		0			0
$15$	−1.32203	+	2.91894i	−0.341348	+	0.753668i
$16$		0			0
$17$		−	1.47660i		−	0.358128i	−0.983837	−	0.179064i	$-0.942693\pi$
$17$		−	1.47660i		−	0.358128i	0.983837	−	0.179064i	$-0.0573069\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$	−0.505665	+	1.11647i	−0.110345	+	0.243633i
$22$		0			0
$23$	−2.68346			−0.559540			−0.279770	−	0.960067i	$-0.590258\pi$
$23$	−2.68346			−0.559540			−0.279770	+	0.960067i	$0.590258\pi$
$24$		0			0
$25$	1.57733			0.315467
$26$		0			0
$27$	−1.51058	−	4.97174i	−0.290711	−	0.956811i
$28$		0			0
$29$			2.44439i			0.453913i	0.973905	+	0.226956i	$0.0728775\pi$
$29$			2.44439i			0.453913i	−0.973905	+	0.226956i	$0.927123\pi$
$30$		0			0
$31$			6.63667i			1.19198i	0.802991	+	0.595991i	$0.203241\pi$
$31$			6.63667i			1.19198i	−0.802991	+	0.595991i	$0.796759\pi$
$32$		0			0
$33$	7.30844	+	3.31010i	1.27224	+	0.576215i
$34$		0			0
$35$	−1.30914			−0.221284
$36$		0			0
$37$	−3.04433			−0.500484			−0.250242	−	0.968183i	$-0.580510\pi$
$37$	−3.04433			−0.500484			−0.250242	+	0.968183i	$0.580510\pi$
$38$		0			0
$39$	2.81195	+	1.27357i	0.450272	+	0.203935i
$40$		0			0
$41$			4.95675i			0.774114i	0.922056	+	0.387057i	$0.126508\pi$
$41$			4.95675i			0.774114i	−0.922056	+	0.387057i	$0.873492\pi$
$42$		0			0
$43$			6.80404i			1.03761i	0.854894	+	0.518803i	$0.173622\pi$
$43$			6.80404i			1.03761i	−0.854894	+	0.518803i	$0.826378\pi$
$44$		0			0
$45$	4.17173	−	3.66070i	0.621884	−	0.545704i
$46$		0			0
$47$	−8.19420			−1.19525			−0.597624	−	0.801777i	$-0.703888\pi$
$47$	−8.19420			−1.19525			−0.597624	+	0.801777i	$0.703888\pi$
$48$		0			0
$49$	6.49927			0.928467
$50$		0			0
$51$	−1.05517	+	2.32973i	−0.147753	+	0.326228i
$52$		0			0
$53$			3.89976i			0.535674i	0.963464	+	0.267837i	$0.0863088\pi$
$53$			3.89976i			0.535674i	−0.963464	+	0.267837i	$0.913691\pi$
$54$		0			0
$55$			8.56966i			1.15553i
$56$		0			0
$57$	−0.714595	+	1.57777i	−0.0946505	+	0.208981i
$58$		0			0
$59$	−8.14537			−1.06044			−0.530218	−	0.847861i	$-0.677890\pi$
$59$	−8.14537			−1.06044			−0.530218	+	0.847861i	$0.677890\pi$
$60$		0			0
$61$	−11.9620			−1.53158			−0.765792	−	0.643089i	$-0.777653\pi$
$61$	−11.9620			−1.53158			−0.765792	+	0.643089i	$0.777653\pi$
$62$		0			0
$63$	1.59564	−	1.40018i	0.201032	−	0.176406i
$64$		0			0
$65$			3.29720i			0.408968i
$66$		0			0
$67$			4.84424i			0.591818i	0.955216	+	0.295909i	$0.0956224\pi$
$67$			4.84424i			0.591818i	−0.955216	+	0.295909i	$0.904378\pi$
$68$		0			0
$69$	4.23388	+	1.91759i	0.509699	+	0.230850i
$70$		0			0
$71$	6.09961			0.723891			0.361945	−	0.932199i	$-0.382113\pi$
$71$	6.09961			0.723891			0.361945	+	0.932199i	$0.382113\pi$
$72$		0			0
$73$	−15.0559			−1.76215			−0.881077	−	0.472973i	$-0.843181\pi$
$73$	−15.0559			−1.76215			−0.881077	+	0.472973i	$0.843181\pi$
$74$		0			0
$75$	−2.48867	−	1.12716i	−0.287367	−	0.130153i
$76$		0			0
$77$			3.27781i			0.373541i
$78$		0			0
$79$		−	2.21403i		−	0.249098i	−0.992213	−	0.124549i	$-0.960252\pi$
$79$		−	2.21403i		−	0.249098i	0.992213	−	0.124549i	$-0.0397485\pi$
$80$		0			0
$81$	−1.16944	+	8.92370i	−0.129937	+	0.991522i
$82$		0			0
$83$	2.65489			0.291412			0.145706	−	0.989328i	$-0.453455\pi$
$83$	2.65489			0.291412			0.145706	+	0.989328i	$0.453455\pi$
$84$		0			0
$85$	−2.73177			−0.296302
$86$		0			0
$87$	1.74675	−	3.85669i	0.187272	−	0.413480i
$88$		0			0
$89$			10.9884i			1.16477i	0.812914	+	0.582384i	$0.197880\pi$
$89$			10.9884i			1.16477i	−0.812914	+	0.582384i	$0.802120\pi$
$90$		0			0
$91$			1.26115i			0.132204i
$92$		0			0
$93$	4.74253	−	10.4711i	0.491778	−	1.08581i
$94$		0			0
$95$	−1.85004			−0.189811
$96$		0			0
$97$	−9.45643			−0.960155			−0.480077	−	0.877226i	$-0.659391\pi$
$97$	−9.45643			−0.960155			−0.480077	+	0.877226i	$0.659391\pi$
$98$		0			0
$99$	−9.16564	−	10.4452i	−0.921182	−	1.04978i
$100$		0			0
$101$		−	3.11881i		−	0.310333i	−0.987888	−	0.155167i	$-0.950409\pi$
$101$		−	3.11881i		−	0.310333i	0.987888	−	0.155167i	$-0.0495915\pi$
$102$		0			0
$103$		−	15.3329i		−	1.51080i	−0.655264	−	0.755400i	$-0.727443\pi$
$103$		−	15.3329i		−	1.51080i	0.655264	−	0.755400i	$-0.272557\pi$
$104$		0			0
$105$	2.06551	+	0.935502i	0.201573	+	0.0912957i
$106$		0			0
$107$	13.4792			1.30308			0.651542	−	0.758613i	$-0.274122\pi$
$107$	13.4792			1.30308			0.651542	+	0.758613i	$0.274122\pi$
$108$		0			0
$109$	4.05566			0.388461			0.194231	−	0.980956i	$-0.437779\pi$
$109$	4.05566			0.388461			0.194231	+	0.980956i	$0.437779\pi$
$110$		0			0
$111$	4.80324	+	2.17546i	0.455904	+	0.206486i
$112$		0			0
$113$		−	9.64136i		−	0.906983i	−0.891261	−	0.453491i	$-0.850178\pi$
$113$		−	9.64136i		−	0.906983i	0.891261	−	0.453491i	$-0.149822\pi$
$114$		0			0
$115$			4.96452i			0.462944i
$116$		0			0
$117$	−3.52651	−	4.01881i	−0.326026	−	0.371539i
$118$		0			0
$119$	−1.04488			−0.0957837
$120$		0			0
$121$	10.4567			0.950608
$122$		0			0
$123$	3.54207	−	7.82060i	0.319378	−	0.705160i
$124$		0			0
$125$		−	12.1684i		−	1.08837i
$126$		0			0
$127$		−	3.70658i		−	0.328905i	−0.986385	−	0.164453i	$-0.947414\pi$
$127$		−	3.70658i		−	0.328905i	0.986385	−	0.164453i	$-0.0525858\pi$
$128$		0			0
$129$	4.86214	−	10.7352i	0.428087	−	0.945182i
$130$		0			0
$131$	−0.0121090			−0.00105797			−0.000528983	−	1.00000i	$-0.500168\pi$
