LMFDB - Embedded newform 912.2.d.b.191.14

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			191.14
Character	$\chi$	$=$	912.191
Dual form			912.2.d.b.191.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.372770 + 1.69146i) q^{3} -4.46369i q^{5} -2.16484i q^{7} +(-2.72208 + 1.26105i) q^{9} +O(q^{10})$	\(q+(0.372770 + 1.69146i) q^{3} -4.46369i q^{5} -2.16484i q^{7} +(-2.72208 + 1.26105i) q^{9} +2.80522 q^{11} -4.91931 q^{13} +(7.55016 - 1.66393i) q^{15} +2.05968i q^{17} -1.00000i q^{19} +(3.66175 - 0.806989i) q^{21} -5.97760 q^{23} -14.9245 q^{25} +(-3.14773 - 4.13422i) q^{27} -7.58678i q^{29} -6.51027i q^{31} +(1.04570 + 4.74493i) q^{33} -9.66318 q^{35} +0.923673 q^{37} +(-1.83377 - 8.32083i) q^{39} -4.75873i q^{41} +7.87563i q^{43} +(5.62894 + 12.1505i) q^{45} +1.70925 q^{47} +2.31346 q^{49} +(-3.48388 + 0.767789i) q^{51} +1.41385i q^{53} -12.5216i q^{55} +(1.69146 - 0.372770i) q^{57} +10.1861 q^{59} -1.72471 q^{61} +(2.72998 + 5.89289i) q^{63} +21.9583i q^{65} -5.78356i q^{67} +(-2.22827 - 10.1109i) q^{69} +4.48172 q^{71} -3.82461 q^{73} +(-5.56341 - 25.2442i) q^{75} -6.07287i q^{77} -1.14728i q^{79} +(5.81949 - 6.86538i) q^{81} +16.6407 q^{83} +9.19379 q^{85} +(12.8328 - 2.82813i) q^{87} -10.6918i q^{89} +10.6495i q^{91} +(11.0119 - 2.42683i) q^{93} -4.46369 q^{95} -4.86696 q^{97} +(-7.63606 + 3.53754i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.372770	+	1.69146i	0.215219	+	0.976566i
$3$	0.372770	+	1.69146i	0.215219	+	0.976566i
$4$		0			0
$5$		−	4.46369i		−	1.99622i	−0.0614408	−	0.998111i	$-0.519570\pi$
$5$		−	4.46369i		−	1.99622i	0.0614408	−	0.998111i	$-0.480430\pi$
$6$		0			0
$7$		−	2.16484i		−	0.818234i	−0.912482	−	0.409117i	$-0.865837\pi$
$7$		−	2.16484i		−	0.818234i	0.912482	−	0.409117i	$-0.134163\pi$
$8$		0			0
$9$	−2.72208	+	1.26105i	−0.907362	+	0.420351i
$10$		0			0
$11$	2.80522			0.845807			0.422904	−	0.906175i	$-0.361011\pi$
$11$	2.80522			0.845807			0.422904	+	0.906175i	$0.361011\pi$
$12$		0			0
$13$	−4.91931			−1.36437			−0.682186	−	0.731179i	$-0.738970\pi$
$13$	−4.91931			−1.36437			−0.682186	+	0.731179i	$0.738970\pi$
$14$		0			0
$15$	7.55016	−	1.66393i	1.94944	−	0.429625i
$16$		0			0
$17$			2.05968i			0.499547i	0.968304	+	0.249773i	$0.0803561\pi$
$17$			2.05968i			0.499547i	−0.968304	+	0.249773i	$0.919644\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$	3.66175	−	0.806989i	0.799059	−	0.176099i
$22$		0			0
$23$	−5.97760			−1.24642			−0.623208	−	0.782056i	$-0.714171\pi$
$23$	−5.97760			−1.24642			−0.623208	+	0.782056i	$0.714171\pi$
$24$		0			0
$25$	−14.9245			−2.98490
$26$		0			0
$27$	−3.14773	−	4.13422i	−0.605782	−	0.795631i
$28$		0			0
$29$		−	7.58678i		−	1.40883i	−0.709788	−	0.704415i	$-0.751209\pi$
$29$		−	7.58678i		−	1.40883i	0.709788	−	0.704415i	$-0.248791\pi$
$30$		0			0
