LMFDB - Embedded newform 912.2.d.b.191.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(191,912)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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			191.21
Character	$\chi$	$=$	912.191
Dual form			912.2.d.b.191.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.57777 - 0.714595i) q^{3} +1.85004i q^{5} -0.707624i q^{7} +(1.97871 - 2.25493i) q^{9} +O(q^{10})$	\(q+(1.57777 - 0.714595i) q^{3} +1.85004i q^{5} -0.707624i q^{7} +(1.97871 - 2.25493i) q^{9} +4.63214 q^{11} -1.78223 q^{13} +(1.32203 + 2.91894i) q^{15} +1.47660i q^{17} -1.00000i q^{19} +(-0.505665 - 1.11647i) q^{21} +2.68346 q^{23} +1.57733 q^{25} +(1.51058 - 4.97174i) q^{27} -2.44439i q^{29} +6.63667i q^{31} +(7.30844 - 3.31010i) q^{33} +1.30914 q^{35} -3.04433 q^{37} +(-2.81195 + 1.27357i) q^{39} -4.95675i q^{41} +6.80404i q^{43} +(4.17173 + 3.66070i) q^{45} +8.19420 q^{47} +6.49927 q^{49} +(1.05517 + 2.32973i) q^{51} -3.89976i q^{53} +8.56966i q^{55} +(-0.714595 - 1.57777i) q^{57} +8.14537 q^{59} -11.9620 q^{61} +(-1.59564 - 1.40018i) q^{63} -3.29720i q^{65} +4.84424i q^{67} +(4.23388 - 1.91759i) q^{69} -6.09961 q^{71} -15.0559 q^{73} +(2.48867 - 1.12716i) q^{75} -3.27781i q^{77} -2.21403i q^{79} +(-1.16944 - 8.92370i) q^{81} -2.65489 q^{83} -2.73177 q^{85} +(-1.74675 - 3.85669i) q^{87} -10.9884i q^{89} +1.26115i q^{91} +(4.74253 + 10.4711i) q^{93} +1.85004 q^{95} -9.45643 q^{97} +(9.16564 - 10.4452i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 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

Display 100 coefficients 10 coefficients

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	$0.135758\pi$
$5$			1.85004i			0.827365i	−0.910421	+	0.413683i	$0.864242\pi$
$6$		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	$-0.253931\pi$
$11$	4.63214			1.39664			0.698321	+	0.715785i	$0.253931\pi$
$12$		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	$0.0573069\pi$
$17$			1.47660i			0.358128i	−0.983837	+	0.179064i	$0.942693\pi$
$18$		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	$-0.409742\pi$
$23$	2.68346			0.559540			0.279770	+	0.960067i	$0.409742\pi$
$24$		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	$-0.927123\pi$
$29$		−	2.44439i		−	0.453913i	0.973905	−	0.226956i	$-0.0728775\pi$
$30$		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	$-0.873492\pi$
$41$		−	4.95675i		−	0.774114i	0.922056	−	0.387057i	$-0.126508\pi$
$42$		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	$-0.296112\pi$
$47$	8.19420			1.19525			0.597624	+	0.801777i	$0.296112\pi$
$48$		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	$-0.913691\pi$
$53$		−	3.89976i		−	0.535674i	0.963464	−	0.267837i	$-0.0863088\pi$
$54$		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	$-0.322110\pi$
$59$	8.14537			1.06044			0.530218	+	0.847861i	$0.322110\pi$
$60$		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	$-0.617887\pi$
$71$	−6.09961			−0.723891			−0.361945	+	0.932199i	$0.617887\pi$
$72$		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	$-0.546545\pi$
$83$	−2.65489			−0.291412			−0.145706	+	0.989328i	$0.546545\pi$
$84$		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	$-0.802120\pi$
$89$		−	10.9884i		−	1.16477i	0.812914	−	0.582384i	$-0.197880\pi$
$90$		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	$0.0495915\pi$
$101$			3.11881i			0.310333i	−0.987888	+	0.155167i	$0.950409\pi$
$102$		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	$-0.725878\pi$
$107$	−13.4792			−1.30308			−0.651542	+	0.758613i	$0.725878\pi$
$108$		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	$0.149822\pi$
$113$			9.64136i			0.906983i	−0.891261	+	0.453491i	$0.850178\pi$
$114$		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	$-0.499832\pi$
$131$	0.0121090			0.00105797			0.000528983		1.00000i	$0.499832\pi$