$131$	−0.0121090			−0.00105797			−0.000528983		1.00000i	$0.500168\pi$
$132$		0			0
$133$	−0.707624			−0.0613588
$134$		0			0
$135$	−9.19793	+	2.79464i	−0.791632	+	0.240524i
$136$		0			0
$137$			16.5314i			1.41237i	0.708028	+	0.706184i	$0.249585\pi$
$137$			16.5314i			1.41237i	−0.708028	+	0.706184i	$0.750415\pi$
$138$		0			0
$139$			4.09796i			0.347585i	0.984782	+	0.173792i	$0.0556022\pi$
$139$			4.09796i			0.347585i	−0.984782	+	0.173792i	$0.944398\pi$
$140$		0			0
$141$	12.9286	+	5.85554i	1.08878	+	0.493125i
$142$		0			0
$143$	8.25553			0.690362
$144$		0			0
$145$	4.52224			0.375552
$146$		0			0
$147$	−10.2543	−	4.64435i	−0.845764	−	0.383059i
$148$		0			0
$149$		−	21.8009i		−	1.78600i	−0.450059	−	0.892999i	$-0.648597\pi$
$149$		−	21.8009i		−	1.78600i	0.450059	−	0.892999i	$-0.351403\pi$
$150$		0			0
$151$			18.6817i			1.52030i	0.649750	+	0.760148i	$0.274873\pi$
$151$			18.6817i			1.52030i	−0.649750	+	0.760148i	$0.725127\pi$
$152$		0			0
$153$	3.32963	−	2.92176i	0.269185	−	0.236210i
$154$		0			0
$155$	12.2781			0.986204
$156$		0			0
$157$	−10.1885			−0.813127			−0.406564	−	0.913622i	$-0.633273\pi$
$157$	−10.1885			−0.813127			−0.406564	+	0.913622i	$0.633273\pi$
$158$		0			0
$159$	2.78675	−	6.15292i	0.221004	−	0.487958i
$160$		0			0
$161$			1.89888i			0.149653i
$162$		0			0
$163$		−	5.10007i		−	0.399469i	−0.979850	−	0.199734i	$-0.935992\pi$
$163$		−	5.10007i		−	0.399469i	0.979850	−	0.199734i	$-0.0640079\pi$
$164$		0			0
$165$	6.12384	−	13.5209i	0.476740	−	1.05260i
$166$		0			0
$167$	12.8667			0.995655			0.497827	−	0.867276i	$-0.334131\pi$
$167$	12.8667			0.995655			0.497827	+	0.867276i	$0.334131\pi$
$168$		0			0
$169$	−9.82366			−0.755666
$170$		0			0
$171$	2.25493	−	1.97871i	0.172439	−	0.151316i
$172$		0			0
$173$		−	11.9225i		−	0.906454i	−0.891395	−	0.453227i	$-0.850273\pi$
$173$		−	11.9225i		−	0.906454i	0.891395	−	0.453227i	$-0.149727\pi$
$174$		0			0
$175$		−	1.11616i		−	0.0843737i
$176$		0			0
$177$	12.8515	+	5.82064i	0.965978	+	0.437506i
$178$		0			0
$179$	−1.28974			−0.0963997			−0.0481998	−	0.998838i	$-0.515348\pi$
$179$	−1.28974			−0.0963997			−0.0481998	+	0.998838i	$0.515348\pi$
$180$		0			0
$181$	−14.7046			−1.09298			−0.546491	−	0.837465i	$-0.684037\pi$
$181$	−14.7046			−1.09298			−0.546491	+	0.837465i	$0.684037\pi$
$182$		0			0
$183$	18.8733	+	8.54802i	1.39516	+	0.631888i
$184$		0			0
$185$			5.63214i			0.414083i
$186$		0			0
$187$			6.83981i			0.500176i
$188$		0			0
$189$	−3.51812	+	1.06892i	−0.255905	+	0.0777525i
$190$		0			0
$191$	−4.82987			−0.349477			−0.174739	−	0.984615i	$-0.555908\pi$
$191$	−4.82987			−0.349477			−0.174739	+	0.984615i	$0.555908\pi$
$192$		0			0
$193$	−2.27363			−0.163659			−0.0818297	−	0.996646i	$-0.526076\pi$
$193$	−2.27363			−0.163659			−0.0818297	+	0.996646i	$0.526076\pi$
$194$		0			0
$195$	2.35617	−	5.20223i	0.168729	−	0.372539i
$196$		0			0
$197$		−	18.5158i		−	1.31919i	−0.751619	−	0.659597i	$-0.770727\pi$
$197$		−	18.5158i		−	1.31919i	0.751619	−	0.659597i	$-0.229273\pi$
$198$		0			0
$199$		−	12.8189i		−	0.908710i	−0.890821	−	0.454355i	$-0.849870\pi$
$199$		−	12.8189i		−	0.908710i	0.890821	−	0.454355i	$-0.150130\pi$
$200$		0			0
$201$	3.46167	−	7.64308i	0.244167	−	0.539101i
$202$		0			0
$203$	1.72971			0.121402
$204$		0			0
$205$	9.17020			0.640475
$206$		0			0
$207$	−5.30978	−	6.05102i	−0.369055	−	0.420575i
$208$		0			0
$209$			4.63214i			0.320412i
$210$		0			0
$211$		−	15.1596i		−	1.04363i	−0.853059	−	0.521815i	$-0.825255\pi$
$211$		−	15.1596i		−	1.04363i	0.853059	−	0.521815i	$-0.174745\pi$
$212$		0			0
$213$	−9.62378	−	4.35875i	−0.659410	−	0.298657i
$214$		0			0
$215$	12.5878			0.858480
$216$		0			0
$217$	4.69627			0.318803
$218$		0			0
$219$	23.7546	+	10.7588i	1.60519	+	0.727015i
$220$		0			0
$221$			2.63164i			0.177023i
$222$		0			0
$223$		−	15.0080i		−	1.00501i	−0.864574	−	0.502505i	$-0.832412\pi$
$223$		−	15.0080i		−	1.00501i	0.864574	−	0.502505i	$-0.167588\pi$
$224$		0			0
$225$	3.12108	+	3.55678i	0.208072	+	0.237119i
$226$		0			0
$227$	19.5512			1.29766			0.648830	−	0.760933i	$-0.275259\pi$
$227$	19.5512			1.29766			0.648830	+	0.760933i	$0.275259\pi$
$228$		0			0
$229$	−14.3435			−0.947845			−0.473923	−	0.880566i	$-0.657162\pi$
$229$	−14.3435			−0.947845			−0.473923	+	0.880566i	$0.657162\pi$
$230$		0			0
$231$	2.34231	−	5.17162i	0.154112	−	0.340268i
$232$		0			0
$233$		−	0.893124i		−	0.0585105i	−0.999572	−	0.0292553i	$-0.990686\pi$
$233$		−	0.893124i		−	0.0585105i	0.999572	−	0.0292553i	$-0.00931357\pi$
$234$		0			0
$235$			15.1596i			0.988906i
$236$		0			0
$237$	−1.58214	+	3.49323i	−0.102771	+	0.226910i
$238$		0			0
$239$	−27.7339			−1.79396			−0.896978	−	0.442075i	$-0.854243\pi$
$239$	−27.7339			−1.79396			−0.896978	+	0.442075i	$0.854243\pi$
$240$		0			0
$241$	−25.8414			−1.66459			−0.832294	−	0.554334i	$-0.812973\pi$
$241$	−25.8414			−1.66459			−0.832294	+	0.554334i	$0.812973\pi$
$242$		0			0
$243$	8.22193	−	13.2439i	0.527437	−	0.849594i
$244$		0			0
$245$		−	12.0239i		−	0.768181i
$246$		0			0
$247$			1.78223i			0.113401i
$248$		0			0
$249$	−4.18881	−	1.89717i	−0.265455	−	0.120229i
$250$		0			0
$251$	3.39616			0.214363			0.107182	−	0.994239i	$-0.465817\pi$
$251$	3.39616			0.214363			0.107182	+	0.994239i	$0.465817\pi$
$252$		0			0