$31$		−	6.51027i		−	1.16928i	−0.811293	−	0.584639i	$-0.801236\pi$
$31$		−	6.51027i		−	1.16928i	0.811293	−	0.584639i	$-0.198764\pi$
$32$		0			0
$33$	1.04570	+	4.74493i	0.182034	+	0.825986i
$34$		0			0
$35$	−9.66318			−1.63338
$36$		0			0
$37$	0.923673			0.151851			0.0759254	−	0.997113i	$-0.475809\pi$
$37$	0.923673			0.151851			0.0759254	+	0.997113i	$0.475809\pi$
$38$		0			0
$39$	−1.83377	−	8.32083i	−0.293639	−	1.33240i
$40$		0			0
$41$		−	4.75873i		−	0.743188i	−0.928395	−	0.371594i	$-0.878811\pi$
$41$		−	4.75873i		−	0.743188i	0.928395	−	0.371594i	$-0.121189\pi$
$42$		0			0
$43$			7.87563i			1.20102i	0.799616	+	0.600511i	$0.205036\pi$
$43$			7.87563i			1.20102i	−0.799616	+	0.600511i	$0.794964\pi$
$44$		0			0
$45$	5.62894	+	12.1505i	0.839113	+	1.81129i
$46$		0			0
$47$	1.70925			0.249320			0.124660	−	0.992200i	$-0.460216\pi$
$47$	1.70925			0.249320			0.124660	+	0.992200i	$0.460216\pi$
$48$		0			0
$49$	2.31346			0.330494
$50$		0			0
$51$	−3.48388	+	0.767789i	−0.487840	+	0.107512i
$52$		0			0
$53$			1.41385i			0.194207i	0.995274	+	0.0971033i	$0.0309577\pi$
$53$			1.41385i			0.194207i	−0.995274	+	0.0971033i	$0.969042\pi$
$54$		0			0
$55$		−	12.5216i		−	1.68842i
$56$		0			0
$57$	1.69146	−	0.372770i	0.224040	−	0.0493746i
$58$		0			0
$59$	10.1861			1.32612			0.663060	−	0.748566i	$-0.269257\pi$
$59$	10.1861			1.32612			0.663060	+	0.748566i	$0.269257\pi$
$60$		0			0
$61$	−1.72471			−0.220827			−0.110413	−	0.993886i	$-0.535218\pi$
$61$	−1.72471			−0.220827			−0.110413	+	0.993886i	$0.535218\pi$
$62$		0			0
$63$	2.72998	+	5.89289i	0.343945	+	0.742434i
$64$		0			0
$65$			21.9583i			2.72359i
$66$		0			0
$67$		−	5.78356i		−	0.706574i	−0.935515	−	0.353287i	$-0.885064\pi$
$67$		−	5.78356i		−	0.706574i	0.935515	−	0.353287i	$-0.114936\pi$
$68$		0			0
$69$	−2.22827	−	10.1109i	−0.268252	−	1.21721i
$70$		0			0
$71$	4.48172			0.531883			0.265941	−	0.963989i	$-0.414317\pi$
$71$	4.48172			0.531883			0.265941	+	0.963989i	$0.414317\pi$
$72$		0			0
$73$	−3.82461			−0.447637			−0.223819	−	0.974631i	$-0.571852\pi$
$73$	−3.82461			−0.447637			−0.223819	+	0.974631i	$0.571852\pi$
$74$		0			0
$75$	−5.56341	−	25.2442i	−0.642407	−	2.91495i
$76$		0			0
$77$		−	6.07287i		−	0.692068i
$78$		0			0
$79$		−	1.14728i		−	0.129079i	−0.997915	−	0.0645393i	$-0.979442\pi$
$79$		−	1.14728i		−	0.129079i	0.997915	−	0.0645393i	$-0.0205578\pi$
$80$		0			0
$81$	5.81949	−	6.86538i	0.646610	−	0.762821i
$82$		0			0
$83$	16.6407			1.82655			0.913275	−	0.407343i	$-0.133545\pi$
$83$	16.6407			1.82655			0.913275	+	0.407343i	$0.133545\pi$
$84$		0			0
$85$	9.19379			0.997206
$86$		0			0
$87$	12.8328	−	2.82813i	1.37582	−	0.303207i
$88$		0			0
$89$		−	10.6918i		−	1.13333i	−0.823947	−	0.566666i	$-0.808233\pi$
$89$		−	10.6918i		−	1.13333i	0.823947	−	0.566666i	$-0.191767\pi$
$90$		0			0
$91$			10.6495i			1.11637i
$92$		0			0