$132$		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	$-0.750415\pi$
$137$		−	16.5314i		−	1.41237i	0.708028	−	0.706184i	$-0.249585\pi$
$138$		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	$0.351403\pi$
$149$			21.8009i			1.78600i	−0.450059	+	0.892999i	$0.648597\pi$
$150$		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	$-0.665869\pi$
$167$	−12.8667			−0.995655			−0.497827	+	0.867276i	$0.665869\pi$
$168$		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	$0.149727\pi$
$173$			11.9225i			0.906454i	−0.891395	+	0.453227i	$0.850273\pi$
$174$		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	$-0.484652\pi$
$179$	1.28974			0.0963997			0.0481998	+	0.998838i	$0.484652\pi$
$180$		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	$-0.444092\pi$
$191$	4.82987			0.349477			0.174739	+	0.984615i	$0.444092\pi$
$192$		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	$0.229273\pi$
$197$			18.5158i			1.31919i	−0.751619	+	0.659597i	$0.770727\pi$
$198$		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	$-0.724741\pi$
$227$	−19.5512			−1.29766			−0.648830	+	0.760933i	$0.724741\pi$
$228$		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	$0.00931357\pi$
$233$			0.893124i			0.0585105i	−0.999572	+	0.0292553i	$0.990686\pi$
$234$		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	$-0.145757\pi$
$239$	27.7339			1.79396			0.896978	+	0.442075i	$0.145757\pi$
$240$		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	$-0.534183\pi$
$251$	−3.39616			−0.214363			−0.107182	+	0.994239i	$0.534183\pi$
$252$		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	$0.279816\pi$
$257$			24.6927i			1.54029i	−0.637868	+	0.770145i	$0.720184\pi$
$258$		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	$-0.318797\pi$
$263$	17.4827			1.07803			0.539014	+	0.842297i	$0.318797\pi$
$264$		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	$-0.760857\pi$
$269$		−	22.3903i		−	1.36516i	0.730809	−	0.682582i	$-0.239143\pi$
$270$		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	$-0.669831\pi$
$281$		−	28.8664i		−	1.72203i	0.508583	−	0.861013i	$-0.330169\pi$
$282$		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	$0.00478059\pi$
$293$			0.514137i			0.0300362i	−0.999887	+	0.0150181i	$0.995219\pi$
$294$		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	$-0.813713\pi$
$311$	−29.4007			−1.66716			−0.833581	+	0.552397i	$0.813713\pi$
$312$		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	$0.0759438\pi$
$317$			8.41538i			0.472655i	−0.971674	+	0.236327i	$0.924056\pi$
$318$		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	$-0.459914\pi$
$347$	4.67942			0.251204			0.125602	+	0.992081i	$0.459914\pi$
$348$		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	$0.206169\pi$
$353$			22.6720i			1.20671i	−0.797474	+	0.603354i	$0.793831\pi$
$354$		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	$-0.325055\pi$
$359$	19.7942			1.04470			0.522350	+	0.852731i	$0.325055\pi$
$360$		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	$-0.384075\pi$
$383$	13.9416			0.712382			0.356191	+	0.934413i	$0.384075\pi$
$384$		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	$-0.925796\pi$
$389$		−	9.11256i		−	0.462025i	0.972951	−	0.231013i	$-0.0742039\pi$
$390$		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	$0.273211\pi$
$401$			30.3076i			1.51349i	−0.653711	+	0.756744i	$0.726789\pi$
$402$		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	$-0.338216\pi$
$419$	19.9232			0.973314			0.486657	+	0.873593i	$0.338216\pi$
$420$		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	$-0.360596\pi$
$431$	17.6084			0.848169			0.424084	+	0.905623i	$0.360596\pi$
$432$		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	$-0.530900\pi$
$443$	−4.08004			−0.193848			−0.0969242	+	0.995292i	$0.530900\pi$