$253$	12.4301			0.781477
$254$		0			0
$255$	4.31011	+	1.95211i	0.269909	+	0.122246i
$256$		0			0
$257$		−	24.6927i		−	1.54029i	−0.637868	−	0.770145i	$-0.720184\pi$
$257$		−	24.6927i		−	1.54029i	0.637868	−	0.770145i	$-0.279816\pi$
$258$		0			0
$259$			2.15424i			0.133858i
$260$		0			0
$261$	−5.51194	+	4.83674i	−0.341181	+	0.299387i
$262$		0			0
$263$	−17.4827			−1.07803			−0.539014	−	0.842297i	$-0.681203\pi$
$263$	−17.4827			−1.07803			−0.539014	+	0.842297i	$0.681203\pi$
$264$		0			0
$265$	7.21473			0.443198
$266$		0			0
$267$	7.85226	−	17.3372i	0.480551	−	1.06102i
$268$		0			0
$269$			22.3903i			1.36516i	0.730809	+	0.682582i	$0.239143\pi$
$269$			22.3903i			1.36516i	−0.730809	+	0.682582i	$0.760857\pi$
$270$		0			0
$271$		−	31.0813i		−	1.88805i	−0.329868	−	0.944027i	$-0.607004\pi$
$271$		−	31.0813i		−	1.88805i	0.329868	−	0.944027i	$-0.392996\pi$
$272$		0			0
$273$	0.901210	−	1.98980i	0.0545437	−	0.120428i
$274$		0			0
$275$	−7.30643			−0.440594
$276$		0			0
$277$	−23.4125			−1.40672			−0.703360	−	0.710834i	$-0.748318\pi$
$277$	−23.4125			−1.40672			−0.703360	+	0.710834i	$0.748318\pi$
$278$		0			0
$279$	−14.9652	+	13.1320i	−0.895946	+	0.786194i
$280$		0			0
$281$			28.8664i			1.72203i	0.508583	+	0.861013i	$0.330169\pi$
$281$			28.8664i			1.72203i	−0.508583	+	0.861013i	$0.669831\pi$
$282$		0			0
$283$			10.8054i			0.642317i	0.947025	+	0.321158i	$0.104072\pi$
$283$			10.8054i			0.642317i	−0.947025	+	0.321158i	$0.895928\pi$
$284$		0			0
$285$	2.91894	+	1.32203i	0.172903	+	0.0783105i
$286$		0			0
$287$	3.50751			0.207042
$288$		0			0
$289$	14.8197			0.871744
$290$		0			0
$291$	14.9201	+	6.75752i	0.874629	+	0.396133i
$292$		0			0
$293$		−	0.514137i		−	0.0300362i	−0.999887	−	0.0150181i	$-0.995219\pi$
$293$		−	0.514137i		−	0.0300362i	0.999887	−	0.0150181i	$-0.00478059\pi$
$294$		0			0
$295$			15.0693i			0.877368i
$296$		0			0
$297$	6.99720	+	23.0298i	0.406019	+	1.33632i
$298$		0			0
$299$	4.78254			0.276581
$300$		0			0
$301$	4.81470			0.277515
$302$		0			0
$303$	−2.22869	+	4.92076i	−0.128035	+	0.282690i
$304$		0			0
$305$			22.1303i			1.26718i
$306$		0			0
$307$			20.2588i			1.15623i	0.815954	+	0.578116i	$0.196212\pi$
$307$			20.2588i			1.15623i	−0.815954	+	0.578116i	$0.803788\pi$
$308$		0			0
$309$	−10.9568	+	24.1918i	−0.623313	+	1.37623i
$310$		0			0
$311$	29.4007			1.66716			0.833581	−	0.552397i	$-0.186287\pi$
$311$	29.4007			1.66716			0.833581	+	0.552397i	$0.186287\pi$
$312$		0			0
$313$	7.82563			0.442331			0.221165	−	0.975236i	$-0.429014\pi$
$313$	7.82563			0.442331			0.221165	+	0.975236i	$0.429014\pi$
$314$		0			0
$315$	−2.59040	−	2.95201i	−0.145952	−	0.166327i
$316$		0			0
$317$		−	8.41538i		−	0.472655i	−0.971674	−	0.236327i	$-0.924056\pi$
$317$		−	8.41538i		−	0.472655i	0.971674	−	0.236327i	$-0.0759438\pi$
$318$		0			0
$319$		−	11.3228i		−	0.633953i
$320$		0			0
$321$	−21.2671	−	9.63218i	−1.18701	−	0.537616i
$322$		0			0
$323$	−1.47660			−0.0821602
$324$		0			0
$325$	−2.81117			−0.155936
$326$		0			0
$327$	−6.39889	−	2.89815i	−0.353859	−	0.160268i
$328$		0			0
$329$			5.79841i			0.319677i
$330$		0			0
$331$			26.7185i			1.46858i	0.678836	+	0.734290i	$0.262485\pi$
$331$			26.7185i			1.46858i	−0.678836	+	0.734290i	$0.737515\pi$
$332$		0			0
$333$	−6.02383	−	6.86475i	−0.330104	−	0.376186i
$334$		0			0
$335$	8.96205			0.489649
$336$		0			0
$337$	−14.1242			−0.769395			−0.384697	−	0.923043i	$-0.625694\pi$
$337$	−14.1242			−0.769395			−0.384697	+	0.923043i	$0.625694\pi$
$338$		0			0
$339$	−6.88967	+	15.2118i	−0.374196	+	0.826194i
$340$		0			0
$341$		−	30.7420i		−	1.66477i
$342$		0			0
$343$		−	9.55240i		−	0.515781i
$344$		0			0
$345$	3.54762	−	7.83286i	0.190998	−	0.421707i
$346$		0			0
$347$	−4.67942			−0.251204			−0.125602	−	0.992081i	$-0.540086\pi$
$347$	−4.67942			−0.251204			−0.125602	+	0.992081i	$0.540086\pi$
$348$		0			0
$349$	1.52611			0.0816911			0.0408455	−	0.999165i	$-0.486995\pi$
$349$	1.52611			0.0816911			0.0408455	+	0.999165i	$0.486995\pi$
$350$		0			0
$351$	2.69220	+	8.86077i	0.143699	+	0.472953i
$352$		0			0
$353$		−	22.6720i		−	1.20671i	−0.797474	−	0.603354i	$-0.793831\pi$
$353$		−	22.6720i		−	1.20671i	0.797474	−	0.603354i	$-0.206169\pi$
$354$		0			0
$355$		−	11.2846i		−	0.598922i
$356$		0			0
$357$	1.64857	+	0.746664i	0.0872517	+	0.0395176i
$358$		0			0
$359$	−19.7942			−1.04470			−0.522350	−	0.852731i	$-0.674945\pi$
$359$	−19.7942			−1.04470			−0.522350	+	0.852731i	$0.674945\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	−16.4982	−	7.47230i	−0.865932	−	0.392194i
$364$		0			0
$365$			27.8540i			1.45794i
$366$		0			0
$367$		−	24.7226i		−	1.29051i	−0.763967	−	0.645255i	$-0.776751\pi$
$367$		−	24.7226i		−	1.29051i	0.763967	−	0.645255i	$-0.223249\pi$
$368$		0			0
$369$	−11.1771	+	9.80795i	−0.581858	+	0.510582i
$370$		0			0
$371$	2.75956			0.143269
$372$		0			0
$373$	27.9194			1.44561			0.722805	−	0.691052i	$-0.242852\pi$
$373$	27.9194			1.44561			0.722805	+	0.691052i	$0.242852\pi$
$374$		0			0
$375$	−8.69545	+	19.1989i	−0.449031	+	0.991425i
$376$		0			0
$377$		−	4.35647i		−	0.224370i
$378$		0			0
$379$			24.4837i			1.25764i	0.777550	+	0.628821i	$0.216462\pi$
$379$			24.4837i			1.25764i	−0.777550	+	0.628821i	$0.783538\pi$
$380$		0			0
$381$	−2.64870	+	5.84812i	−0.135697	+	0.299608i