$93$	11.0119	−	2.42683i	1.14188	−	0.251651i
$94$		0			0
$95$	−4.46369			−0.457965
$96$		0			0
$97$	−4.86696			−0.494165			−0.247082	−	0.968995i	$-0.579472\pi$
$97$	−4.86696			−0.494165			−0.247082	+	0.968995i	$0.579472\pi$
$98$		0			0
$99$	−7.63606	+	3.53754i	−0.767453	+	0.355536i
$100$		0			0
$101$			10.9361i			1.08818i	0.839026	+	0.544091i	$0.183125\pi$
$101$			10.9361i			1.08818i	−0.839026	+	0.544091i	$0.816875\pi$
$102$		0			0
$103$			16.6040i			1.63604i	0.575191	+	0.818019i	$0.304928\pi$
$103$			16.6040i			1.63604i	−0.575191	+	0.818019i	$0.695072\pi$
$104$		0			0
$105$	−3.60214	−	16.3449i	−0.351533	−	1.59510i
$106$		0			0
$107$	6.45093			0.623635			0.311817	−	0.950142i	$-0.399062\pi$
$107$	6.45093			0.623635			0.311817	+	0.950142i	$0.399062\pi$
$108$		0			0
$109$	−8.24717			−0.789936			−0.394968	−	0.918695i	$-0.629244\pi$
$109$	−8.24717			−0.789936			−0.394968	+	0.918695i	$0.629244\pi$
$110$		0			0
$111$	0.344318	+	1.56236i	0.0326812	+	0.148292i
$112$		0			0
$113$		−	15.7853i		−	1.48496i	−0.669869	−	0.742479i	$-0.733650\pi$
$113$		−	15.7853i		−	1.48496i	0.669869	−	0.742479i	$-0.266350\pi$
$114$		0			0
$115$			26.6821i			2.48812i
$116$		0			0
$117$	13.3908	−	6.20351i	1.23798	−	0.573515i
$118$		0			0
$119$	4.45889			0.408746
$120$		0			0
$121$	−3.13071			−0.284610
$122$		0			0
$123$	8.04920	−	1.77391i	0.725772	−	0.159948i
$124$		0			0
$125$			44.2999i			3.96230i
$126$		0			0
$127$		−	11.0378i		−	0.979450i	−0.871877	−	0.489725i	$-0.837097\pi$
$127$		−	11.0378i		−	0.979450i	0.871877	−	0.489725i	$-0.162903\pi$
$128$		0			0
$129$	−13.3213	+	2.93580i	−1.17288	+	0.258483i
$130$		0			0
$131$	1.71368			0.149725			0.0748624	−	0.997194i	$-0.476148\pi$
$131$	1.71368			0.149725			0.0748624	+	0.997194i	$0.476148\pi$
$132$		0			0
$133$	−2.16484			−0.187716
$134$		0			0
$135$	−18.4539	+	14.0505i	−1.58826	+	1.20927i
$136$		0			0
$137$		−	14.1694i		−	1.21057i	−0.796009	−	0.605285i	$-0.793059\pi$
$137$		−	14.1694i		−	1.21057i	0.796009	−	0.605285i	$-0.206941\pi$
$138$		0			0
$139$		−	10.7288i		−	0.910008i	−0.890489	−	0.455004i	$-0.849638\pi$
$139$		−	10.7288i		−	0.910008i	0.890489	−	0.455004i	$-0.150362\pi$
$140$		0			0
$141$	0.637157	+	2.89113i	0.0536583	+	0.243477i
$142$		0			0
$143$	−13.7998			−1.15400
$144$		0			0
$145$	−33.8650			−2.81234
$146$		0			0
$147$	0.862387	+	3.91312i	0.0711285	+	0.322749i
$148$		0			0
$149$		−	6.92527i		−	0.567340i	−0.958922	−	0.283670i	$-0.908448\pi$
$149$		−	6.92527i		−	0.567340i	0.958922	−	0.283670i	$-0.0915521\pi$
$150$		0			0
$151$			1.72975i			0.140765i	0.997520	+	0.0703824i	$0.0224220\pi$
$151$			1.72975i			0.140765i	−0.997520	+	0.0703824i	$0.977578\pi$
$152$		0			0
$153$	−2.59737	−	5.60664i	−0.209985	−	0.453270i
$154$		0			0
$155$	−29.0598			−2.33414
$156$		0			0
$157$	15.3245			1.22303			0.611514	−	0.791233i	$-0.290561\pi$