$444$		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	$-0.777893\pi$
$449$		−	27.2291i		−	1.28502i	0.766277	−	0.642511i	$-0.222107\pi$
$450$		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	$-0.863493\pi$
$461$		−	17.8562i		−	0.831648i	0.909445	−	0.415824i	$-0.136507\pi$
$462$		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	$-0.108535\pi$
$467$	40.7322			1.88486			0.942430	+	0.334404i	$0.108535\pi$
$468$		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	$-0.514178\pi$
$479$	−1.94900			−0.0890519			−0.0445260	+	0.999008i	$0.514178\pi$
$480$		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	$-0.438967\pi$
$491$	8.44538			0.381134			0.190567	+	0.981674i	$0.438967\pi$
$492$		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	$-0.423474\pi$
$503$	10.6802			0.476209			0.238104	+	0.971240i	$0.423474\pi$
$504$		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	$-0.664749\pi$
$509$		−	39.2121i		−	1.73804i	0.494773	−	0.869022i	$-0.335251\pi$
$510$		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	$-0.711200\pi$
$521$		−	35.9655i		−	1.57568i	0.615881	−	0.787839i	$-0.288800\pi$
$522$		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	$-0.613206\pi$
$557$		−	44.2479i		−	1.87484i	0.348197	−	0.937421i	$-0.386794\pi$
$558$		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	$-0.775270\pi$
$563$	−36.1114			−1.52191			−0.760956	+	0.648803i	$0.775270\pi$
$564$		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	$0.0554885\pi$
$569$			8.27441i			0.346881i	−0.984844	+	0.173441i	$0.944511\pi$
$570$		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	$-0.336798\pi$
$587$	23.7699			0.981089			0.490545	+	0.871416i	$0.336798\pi$
$588$		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	$0.188386\pi$
$593$			27.1707i			1.11577i	−0.829920	+	0.557883i	$0.811614\pi$
$594$		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	$-0.399483\pi$
$599$	15.2016			0.621122			0.310561	+	0.950554i	$0.399483\pi$
$600$		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	$-0.949874\pi$
$617$		−	7.79096i		−	0.313652i	0.987626	−	0.156826i	$-0.0501262\pi$
$618$		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	$-0.938972\pi$
$641$		−	9.64878i		−	0.381104i	0.981677	−	0.190552i	$-0.0610278\pi$
$642$		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	$-0.777670\pi$
$647$	−38.9594			−1.53165			−0.765826	+	0.643048i	$0.777670\pi$
$648$		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	$-0.938752\pi$
$653$		−	9.77338i		−	0.382462i	0.981545	−	0.191231i	$-0.0612479\pi$
$654$		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	$-0.550127\pi$
$659$	−8.05188			−0.313657			−0.156828	+	0.987626i	$0.550127\pi$
$660$		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	$-0.687005\pi$
$677$		−	43.3133i		−	1.66466i	0.554277	−	0.832332i	$-0.312995\pi$
$678$		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	$-0.461055\pi$
$683$	6.37903			0.244087			0.122043	+	0.992525i	$0.461055\pi$
$684$		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	$0.234350\pi$
$701$			35.5578i			1.34300i	−0.741004	+	0.671500i	$0.765650\pi$
$702$		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	$-0.319149\pi$
$719$	28.8564			1.07616			0.538082	+	0.842893i	$0.319149\pi$
$720$		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	$-0.543786\pi$
$743$	−7.47556			−0.274252			−0.137126	+	0.990554i	$0.543786\pi$
$744$		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	$-0.928244\pi$
$761$		−	12.3323i		−	0.447046i	0.974699	−	0.223523i	$-0.0717558\pi$
$762$		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	$0.258055\pi$
$773$			40.3015i			1.44954i	−0.688989	+	0.724771i	$0.741945\pi$
$774$		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	$0.161347\pi$
$797$			27.4101i			0.970915i	−0.874260	+	0.485457i	$0.838653\pi$