$382$		0			0
$383$	−13.9416			−0.712382			−0.356191	−	0.934413i	$-0.615925\pi$
$383$	−13.9416			−0.712382			−0.356191	+	0.934413i	$0.615925\pi$
$384$		0			0
$385$	6.06409			0.309055
$386$		0			0
$387$	−15.3427	+	13.4632i	−0.779911	+	0.684373i
$388$		0			0
$389$			9.11256i			0.462025i	0.972951	+	0.231013i	$0.0742039\pi$
$389$			9.11256i			0.462025i	−0.972951	+	0.231013i	$0.925796\pi$
$390$		0			0
$391$			3.96239i			0.200387i
$392$		0			0
$393$	0.0191051	+	0.00865301i	0.000963727	+	0.000436487i
$394$		0			0
$395$	−4.09606			−0.206095
$396$		0			0
$397$	7.45405			0.374108			0.187054	−	0.982350i	$-0.440106\pi$
$397$	7.45405			0.374108			0.187054	+	0.982350i	$0.440106\pi$
$398$		0			0
$399$	1.11647	+	0.505665i	0.0558932	+	0.0253149i
$400$		0			0
$401$		−	30.3076i		−	1.51349i	−0.653711	−	0.756744i	$-0.726789\pi$
$401$		−	30.3076i		−	1.51349i	0.653711	−	0.756744i	$-0.273211\pi$
$402$		0			0
$403$		−	11.8281i		−	0.589198i
$404$		0			0
$405$	16.5092	+	2.16351i	0.820351	+	0.107506i
$406$		0			0
$407$	14.1017			0.698997
$408$		0			0
$409$	34.7359			1.71758			0.858790	−	0.512328i	$-0.171217\pi$
$409$	34.7359			1.71758			0.858790	+	0.512328i	$0.171217\pi$
$410$		0			0
$411$	11.8132	−	26.0826i	0.582703	−	1.28656i
$412$		0			0
$413$			5.76385i			0.283621i
$414$		0			0
$415$		−	4.91167i		−	0.241105i
$416$		0			0
$417$	2.92839	−	6.46564i	0.143404	−	0.316624i
$418$		0			0
$419$	−19.9232			−0.973314			−0.486657	−	0.873593i	$-0.661784\pi$
$419$	−19.9232			−0.973314			−0.486657	+	0.873593i	$0.661784\pi$
$420$		0			0
$421$	−9.25156			−0.450893			−0.225447	−	0.974256i	$-0.572384\pi$
$421$	−9.25156			−0.450893			−0.225447	+	0.974256i	$0.572384\pi$
$422$		0			0
$423$	−16.2139	−	18.4774i	−0.788348	−	0.898400i
$424$		0			0
$425$		−	2.32909i		−	0.112977i
$426$		0			0
$427$			8.46463i			0.409632i
$428$		0			0
$429$	−13.0253	−	5.89936i	−0.628868	−	0.284824i
$430$		0			0
$431$	−17.6084			−0.848169			−0.424084	−	0.905623i	$-0.639404\pi$
$431$	−17.6084			−0.848169			−0.424084	+	0.905623i	$0.639404\pi$
$432$		0			0
$433$	23.4409			1.12650			0.563248	−	0.826288i	$-0.309551\pi$
$433$	23.4409			1.12650			0.563248	+	0.826288i	$0.309551\pi$
$434$		0			0
$435$	−7.13505	−	3.23157i	−0.342099	−	0.154942i
$436$		0			0
$437$			2.68346i			0.128367i
$438$		0			0
$439$			10.9049i			0.520461i	0.965547	+	0.260231i	$0.0837986\pi$
$439$			10.9049i			0.520461i	−0.965547	+	0.260231i	$0.916201\pi$
$440$		0			0
$441$	12.8601	+	14.6554i	0.612388	+	0.697877i
$442$		0			0
$443$	4.08004			0.193848			0.0969242	−	0.995292i	$-0.469100\pi$
$443$	4.08004			0.193848			0.0969242	+	0.995292i	$0.469100\pi$
$444$		0			0
$445$	20.3290			0.963689
$446$		0			0
$447$	−15.5788	+	34.3967i	−0.736852	+	1.62691i
$448$		0			0
$449$			27.2291i			1.28502i	0.766277	+	0.642511i	$0.222107\pi$
$449$			27.2291i			1.28502i	−0.766277	+	0.642511i	$0.777893\pi$
$450$		0			0
$451$		−	22.9603i		−	1.08116i
$452$		0			0
$453$	13.3499	−	29.4754i	0.627231	−	1.38487i
$454$		0			0
$455$	2.33318			0.109381
$456$		0			0
$457$	6.10668			0.285658			0.142829	−	0.989747i	$-0.454380\pi$
$457$	6.10668			0.285658			0.142829	+	0.989747i	$0.454380\pi$
$458$		0			0
$459$	−7.34126	+	2.23052i	−0.342661	+	0.104112i
$460$		0			0
$461$			17.8562i			0.831648i	0.909445	+	0.415824i	$0.136507\pi$
$461$			17.8562i			0.831648i	−0.909445	+	0.415824i	$0.863493\pi$
$462$		0			0
$463$			14.1434i			0.657299i	0.944452	+	0.328649i	$0.106593\pi$
$463$			14.1434i			0.657299i	−0.944452	+	0.328649i	$0.893407\pi$
$464$		0			0
$465$	−19.3721	−	8.77390i	−0.898358	−	0.406880i
$466$		0			0
$467$	−40.7322			−1.88486			−0.942430	−	0.334404i	$-0.891465\pi$
$467$	−40.7322			−1.88486			−0.942430	+	0.334404i	$0.891465\pi$
$468$		0			0
$469$	3.42790			0.158286
$470$		0			0
$471$	16.0750	+	7.28062i	0.740698	+	0.335473i
$472$		0			0
$473$		−	31.5172i		−	1.44916i
$474$		0			0
$475$		−	1.57733i		−	0.0723730i
$476$		0			0
$477$	−8.79370	+	7.71649i	−0.402636	+	0.353314i
$478$		0			0
$479$	1.94900			0.0890519			0.0445260	−	0.999008i	$-0.485822\pi$
$479$	1.94900			0.0890519			0.0445260	+	0.999008i	$0.485822\pi$
$480$		0			0
$481$	5.42569			0.247390
$482$		0			0
$483$	1.35693	−	2.99599i	0.0617425	−	0.136322i
$484$		0			0
$485$			17.4948i			0.794399i
$486$		0			0
$487$			1.97691i			0.0895824i	0.998996	+	0.0447912i	$0.0142622\pi$
$487$			1.97691i			0.0895824i	−0.998996	+	0.0447912i	$0.985738\pi$
$488$		0			0
$489$	−3.64449	+	8.04674i	−0.164809	+	0.363886i
$490$		0			0
$491$	−8.44538			−0.381134			−0.190567	−	0.981674i	$-0.561033\pi$
$491$	−8.44538			−0.381134			−0.190567	+	0.981674i	$0.561033\pi$
$492$		0			0
$493$	3.60939			0.162559
$494$		0			0
$495$	−19.3240	+	16.9568i	−0.868549	+	0.762154i
$496$		0			0
$497$		−	4.31623i		−	0.193609i
$498$		0			0
$499$			17.5057i			0.783664i	0.920037	+	0.391832i	$0.128159\pi$
$499$			17.5057i			0.783664i	−0.920037	+	0.391832i	$0.871841\pi$
$500$		0			0
$501$	−20.3007	−	9.19448i	−0.906967	−	0.410779i
$502$		0			0
$503$	−10.6802			−0.476209			−0.238104	−	0.971240i	$-0.576526\pi$
$503$	−10.6802			−0.476209			−0.238104	+	0.971240i	$0.576526\pi$
$504$		0			0
$505$	−5.76994			−0.256759
$506$		0			0
$507$	15.4995	+	7.01994i	0.688355	+	0.311766i
$508$		0			0
$509$			39.2121i			1.73804i	0.494773	+	0.869022i	$0.335251\pi$