$157$	15.3245			1.22303			0.611514	+	0.791233i	$0.290561\pi$
$158$		0			0
$159$	−2.39147	+	0.527039i	−0.189656	+	0.0417969i
$160$		0			0
$161$			12.9406i			1.01986i
$162$		0			0
$163$			8.20985i			0.643045i	0.946902	+	0.321523i	$0.104195\pi$
$163$			8.20985i			0.643045i	−0.946902	+	0.321523i	$0.895805\pi$
$164$		0			0
$165$	21.1799	−	4.66770i	1.64885	−	0.363380i
$166$		0			0
$167$	−13.6368			−1.05525			−0.527624	−	0.849478i	$-0.676917\pi$
$167$	−13.6368			−1.05525			−0.527624	+	0.849478i	$0.676917\pi$
$168$		0			0
$169$	11.1996			0.861509
$170$		0			0
$171$	1.26105	+	2.72208i	0.0964351	+	0.208163i
$172$		0			0
$173$			0.190545i			0.0144868i	0.999974	+	0.00724342i	$0.00230567\pi$
$173$			0.190545i			0.0144868i	−0.999974	+	0.00724342i	$0.997694\pi$
$174$		0			0
$175$			32.3092i			2.44235i
$176$		0			0
$177$	3.79708	+	17.2294i	0.285406	+	1.29504i
$178$		0			0
$179$	18.3680			1.37289			0.686443	−	0.727184i	$-0.259171\pi$
$179$	18.3680			1.37289			0.686443	+	0.727184i	$0.259171\pi$
$180$		0			0
$181$	10.1869			0.757188			0.378594	−	0.925563i	$-0.376408\pi$
$181$	10.1869			0.757188			0.378594	+	0.925563i	$0.376408\pi$
$182$		0			0
$183$	−0.642922	−	2.91729i	−0.0475261	−	0.215652i
$184$		0			0
$185$		−	4.12299i		−	0.303128i
$186$		0			0
$187$			5.77788i			0.422520i
$188$		0			0
$189$	−8.94993	+	6.81435i	−0.651012	+	0.495671i
$190$		0			0
$191$	−15.8965			−1.15023			−0.575113	−	0.818074i	$-0.695042\pi$
$191$	−15.8965			−1.15023			−0.575113	+	0.818074i	$0.695042\pi$
$192$		0			0
$193$	4.43616			0.319322			0.159661	−	0.987172i	$-0.448960\pi$
$193$	4.43616			0.319322			0.159661	+	0.987172i	$0.448960\pi$
$194$		0			0
$195$	−37.1416	+	8.18538i	−2.65976	+	0.586168i
$196$		0			0
$197$			16.7467i			1.19315i	0.802557	+	0.596576i	$0.203472\pi$
$197$			16.7467i			1.19315i	−0.802557	+	0.596576i	$0.796528\pi$
$198$		0			0
$199$			10.3295i			0.732239i	0.930568	+	0.366119i	$0.119314\pi$
$199$			10.3295i			0.732239i	−0.930568	+	0.366119i	$0.880686\pi$
$200$		0			0
$201$	9.78267	−	2.15594i	0.690016	−	0.152068i
$202$		0			0
$203$	−16.4242			−1.15275
$204$		0			0
$205$	−21.2415			−1.48357
$206$		0			0
$207$	16.2715	−	7.53807i	1.13095	−	0.523932i
$208$		0			0
$209$		−	2.80522i		−	0.194041i
$210$		0			0
$211$		−	14.1692i		−	0.975450i	−0.872997	−	0.487725i	$-0.837827\pi$
$211$		−	14.1692i		−	0.975450i	0.872997	−	0.487725i	$-0.162173\pi$
$212$		0			0
$213$	1.67065	+	7.58066i	0.114471	+	0.519419i
$214$		0			0
$215$	35.1544			2.39751
$216$		0			0
$217$	−14.0937			−0.956743
$218$		0			0
$219$	−1.42570	−	6.46919i	−0.0963400	−	0.437147i
$220$		0			0
$221$		−	10.1322i		−	0.681568i
$222$		0			0
$223$			10.6776i			0.715025i	0.933908	+	0.357512i	$0.116375\pi$
$223$			10.6776i			0.715025i	−0.933908	+	0.357512i	$0.883625\pi$
$224$		0			0
$225$	40.6258	−	18.8206i	2.70838	−	1.25471i
$226$		0			0