$798$		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	$0.159155\pi$
$809$			27.2725i			0.958849i	−0.877583	+	0.479425i	$0.840845\pi$
$810$		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	$0.0781875\pi$
$821$			13.9352i			0.486341i	−0.969984	+	0.243171i	$0.921812\pi$
$822$		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	$-0.432377\pi$
$827$	12.1270			0.421697			0.210849	+	0.977519i	$0.432377\pi$
$828$		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	$-0.109803\pi$
$839$	54.5183			1.88218			0.941090	+	0.338156i	$0.109803\pi$
$840$		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	$0.464316\pi$
$857$			58.1816i			1.98745i	−0.111869	+	0.993723i	$0.535684\pi$
$858$		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	$-0.900967\pi$
$863$	−55.9330			−1.90398			−0.951991	+	0.306126i	$0.900967\pi$
$864$		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	$0.326638\pi$
$881$			50.7744i			1.71063i	−0.518105	+	0.855317i	$0.673362\pi$
$882$		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	$-0.457224\pi$
$887$	7.98064			0.267964			0.133982	+	0.990984i	$0.457224\pi$
$888$		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	$-0.251248\pi$
$911$	42.5173			1.40866			0.704330	+	0.709873i	$0.251248\pi$
$912$		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	$0.291527\pi$
$929$			48.3458i			1.58617i	−0.609109	+	0.793086i	$0.708473\pi$
$930$		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	$-0.868549\pi$
$941$		−	24.6221i		−	0.802657i	0.915934	−	0.401328i	$-0.131451\pi$
$942$		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	$-0.456953\pi$
$947$	8.29796			0.269647			0.134824	+	0.990870i	$0.456953\pi$
$948$		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	$0.0545494\pi$
$953$			10.5290i			0.341069i	−0.985352	+	0.170534i	$0.945451\pi$
$954$		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	$-0.726863\pi$
$971$	−40.7513			−1.30777			−0.653886	+	0.756593i	$0.726863\pi$
$972$		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	$-0.706722\pi$
$977$		−	49.7877i		−	1.59285i	0.604737	−	0.796425i	$-0.293278\pi$
$978$		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	$-0.603893\pi$
$983$	−20.1050			−0.641250			−0.320625	+	0.947206i	$0.603893\pi$
$984$		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.191.21	yes	24
3.2	odd	2	inner	912.2.d.b.191.3	✓	24
4.3	odd	2	inner	912.2.d.b.191.4	yes	24
12.11	even	2	inner	912.2.d.b.191.22	yes	24

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

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

\(f(q)\)	\(=\)	\(q+(1.57777 - 0.714595i) q^{3} +1.85004i q^{5} -0.707624i q^{7} +(1.97871 - 2.25493i) q^{9} +O(q^{10})\)	\(q+(1.57777 - 0.714595i) q^{3} +1.85004i q^{5} -0.707624i q^{7} +(1.97871 - 2.25493i) q^{9} +4.63214 q^{11} -1.78223 q^{13} +(1.32203 + 2.91894i) q^{15} +1.47660i q^{17} -1.00000i q^{19} +(-0.505665 - 1.11647i) q^{21} +2.68346 q^{23} +1.57733 q^{25} +(1.51058 - 4.97174i) q^{27} -2.44439i q^{29} +6.63667i q^{31} +(7.30844 - 3.31010i) q^{33} +1.30914 q^{35} -3.04433 q^{37} +(-2.81195 + 1.27357i) q^{39} -4.95675i q^{41} +6.80404i q^{43} +(4.17173 + 3.66070i) q^{45} +8.19420 q^{47} +6.49927 q^{49} +(1.05517 + 2.32973i) q^{51} -3.89976i q^{53} +8.56966i q^{55} +(-0.714595 - 1.57777i) q^{57} +8.14537 q^{59} -11.9620 q^{61} +(-1.59564 - 1.40018i) q^{63} -3.29720i q^{65} +4.84424i q^{67} +(4.23388 - 1.91759i) q^{69} -6.09961 q^{71} -15.0559 q^{73} +(2.48867 - 1.12716i) q^{75} -3.27781i q^{77} -2.21403i q^{79} +(-1.16944 - 8.92370i) q^{81} -2.65489 q^{83} -2.73177 q^{85} +(-1.74675 - 3.85669i) q^{87} -10.9884i q^{89} +1.26115i q^{91} +(4.74253 + 10.4711i) q^{93} +1.85004 q^{95} -9.45643 q^{97} +(9.16564 - 10.4452i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) \) 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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