$509$			39.2121i			1.73804i	−0.494773	+	0.869022i	$0.664749\pi$
$510$		0			0
$511$			10.6539i			0.471300i
$512$		0			0
$513$	−4.97174	+	1.51058i	−0.219507	+	0.0666937i
$514$		0			0
$515$	−28.3666			−1.24998
$516$		0			0
$517$	37.9567			1.66933
$518$		0			0
$519$	−8.51979	+	18.8110i	−0.373977	+	0.825711i
$520$		0			0
$521$			35.9655i			1.57568i	0.615881	+	0.787839i	$0.288800\pi$
$521$			35.9655i			1.57568i	−0.615881	+	0.787839i	$0.711200\pi$
$522$		0			0
$523$		−	20.0443i		−	0.876477i	−0.898859	−	0.438238i	$-0.855603\pi$
$523$		−	20.0443i		−	0.876477i	0.898859	−	0.438238i	$-0.144397\pi$
$524$		0			0
$525$	−0.797602	+	1.76104i	−0.0348102	+	0.0768581i
$526$		0			0
$527$	9.79970			0.426882
$528$		0			0
$529$	−15.7990			−0.686915
$530$		0			0
$531$	−16.1173	−	18.3672i	−0.699431	−	0.797071i
$532$		0			0
$533$		−	8.83406i		−	0.382646i
$534$		0			0
$535$		−	24.9371i		−	1.07813i
$536$		0			0
$537$	2.03491	+	0.921642i	0.0878129	+	0.0397718i
$538$		0			0
$539$	−30.1055			−1.29674
$540$		0			0
$541$	−4.27173			−0.183656			−0.0918280	−	0.995775i	$-0.529271\pi$
$541$	−4.27173			−0.183656			−0.0918280	+	0.995775i	$0.529271\pi$
$542$		0			0
$543$	23.2004	+	10.5078i	0.995624	+	0.450933i
$544$		0			0
$545$		−	7.50314i		−	0.321399i
$546$		0			0
$547$			7.68080i			0.328407i	0.986426	+	0.164204i	$0.0525054\pi$
$547$			7.68080i			0.328407i	−0.986426	+	0.164204i	$0.947495\pi$
$548$		0			0
$549$	−23.6694	−	26.9736i	−1.01018	−	1.15121i
$550$		0			0
$551$	2.44439			0.104135
$552$		0			0
$553$	−1.56670			−0.0666230
$554$		0			0
$555$	4.02470	−	8.88621i	0.170839	−	0.377199i
$556$		0			0
$557$			44.2479i			1.87484i	0.348197	+	0.937421i	$0.386794\pi$
$557$			44.2479i			1.87484i	−0.348197	+	0.937421i	$0.613206\pi$
$558$		0			0
$559$		−	12.1264i		−	0.512891i
$560$		0			0
$561$	4.88769	−	10.7916i	0.206359	−	0.455623i
$562$		0			0
$563$	36.1114			1.52191			0.760956	−	0.648803i	$-0.224730\pi$
$563$	36.1114			1.52191			0.760956	+	0.648803i	$0.224730\pi$
$564$		0			0
$565$	−17.8370			−0.750406
$566$		0			0
$567$	6.31462	+	0.827521i	0.265189	+	0.0347526i
$568$		0			0
$569$		−	8.27441i		−	0.346881i	−0.984844	−	0.173441i	$-0.944511\pi$
$569$		−	8.27441i		−	0.346881i	0.984844	−	0.173441i	$-0.0554885\pi$
$570$		0			0
$571$		−	27.3137i		−	1.14304i	−0.820587	−	0.571521i	$-0.806353\pi$
$571$		−	27.3137i		−	1.14304i	0.820587	−	0.571521i	$-0.193647\pi$
$572$		0			0
$573$	7.62042	+	3.45140i	0.318347	+	0.144184i
$574$		0			0
$575$	−4.23271			−0.176516
$576$		0			0
$577$	−11.1555			−0.464408			−0.232204	−	0.972667i	$-0.574594\pi$
$577$	−11.1555			−0.464408			−0.232204	+	0.972667i	$0.574594\pi$
$578$		0			0
$579$	3.58726	+	1.62472i	0.149081	+	0.0675212i
$580$		0			0
$581$		−	1.87867i		−	0.0779402i
$582$		0			0
$583$		−	18.0642i		−	0.748144i
$584$		0			0
$585$	−7.43497	+	6.52420i	−0.307398	+	0.269743i
$586$		0			0
$587$	−23.7699			−0.981089			−0.490545	−	0.871416i	$-0.663202\pi$
$587$	−23.7699			−0.981089			−0.490545	+	0.871416i	$0.663202\pi$
$588$		0			0
$589$	6.63667			0.273459
$590$		0			0
$591$	−13.2313	+	29.2136i	−0.544262	+	1.20169i
$592$		0			0
$593$		−	27.1707i		−	1.11577i	−0.829920	−	0.557883i	$-0.811614\pi$
$593$		−	27.1707i		−	1.11577i	0.829920	−	0.557883i	$-0.188386\pi$
$594$		0			0
$595$			1.93307i			0.0792481i
$596$		0			0
$597$	−9.16035	+	20.2253i	−0.374908	+	0.827767i
$598$		0			0
$599$	−15.2016			−0.621122			−0.310561	−	0.950554i	$-0.600517\pi$
$599$	−15.2016			−0.621122			−0.310561	+	0.950554i	$0.600517\pi$
$600$		0			0
$601$	17.7396			0.723614			0.361807	−	0.932253i	$-0.382160\pi$
$601$	17.7396			0.723614			0.361807	+	0.932253i	$0.382160\pi$
$602$		0			0
$603$	−10.9234	+	9.58532i	−0.444836	+	0.390345i
$604$		0			0
$605$		−	19.3453i		−	0.786500i
$606$		0			0
$607$		−	1.00873i		−	0.0409429i	−0.999790	−	0.0204715i	$-0.993483\pi$
$607$		−	1.00873i		−	0.0409429i	0.999790	−	0.0204715i	$-0.00651672\pi$
$608$		0			0
$609$	−2.72908	−	1.23604i	−0.110588	−	0.0500870i
$610$		0			0
$611$	14.6039			0.590813
$612$		0			0
$613$	45.7234			1.84675			0.923375	−	0.383899i	$-0.125419\pi$
$613$	45.7234			1.84675			0.923375	+	0.383899i	$0.125419\pi$
$614$		0			0
$615$	−14.4685	−	6.55299i	−0.583425	−	0.264242i
$616$		0			0
$617$			7.79096i			0.313652i	0.987626	+	0.156826i	$0.0501262\pi$
$617$			7.79096i			0.313652i	−0.987626	+	0.156826i	$0.949874\pi$
$618$		0			0
$619$			18.8993i			0.759628i	0.925063	+	0.379814i	$0.124012\pi$
$619$			18.8993i			0.759628i	−0.925063	+	0.379814i	$0.875988\pi$
$620$		0			0
$621$	4.05358	+	13.3415i	0.162664	+	0.535374i
$622$		0			0
$623$	7.77565			0.311525
$624$		0			0
$625$	−14.6253			−0.585014
$626$		0			0
$627$	3.31010	−	7.30844i	0.132193	−	0.291871i
$628$		0			0
$629$			4.49525i			0.179237i
$630$		0			0
$631$		−	12.4573i		−	0.495918i	−0.968771	−	0.247959i	$-0.920240\pi$
$631$		−	12.4573i		−	0.495918i	0.968771	−	0.247959i	$-0.0797598\pi$
$632$		0			0
$633$	−10.8330	+	23.9183i	−0.430572	+	0.950668i
$634$		0			0
$635$	−6.85733			−0.272125
$636$		0			0
$637$	−11.5832			−0.458943
$638$		0			0
$639$	12.0693	+	13.7542i	0.477456	+	0.544108i
$640$		0			0
$641$			9.64878i			0.381104i	0.981677	+	0.190552i	$0.0610278\pi$
$641$			9.64878i			0.381104i	−0.981677	+	0.190552i	$0.938972\pi$
$642$		0			0