$227$	25.5808			1.69786			0.848928	−	0.528509i	$-0.177249\pi$
$227$	25.5808			1.69786			0.848928	+	0.528509i	$0.177249\pi$
$228$		0			0
$229$	0.470101			0.0310652			0.0155326	−	0.999879i	$-0.495056\pi$
$229$	0.470101			0.0310652			0.0155326	+	0.999879i	$0.495056\pi$
$230$		0			0
$231$	10.2720	−	2.26378i	0.675850	−	0.148946i
$232$		0			0
$233$		−	12.3842i		−	0.811315i	−0.914025	−	0.405658i	$-0.867043\pi$
$233$		−	12.3842i		−	0.811315i	0.914025	−	0.405658i	$-0.132957\pi$
$234$		0			0
$235$		−	7.62955i		−	0.497697i
$236$		0			0
$237$	1.94057	−	0.427670i	0.126054	−	0.0277802i
$238$		0			0
$239$	−14.2387			−0.921023			−0.460511	−	0.887654i	$-0.652334\pi$
$239$	−14.2387			−0.921023			−0.460511	+	0.887654i	$0.652334\pi$
$240$		0			0
$241$	10.0635			0.648250			0.324125	−	0.946014i	$-0.394930\pi$
$241$	10.0635			0.648250			0.324125	+	0.946014i	$0.394930\pi$
$242$		0			0
$243$	13.7819	+	7.28424i	0.884107	+	0.467284i
$244$		0			0
$245$		−	10.3265i		−	0.659739i
$246$		0			0
$247$			4.91931i			0.313008i
$248$		0			0
$249$	6.20314	+	28.1471i	0.393108	+	1.78375i
$250$		0			0
$251$	5.45735			0.344465			0.172233	−	0.985056i	$-0.444902\pi$
$251$	5.45735			0.344465			0.172233	+	0.985056i	$0.444902\pi$
$252$		0			0
$253$	−16.7685			−1.05423
$254$		0			0
$255$	3.42717	+	15.5509i	0.214618	+	0.973838i
$256$		0			0
$257$		−	14.8964i		−	0.929214i	−0.885517	−	0.464607i	$-0.846196\pi$
$257$		−	14.8964i		−	0.929214i	0.885517	−	0.464607i	$-0.153804\pi$
$258$		0			0
$259$		−	1.99961i		−	0.124249i
$260$		0			0
$261$	9.56733	+	20.6519i	0.592203	+	1.27832i
$262$		0			0
$263$	10.7471			0.662692			0.331346	−	0.943509i	$-0.392497\pi$
$263$	10.7471			0.662692			0.331346	+	0.943509i	$0.392497\pi$
$264$		0			0
$265$	6.31096			0.387679
$266$		0			0
$267$	18.0848	−	3.98560i	1.10677	−	0.243915i
$268$		0			0
$269$		−	15.8905i		−	0.968862i	−0.874829	−	0.484431i	$-0.839027\pi$
$269$		−	15.8905i		−	0.968862i	0.874829	−	0.484431i	$-0.160973\pi$
$270$		0			0
$271$			16.0632i			0.975771i	0.872908	+	0.487885i	$0.162232\pi$
$271$			16.0632i			0.975771i	−0.872908	+	0.487885i	$0.837768\pi$
$272$		0			0
$273$	−18.0133	+	3.96983i	−1.09021	+	0.240265i
$274$		0			0
$275$	−41.8666			−2.52465
$276$		0			0
$277$	25.2121			1.51485			0.757424	−	0.652923i	$-0.226458\pi$
$277$	25.2121			1.51485			0.757424	+	0.652923i	$0.226458\pi$
$278$		0			0
$279$	8.20979	+	17.7215i	0.491507	+	1.06096i
$280$		0			0
$281$			16.4958i			0.984056i	0.870579	+	0.492028i	$0.163744\pi$
$281$			16.4958i			0.984056i	−0.870579	+	0.492028i	$0.836256\pi$
$282$		0			0
$283$			12.2030i			0.725393i	0.931907	+	0.362697i	$0.118144\pi$
$283$			12.2030i			0.725393i	−0.931907	+	0.362697i	$0.881856\pi$
$284$		0			0
$285$	−1.66393	−	7.55016i	−0.0985627	−	0.447233i
$286$		0			0
$287$	−10.3019			−0.608102
$288$		0			0
$289$	12.7577			0.750453