$643$		−	11.1740i		−	0.440659i	−0.975426	−	0.220329i	$-0.929287\pi$
$643$		−	11.1740i		−	0.440659i	0.975426	−	0.220329i	$-0.0707132\pi$
$644$		0			0
$645$	−19.8606	−	8.99517i	−0.782011	−	0.354185i
$646$		0			0
$647$	38.9594			1.53165			0.765826	−	0.643048i	$-0.222330\pi$
$647$	38.9594			1.53165			0.765826	+	0.643048i	$0.222330\pi$
$648$		0			0
$649$	37.7304			1.48105
$650$		0			0
$651$	−7.40962	−	3.35593i	−0.290406	−	0.131529i
$652$		0			0
$653$			9.77338i			0.382462i	0.981545	+	0.191231i	$0.0612479\pi$
$653$			9.77338i			0.382462i	−0.981545	+	0.191231i	$0.938752\pi$
$654$		0			0
$655$			0.0224021i			0.000875324i
$656$		0			0
$657$	−29.7911	−	33.9499i	−1.16226	−	1.32451i
$658$		0			0
$659$	8.05188			0.313657			0.156828	−	0.987626i	$-0.449873\pi$
$659$	8.05188			0.313657			0.156828	+	0.987626i	$0.449873\pi$
$660$		0			0
$661$	13.6768			0.531967			0.265984	−	0.963978i	$-0.414303\pi$
$661$	13.6768			0.531967			0.265984	+	0.963978i	$0.414303\pi$
$662$		0			0
$663$	1.88056	−	4.15212i	0.0730348	−	0.161255i
$664$		0			0
$665$			1.30914i			0.0507661i
$666$		0			0
$667$		−	6.55943i		−	0.253982i
$668$		0			0
$669$	−10.7247	+	23.6792i	−0.414639	+	0.915489i
$670$		0			0
$671$	55.4098			2.13907
$672$		0			0
$673$	−34.9298			−1.34644			−0.673221	−	0.739441i	$-0.735090\pi$
$673$	−34.9298			−1.34644			−0.673221	+	0.739441i	$0.735090\pi$
$674$		0			0
$675$	−2.38269	−	7.84209i	−0.0917096	−	0.301842i
$676$		0			0
$677$			43.3133i			1.66466i	0.554277	+	0.832332i	$0.312995\pi$
$677$			43.3133i			1.66466i	−0.554277	+	0.832332i	$0.687005\pi$
$678$		0			0
$679$			6.69159i			0.256800i
$680$		0			0
$681$	−30.8473	−	13.9712i	−1.18207	−	0.535378i
$682$		0			0
$683$	−6.37903			−0.244087			−0.122043	−	0.992525i	$-0.538945\pi$
$683$	−6.37903			−0.244087			−0.122043	+	0.992525i	$0.538945\pi$
$684$		0			0
$685$	30.5837			1.16854
$686$		0			0
$687$	22.6307	+	10.2498i	0.863416	+	0.391054i
$688$		0			0
$689$		−	6.95027i		−	0.264784i
$690$		0			0
$691$			32.0940i			1.22091i	0.792050	+	0.610456i	$0.209014\pi$
$691$			32.0940i			1.22091i	−0.792050	+	0.610456i	$0.790986\pi$
$692$		0			0
$693$	−7.39124	+	6.48582i	−0.280770	+	0.246376i
$694$		0			0
$695$	7.58142			0.287580
$696$		0			0
$697$	7.31913			0.277232
$698$		0			0
$699$	−0.638222	+	1.40914i	−0.0241398	+	0.0532987i
$700$		0			0
$701$		−	35.5578i		−	1.34300i	−0.741004	−	0.671500i	$-0.765650\pi$
$701$		−	35.5578i		−	1.34300i	0.741004	−	0.671500i	$-0.234350\pi$
$702$		0			0
$703$			3.04433i			0.114819i
$704$		0			0
$705$	10.8330	−	23.9184i	0.407995	−	0.900819i
$706$		0			0
$707$	−2.20694			−0.0830007
$708$		0			0
$709$	−8.04768			−0.302237			−0.151118	−	0.988516i	$-0.548287\pi$
$709$	−8.04768			−0.302237			−0.151118	+	0.988516i	$0.548287\pi$
$710$		0			0
$711$	4.99250	−	4.38093i	0.187233	−	0.164298i
$712$		0			0
$713$		−	17.8092i		−	0.666961i
$714$		0			0
$715$		−	15.2731i		−	0.571182i
$716$		0			0
$717$	43.7577	+	19.8185i	1.63416	+	0.740136i
$718$		0			0
$719$	−28.8564			−1.07616			−0.538082	−	0.842893i	$-0.680851\pi$
$719$	−28.8564			−1.07616			−0.538082	+	0.842893i	$0.680851\pi$
$720$		0			0
$721$	−10.8500			−0.404073
$722$		0			0
$723$	40.7717	+	18.4661i	1.51632	+	0.686762i
$724$		0			0
$725$			3.85563i			0.143194i
$726$		0			0
$727$			33.8395i			1.25504i	0.778602	+	0.627518i	$0.215929\pi$
$727$			33.8395i			1.25504i	−0.778602	+	0.627518i	$0.784071\pi$
$728$		0			0
$729$	−22.4363	+	15.0204i	−0.830974	+	0.556311i
$730$		0			0
$731$	10.0468			0.371596
$732$		0			0
$733$	−23.6799			−0.874639			−0.437319	−	0.899306i	$-0.644072\pi$
$733$	−23.6799			−0.874639			−0.437319	+	0.899306i	$0.644072\pi$
$734$		0			0
$735$	−8.59225	+	18.9710i	−0.316930	+	0.699756i
$736$		0			0
$737$		−	22.4392i		−	0.826557i
$738$		0			0
$739$			37.9341i			1.39543i	0.716375	+	0.697715i	$0.245800\pi$
$739$			37.9341i			1.39543i	−0.716375	+	0.697715i	$0.754200\pi$
$740$		0			0
$741$	1.27357	−	2.81195i	0.0467859	−	0.103299i
$742$		0			0
$743$	7.47556			0.274252			0.137126	−	0.990554i	$-0.456214\pi$
$743$	7.47556			0.274252			0.137126	+	0.990554i	$0.456214\pi$
$744$		0			0
$745$	−40.3326			−1.47767
$746$		0			0
$747$	5.25326	+	5.98660i	0.192207	+	0.219038i
$748$		0			0
$749$		−	9.53820i		−	0.348518i
$750$		0			0
$751$		−	52.8763i		−	1.92948i	−0.263198	−	0.964742i	$-0.584777\pi$
$751$		−	52.8763i		−	1.92948i	0.263198	−	0.964742i	$-0.415223\pi$
$752$		0			0
$753$	−5.35835	−	2.42688i	−0.195269	−	0.0884403i
$754$		0			0
$755$	34.5620			1.25784
$756$		0			0
$757$	3.02847			0.110071			0.0550357	−	0.998484i	$-0.482473\pi$
$757$	3.02847			0.110071			0.0550357	+	0.998484i	$0.482473\pi$
$758$		0			0
$759$	−19.6119	−	8.88253i	−0.711867	−	0.322415i
$760$		0			0
$761$			12.3323i			0.447046i	0.974699	+	0.223523i	$0.0717558\pi$
$761$			12.3323i			0.447046i	−0.974699	+	0.223523i	$0.928244\pi$
$762$		0			0
$763$		−	2.86988i		−	0.103897i
$764$		0			0
$765$	−5.40538	−	6.15996i	−0.195432	−	0.222714i
$766$		0			0
$767$	14.5169			0.524175
$768$		0			0
$769$	35.7181			1.28803			0.644015	−	0.765013i	$-0.277268\pi$
$769$	35.7181			1.28803			0.644015	+	0.765013i	$0.277268\pi$
$770$		0			0
$771$	−17.6453	+	38.9594i	−0.635481	+	1.40309i
$772$		0			0
$773$		−	40.3015i		−	1.44954i	−0.688989	−	0.724771i	$-0.741945\pi$