$290$		0			0
$291$	−1.81426	−	8.23227i	−0.106354	−	0.482584i
$292$		0			0
$293$		−	25.2421i		−	1.47466i	−0.675532	−	0.737330i	$-0.736086\pi$
$293$		−	25.2421i		−	1.47466i	0.675532	−	0.737330i	$-0.263914\pi$
$294$		0			0
$295$		−	45.4677i		−	2.64723i
$296$		0			0
$297$	−8.83010	−	11.5974i	−0.512374	−	0.672950i
$298$		0			0
$299$	29.4057			1.70057
$300$		0			0
$301$	17.0495			0.982717
$302$		0			0
$303$	−18.4980	+	4.07665i	−1.06268	+	0.234197i
$304$		0			0
$305$			7.69858i			0.440820i
$306$		0			0
$307$			28.2212i			1.61067i	0.592820	+	0.805335i	$0.298015\pi$
$307$			28.2212i			1.61067i	−0.592820	+	0.805335i	$0.701985\pi$
$308$		0			0
$309$	−28.0850	+	6.18946i	−1.59770	+	0.352106i
$310$		0			0
$311$	17.2665			0.979093			0.489546	−	0.871977i	$-0.337162\pi$
$311$	17.2665			0.979093			0.489546	+	0.871977i	$0.337162\pi$
$312$		0			0
$313$	−22.8219			−1.28997			−0.644984	−	0.764196i	$-0.723136\pi$
$313$	−22.8219			−1.28997			−0.644984	+	0.764196i	$0.723136\pi$
$314$		0			0
$315$	26.3040	−	12.1858i	1.48206	−	0.686591i
$316$		0			0
$317$		−	5.81008i		−	0.326327i	−0.986599	−	0.163163i	$-0.947830\pi$
$317$		−	5.81008i		−	0.326327i	0.986599	−	0.163163i	$-0.0521698\pi$
$318$		0			0
$319$		−	21.2826i		−	1.19160i
$320$		0			0
$321$	2.40471	+	10.9115i	0.134218	+	0.609021i
$322$		0			0
$323$	2.05968			0.114604
$324$		0			0
$325$	73.4183			4.07251
$326$		0			0
$327$	−3.07430	−	13.9498i	−0.170009	−	0.771424i
$328$		0			0
$329$		−	3.70026i		−	0.204002i
$330$		0			0
$331$			15.6647i			0.861011i	0.902588	+	0.430505i	$0.141665\pi$
$331$			15.6647i			0.861011i	−0.902588	+	0.430505i	$0.858335\pi$
$332$		0			0
$333$	−2.51432	+	1.16480i	−0.137784	+	0.0638306i
$334$		0			0
$335$	−25.8160			−1.41048
$336$		0			0
$337$	4.67542			0.254686			0.127343	−	0.991859i	$-0.459355\pi$
$337$	4.67542			0.254686			0.127343	+	0.991859i	$0.459355\pi$
$338$		0			0
$339$	26.7003	−	5.88430i	1.45016	−	0.319591i
$340$		0			0
$341$		−	18.2628i		−	0.988984i
$342$		0			0
$343$		−	20.1622i		−	1.08865i
$344$		0			0
$345$	−45.1318	+	9.94630i	−2.42981	+	0.535491i
$346$		0			0
$347$	−13.0331			−0.699655			−0.349828	−	0.936814i	$-0.613760\pi$
$347$	−13.0331			−0.699655			−0.349828	+	0.936814i	$0.613760\pi$
$348$		0			0
$349$	−4.58023			−0.245174			−0.122587	−	0.992458i	$-0.539119\pi$
$349$	−4.58023			−0.245174			−0.122587	+	0.992458i	$0.539119\pi$
$350$		0			0
$351$	15.4847	+	20.3375i	0.826511	+	1.08554i
$352$		0			0
$353$			11.5705i			0.615837i	0.951413	+	0.307918i	$0.0996324\pi$
$353$			11.5705i			0.615837i	−0.951413	+	0.307918i	$0.900368\pi$
$354$		0			0
$355$		−	20.0050i		−	1.06176i
$356$		0			0
$357$	1.66214	+	7.54205i	0.0879699	+	0.399167i
$358$		0			0
$359$	−16.5779			−0.874946			−0.437473	−	0.899232i	$-0.644126\pi$
$359$	−16.5779			−0.874946			−0.437473	+	0.899232i	$0.644126\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	−1.16704	−	5.29548i	−0.0612535	−	0.277941i
$364$		0			0
$365$			17.0719i			0.893583i
$366$		0			0
$367$			28.7461i			1.50053i	0.661136	+	0.750266i	$0.270075\pi$
$367$			28.7461i			1.50053i	−0.661136	+	0.750266i	$0.729925\pi$
$368$		0			0
$369$	6.00100	+	12.9537i	0.312400	+	0.674340i
$370$		0			0
$371$	3.06075			0.158906
$372$		0			0
$373$	−10.2320			−0.529795			−0.264898	−	0.964277i	$-0.585338\pi$
$373$	−10.2320			−0.529795			−0.264898	+	0.964277i	$0.585338\pi$
$374$		0			0
$375$	−74.9315	+	16.5137i	−3.86945	+	0.852762i
$376$		0			0
$377$			37.3217i			1.92217i
$378$		0			0
$379$		−	26.1932i		−	1.34545i	−0.739891	−	0.672727i	$-0.765123\pi$
$379$		−	26.1932i		−	1.34545i	0.739891	−	0.672727i	$-0.234877\pi$
$380$		0			0
$381$	18.6701	−	4.11458i	0.956497	−	0.210796i
$382$		0			0
$383$	14.8112			0.756817			0.378409	−	0.925639i	$-0.376471\pi$
$383$	14.8112			0.756817			0.378409	+	0.925639i	$0.376471\pi$
$384$		0			0
$385$	−27.1074			−1.38152
$386$		0			0
$387$	−9.93159	−	21.4381i	−0.504851	−	1.08976i
$388$		0			0
$389$			19.2711i			0.977086i	0.872540	+	0.488543i	$0.162471\pi$
$389$			19.2711i			0.977086i	−0.872540	+	0.488543i	$0.837529\pi$
$390$		0			0
$391$		−	12.3120i		−	0.622643i
$392$		0			0
$393$	0.638808	+	2.89862i	0.0322236	+	0.146216i
$394$		0			0
$395$	−5.12108			−0.257670
$396$		0			0
$397$	−2.91933			−0.146517			−0.0732585	−	0.997313i	$-0.523340\pi$
$397$	−2.91933			−0.146517			−0.0732585	+	0.997313i	$0.523340\pi$
$398$		0			0
$399$	−0.806989	−	3.66175i	−0.0404000	−	0.183317i
$400$		0			0
$401$		−	4.97620i		−	0.248499i	−0.992251	−	0.124250i	$-0.960348\pi$
$401$		−	4.97620i		−	0.248499i	0.992251	−	0.124250i	$-0.0396524\pi$
$402$		0			0
$403$			32.0260i			1.59533i
$404$		0			0
$405$	−30.6449	−	25.9764i	−1.52276	−	1.29078i
$406$		0			0
$407$	2.59111			0.128437
$408$		0			0
$409$	−10.7836			−0.533213			−0.266607	−	0.963805i	$-0.585902\pi$
$409$	−10.7836			−0.533213			−0.266607	+	0.963805i	$0.585902\pi$
$410$		0			0
$411$	23.9669	−	5.28191i	1.18220	−	0.260537i
$412$		0			0
$413$		−	22.0514i		−	1.08508i
$414$		0			0
$415$		−	74.2787i		−	3.64620i
$416$		0			0
$417$	18.1474	−	3.99939i	0.888683	−	0.195851i
$418$		0			0
$419$	−17.1097			−0.835864			−0.417932	−	0.908478i	$-0.637245\pi$
$419$	−17.1097			−0.835864			−0.417932	+	0.908478i	$0.637245\pi$
$420$		0			0
$421$	−13.6052			−0.663076			−0.331538	−	0.943442i	$-0.607568\pi$
$421$	−13.6052			−0.663076			−0.331538	+	0.943442i	$0.607568\pi$
$422$		0			0
$423$	−4.65272	+	2.15545i	−0.226223	+	0.104802i
$424$		0			0
$425$		−	30.7398i		−	1.49110i
$426$		0			0
$427$			3.73373i			0.180688i
$428$		0			0
$429$	−5.14414	−	23.3418i	−0.248362	−	1.12695i
$430$		0			0
$431$	14.4189			0.694534			0.347267	−	0.937766i	$-0.387110\pi$
$431$	14.4189			0.694534			0.347267	+	0.937766i	$0.387110\pi$