$773$		−	40.3015i		−	1.44954i	0.688989	−	0.724771i	$-0.258055\pi$
$774$		0			0
$775$			10.4682i			0.376031i
$776$		0			0
$777$	1.53941	−	3.39889i	0.0552260	−	0.121934i
$778$		0			0
$779$	4.95675			0.177594
$780$		0			0
$781$	−28.2542			−1.01102
$782$		0			0
$783$	12.1529	−	3.69245i	0.434309	−	0.131957i
$784$		0			0
$785$			18.8491i			0.672753i
$786$		0			0
$787$		−	26.5594i		−	0.946742i	−0.880863	−	0.473371i	$-0.843037\pi$
$787$		−	26.5594i		−	0.946742i	0.880863	−	0.473371i	$-0.156963\pi$
$788$		0			0
$789$	27.5836	+	12.4930i	0.982004	+	0.444764i
$790$		0			0
$791$	−6.82246			−0.242579
$792$		0			0
$793$	21.3191			0.757064
$794$		0			0
$795$	−11.3832	−	5.15562i	−0.403720	−	0.182851i
$796$		0			0
$797$		−	27.4101i		−	0.970915i	−0.874260	−	0.485457i	$-0.838653\pi$
$797$		−	27.4101i		−	0.970915i	0.874260	−	0.485457i	$-0.161347\pi$
$798$		0			0
$799$			12.0995i			0.428051i
$800$		0			0
$801$	−24.7781	+	21.7428i	−0.875491	+	0.768245i
$802$		0			0
$803$	69.7407			2.46110
$804$		0			0
$805$	3.51301			0.123817
$806$		0			0
$807$	16.0000	−	35.3268i	0.563228	−	1.24356i
$808$		0			0
$809$		−	27.2725i		−	0.958849i	−0.877583	−	0.479425i	$-0.840845\pi$
$809$		−	27.2725i		−	0.958849i	0.877583	−	0.479425i	$-0.159155\pi$
$810$		0			0
$811$		−	18.6173i		−	0.653740i	−0.945069	−	0.326870i	$-0.894006\pi$
$811$		−	18.6173i		−	0.653740i	0.945069	−	0.326870i	$-0.105994\pi$
$812$		0			0
$813$	−22.2105	+	49.0391i	−0.778958	+	1.71988i
$814$		0			0
$815$	−9.43536			−0.330506
$816$		0			0
$817$	6.80404			0.238043
$818$		0			0
$819$	−2.84380	+	2.49544i	−0.0993705	+	0.0871978i
$820$		0			0
$821$		−	13.9352i		−	0.486341i	−0.969984	−	0.243171i	$-0.921812\pi$
$821$		−	13.9352i		−	0.486341i	0.969984	−	0.243171i	$-0.0781875\pi$
$822$		0			0
$823$		−	49.9501i		−	1.74115i	−0.492035	−	0.870575i	$-0.663747\pi$
$823$		−	49.9501i		−	1.74115i	0.492035	−	0.870575i	$-0.336253\pi$
$824$		0			0
$825$	11.5278	+	5.22114i	0.401348	+	0.181777i
$826$		0			0
$827$	−12.1270			−0.421697			−0.210849	−	0.977519i	$-0.567623\pi$
$827$	−12.1270			−0.421697			−0.210849	+	0.977519i	$0.567623\pi$
$828$		0			0
$829$	18.3614			0.637717			0.318859	−	0.947802i	$-0.396700\pi$
$829$	18.3614			0.637717			0.318859	+	0.947802i	$0.396700\pi$
$830$		0			0
$831$	36.9395	+	16.7305i	1.28142	+	0.580373i
$832$		0			0
$833$		−	9.59681i		−	0.332510i
$834$		0			0
$835$		−	23.8040i		−	0.823770i
$836$		0			0
$837$	32.9958	−	10.0252i	1.14050	−	0.346522i
$838$		0			0
$839$	−54.5183			−1.88218			−0.941090	−	0.338156i	$-0.890197\pi$
$839$	−54.5183			−1.88218			−0.941090	+	0.338156i	$0.890197\pi$
$840$		0			0
$841$	23.0249			0.793963
$842$		0			0
$843$	20.6278	−	45.5445i	0.710459	−	1.56864i
$844$		0			0
$845$			18.1742i			0.625212i
$846$		0			0
$847$		−	7.39940i		−	0.254246i
$848$		0			0
$849$	7.72152	−	17.0485i	0.265002	−	0.585103i
$850$		0			0
$851$	8.16933			0.280041
$852$		0			0
$853$	3.62135			0.123993			0.0619964	−	0.998076i	$-0.480253\pi$
$853$	3.62135			0.123993			0.0619964	+	0.998076i	$0.480253\pi$
$854$		0			0
$855$	−3.66070	−	4.17173i	−0.125193	−	0.142670i
$856$		0			0
$857$		−	58.1816i		−	1.98745i	−0.111869	−	0.993723i	$-0.535684\pi$
$857$		−	58.1816i		−	1.98745i	0.111869	−	0.993723i	$-0.464316\pi$
$858$		0			0
$859$			11.0241i			0.376136i	0.982156	+	0.188068i	$0.0602226\pi$
$859$			11.0241i			0.376136i	−0.982156	+	0.188068i	$0.939777\pi$
$860$		0			0
$861$	−5.53404	−	2.50645i	−0.188600	−	0.0854196i
$862$		0			0
$863$	55.9330			1.90398			0.951991	−	0.306126i	$-0.0990330\pi$
$863$	55.9330			1.90398			0.951991	+	0.306126i	$0.0990330\pi$
$864$		0			0
$865$	−22.0572			−0.749968
$866$		0			0
$867$	−23.3820	−	10.5901i	−0.794094	−	0.359657i
$868$		0			0
$869$			10.2557i			0.347901i
$870$		0			0
$871$		−	8.63354i		−	0.292536i
$872$		0			0
$873$	−18.7115	−	21.3236i	−0.633288	−	0.721694i
$874$		0			0
$875$	−8.61062			−0.291092
$876$		0			0
$877$	33.6467			1.13617			0.568084	−	0.822971i	$-0.307685\pi$
$877$	33.6467			1.13617			0.568084	+	0.822971i	$0.307685\pi$
$878$		0			0
$879$	−0.367400	+	0.811190i	−0.0123921	+	0.0273608i
$880$		0			0
$881$		−	50.7744i		−	1.71063i	−0.518105	−	0.855317i	$-0.673362\pi$
$881$		−	50.7744i		−	1.71063i	0.518105	−	0.855317i	$-0.326638\pi$
$882$		0			0
$883$			15.6602i			0.527007i	0.964658	+	0.263504i	$0.0848781\pi$
$883$			15.6602i			0.527007i	−0.964658	+	0.263504i	$0.915122\pi$
$884$		0			0
$885$	10.7684	−	23.7759i	0.361977	−	0.799217i
$886$		0			0
$887$	−7.98064			−0.267964			−0.133982	−	0.990984i	$-0.542776\pi$
$887$	−7.98064			−0.267964			−0.133982	+	0.990984i	$0.542776\pi$
$888$		0			0
$889$	−2.62286			−0.0879679
$890$		0			0
$891$	5.41699	−	41.3358i	0.181476	−	1.38480i
$892$		0			0
$893$			8.19420i			0.274208i
$894$		0			0
$895$			2.38608i			0.0797577i
$896$		0			0
$897$	−7.54574	−	3.41758i	−0.251945	−	0.114110i
$898$		0			0
$899$	−16.2226			−0.541055
$900$		0			0
$901$	5.75838			0.191840
$902$		0			0
$903$	−7.59648	−	3.44056i	−0.252795	−	0.114495i
$904$		0			0
$905$			27.2041i			0.904295i
$906$		0			0
$907$		−	21.9105i		−	0.727527i	−0.931491	−	0.363764i	$-0.881492\pi$
$907$		−	21.9105i		−	0.727527i	0.931491	−	0.363764i	$-0.118508\pi$
$908$		0			0
$909$	7.03271	−	6.17121i	0.233260	−	0.204686i
$910$		0			0
$911$	−42.5173			−1.40866			−0.704330	−	0.709873i	$-0.748752\pi$
$911$	−42.5173			−1.40866			−0.704330	+	0.709873i	$0.748752\pi$
$912$		0			0
$913$	−12.2978			−0.406999
$914$		0			0
$915$	15.8142	−	34.9165i	0.522802	−	1.15430i
$916$		0			0
$917$			0.00856859i			0.000282960i
$918$		0			0
$919$		−	20.2272i		−	0.667234i	−0.942709	−	0.333617i	$-0.891731\pi$
$919$		−	20.2272i		−	0.667234i	0.942709	−	0.333617i	$-0.108269\pi$
$920$		0			0
$921$	14.4769	−	31.9637i	0.477029	−	1.05324i
$922$		0			0
$923$	−10.8709			−0.357820
$924$		0			0
$925$	−4.80192			−0.157886
$926$		0			0
$927$	34.5747	−	30.3394i	1.13558	−	0.996476i
$928$		0			0
$929$		−	48.3458i		−	1.58617i	−0.609109	−	0.793086i	$-0.708473\pi$
$929$		−	48.3458i		−	1.58617i	0.609109	−	0.793086i	$-0.291527\pi$
$930$		0			0
$931$		−	6.49927i		−	0.213005i
$932$		0			0
$933$	−46.3875	−	21.0096i	−1.51866	−	0.687824i
$934$		0			0
$935$	12.6539			0.413828
$936$		0			0
$937$	−49.7559			−1.62545			−0.812727	−	0.582645i	$-0.802018\pi$
$937$	−49.7559			−1.62545			−0.812727	+	0.582645i	$0.802018\pi$
$938$		0			0
$939$	−12.3470	−	5.59216i	−0.402930	−	0.182493i
$940$		0			0
$941$			24.6221i			0.802657i	0.915934	+	0.401328i	$0.131451\pi$
$941$			24.6221i			0.802657i	−0.915934	+	0.401328i	$0.868549\pi$
$942$		0			0
$943$		−	13.3012i		−	0.433148i
$944$		0			0
$945$	1.97755	+	6.50868i	0.0643298	+	0.211727i
$946$		0			0
$947$	−8.29796			−0.269647			−0.134824	−	0.990870i	$-0.543047\pi$
$947$	−8.29796			−0.269647			−0.134824	+	0.990870i	$0.543047\pi$
$948$		0			0
$949$	26.8330			0.871035
$950$		0			0
$951$	−6.01359	+	13.2775i	−0.195004	+	0.430553i
$952$		0			0
$953$		−	10.5290i		−	0.341069i	−0.985352	−	0.170534i	$-0.945451\pi$
$953$		−	10.5290i		−	0.341069i	0.985352	−	0.170534i	$-0.0545494\pi$
$954$		0			0
$955$			8.93548i			0.289145i
$956$		0			0
$957$	−8.09120	+	17.8647i	−0.261551	+	0.577484i
$958$		0			0
$959$	11.6980			0.377747
$960$		0			0
$961$	−13.0454			−0.420820
$962$		0			0
$963$	26.6714	+	30.3947i	0.859474	+	0.979455i
$964$		0			0
$965$			4.20631i			0.135406i
$966$		0			0
$967$			49.6937i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$967$			49.6937i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$968$		0			0
$969$	2.32973	+	1.05517i	0.0748417	+	0.0338970i
$970$		0			0
$971$	40.7513			1.30777			0.653886	−	0.756593i	$-0.273137\pi$
$971$	40.7513			1.30777			0.653886	+	0.756593i	$0.273137\pi$
$972$		0			0
$973$	2.89982			0.0929639
$974$		0			0
$975$	4.43538	+	2.00885i	0.142046	+	0.0643347i
$976$		0			0
$977$			49.7877i			1.59285i	0.604737	+	0.796425i	$0.293278\pi$
$977$			49.7877i			1.59285i	−0.604737	+	0.796425i	$0.706722\pi$
$978$		0			0
$979$		−	50.8998i		−	1.62676i
$980$		0			0
$981$	8.02495	+	9.14523i	0.256217	+	0.291985i
$982$		0			0
$983$	20.1050			0.641250			0.320625	−	0.947206i	$-0.396107\pi$
$983$	20.1050			0.641250			0.320625	+	0.947206i	$0.396107\pi$
$984$		0			0
$985$	−34.2550			−1.09146
$986$		0			0
$987$	4.14352	−	9.14855i	0.131890	−	0.291202i
$988$		0			0
$989$		−	18.2584i		−	0.580582i
$990$		0			0
$991$		−	4.64260i		−	0.147477i	−0.997278	−	0.0737384i	$-0.976507\pi$
$991$		−	4.64260i		−	0.147477i	0.997278	−	0.0737384i	$-0.0234930\pi$
$992$		0			0
$993$	19.0929	−	42.1556i	0.605895	−	1.33777i
$994$		0			0
$995$	−23.7156			−0.751835
$996$		0			0
$997$	−46.1056			−1.46018			−0.730089	−	0.683352i	$-0.760521\pi$
$997$	−46.1056			−1.46018			−0.730089	+	0.683352i	$0.760521\pi$
$998$		0			0
$999$	4.59869	+	15.1356i	0.145496	+	0.478869i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.b.191.3	✓	24
3.2	odd	2	inner	912.2.d.b.191.21	yes	24
4.3	odd	2	inner	912.2.d.b.191.22	yes	24
12.11	even	2	inner	912.2.d.b.191.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.b.191.3	✓	24	1.1	even	1	trivial
912.2.d.b.191.4	yes	24	12.11	even	2	inner
912.2.d.b.191.21	yes	24	3.2	odd	2	inner
912.2.d.b.191.22	yes	24	4.3	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.57777 - 0.714595i) q^{3} -1.85004i q^{5} -0.707624i q^{7} +(1.97871 + 2.25493i) q^{9} -4.63214 q^{11} -1.78223 q^{13} +(-1.32203 + 2.91894i) q^{15} -1.47660i q^{17} -1.00000i q^{19} +(-0.505665 + 1.11647i) q^{21} -2.68346 q^{23} +1.57733 q^{25} +(-1.51058 - 4.97174i) q^{27} +2.44439i q^{29} +6.63667i q^{31} +(7.30844 + 3.31010i) q^{33} -1.30914 q^{35} -3.04433 q^{37} +(2.81195 + 1.27357i) q^{39} +4.95675i q^{41} +6.80404i q^{43} +(4.17173 - 3.66070i) q^{45} -8.19420 q^{47} +6.49927 q^{49} +(-1.05517 + 2.32973i) q^{51} +3.89976i q^{53} +8.56966i q^{55} +(-0.714595 + 1.57777i) q^{57} -8.14537 q^{59} -11.9620 q^{61} +(1.59564 - 1.40018i) q^{63} +3.29720i q^{65} +4.84424i q^{67} +(4.23388 + 1.91759i) q^{69} +6.09961 q^{71} -15.0559 q^{73} +(-2.48867 - 1.12716i) q^{75} +3.27781i q^{77} -2.21403i q^{79} +(-1.16944 + 8.92370i) q^{81} +2.65489 q^{83} -2.73177 q^{85} +(1.74675 - 3.85669i) q^{87} +10.9884i q^{89} +1.26115i q^{91} +(4.74253 - 10.4711i) q^{93} -1.85004 q^{95} -9.45643 q^{97} +(-9.16564 - 10.4452i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) \) 24 * q + 4 * q^9 - 12 * q^21 - 64 * q^25 + 12 * q^33 + 64 * q^37 - 16 * q^45 - 8 * q^49 - 24 * q^61 - 8 * q^69 - 16 * q^73 - 4 * q^81 - 8 * q^85 + 32 * q^93 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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