$432$		0			0
$433$	0.942348			0.0452863			0.0226432	−	0.999744i	$-0.492792\pi$
$433$	0.942348			0.0452863			0.0226432	+	0.999744i	$0.492792\pi$
$434$		0			0
$435$	−12.6239	−	57.2814i	−0.605268	−	2.74643i
$436$		0			0
$437$			5.97760i			0.285947i
$438$		0			0
$439$		−	18.8985i		−	0.901977i	−0.892530	−	0.450989i	$-0.851071\pi$
$439$		−	18.8985i		−	0.901977i	0.892530	−	0.450989i	$-0.148929\pi$
$440$		0			0
$441$	−6.29742	+	2.91739i	−0.299877	+	0.138923i
$442$		0			0
$443$	−34.1308			−1.62160			−0.810802	−	0.585320i	$-0.800969\pi$
$443$	−34.1308			−1.62160			−0.810802	+	0.585320i	$0.800969\pi$
$444$		0			0
$445$	−47.7250			−2.26238
$446$		0			0
$447$	11.7138	−	2.58153i	0.554045	−	0.122102i
$448$		0			0
$449$			6.78994i			0.320437i	0.987082	+	0.160219i	$0.0512199\pi$
$449$			6.78994i			0.320437i	−0.987082	+	0.160219i	$0.948780\pi$
$450$		0			0
$451$		−	13.3493i		−	0.628594i
$452$		0			0
$453$	−2.92580	+	0.644798i	−0.137466	+	0.0302952i
$454$		0			0
$455$	47.5362			2.22853
$456$		0			0
$457$	−17.6632			−0.826251			−0.413125	−	0.910674i	$-0.635563\pi$
$457$	−17.6632			−0.826251			−0.413125	+	0.910674i	$0.635563\pi$
$458$		0			0
$459$	8.51519	−	6.48334i	0.397455	−	0.302616i
$460$		0			0
$461$			8.65256i			0.402990i	0.979490	+	0.201495i	$0.0645799\pi$
$461$			8.65256i			0.402990i	−0.979490	+	0.201495i	$0.935420\pi$
$462$		0			0
$463$		−	2.34526i		−	0.108993i	−0.998514	−	0.0544966i	$-0.982645\pi$
$463$		−	2.34526i		−	0.108993i	0.998514	−	0.0544966i	$-0.0173554\pi$
$464$		0			0
$465$	−10.8326	−	49.1535i	−0.502351	−	2.27944i
$466$		0			0
$467$	−28.6672			−1.32656			−0.663281	−	0.748371i	$-0.730836\pi$
$467$	−28.6672			−1.32656			−0.663281	+	0.748371i	$0.730836\pi$
$468$		0			0
$469$	−12.5205			−0.578143
$470$		0			0
$471$	5.71252	+	25.9208i	0.263219	+	1.19437i
$472$		0			0
$473$			22.0929i			1.01583i
$474$		0			0
$475$			14.9245i			0.684783i
$476$		0			0
$477$	−1.78293	−	3.84861i	−0.0816349	−	0.176216i
$478$		0			0
$479$	−12.0372			−0.549995			−0.274997	−	0.961445i	$-0.588677\pi$
$479$	−12.0372			−0.549995			−0.274997	+	0.961445i	$0.588677\pi$
$480$		0			0
$481$	−4.54383			−0.207181
$482$		0			0
$483$	−21.8885	+	4.82385i	−0.995959	+	0.219493i
$484$		0			0
$485$			21.7246i			0.986462i
$486$		0			0
$487$			22.7506i			1.03093i	0.856911	+	0.515464i	$0.172381\pi$
$487$			22.7506i			1.03093i	−0.856911	+	0.515464i	$0.827619\pi$
$488$		0			0
$489$	−13.8867	+	3.06039i	−0.627976	+	0.138396i
$490$		0			0
$491$	−10.1415			−0.457677			−0.228839	−	0.973464i	$-0.573493\pi$
$491$	−10.1415			−0.457677			−0.228839	+	0.973464i	$0.573493\pi$
$492$		0			0
$493$	15.6264			0.703777
$494$		0			0
$495$	15.7905	+	34.0850i	0.709728	+	1.53201i
$496$		0			0
$497$		−	9.70223i		−	0.435204i
$498$		0			0
$499$		−	5.58305i		−	0.249932i	−0.992161	−	0.124966i	$-0.960118\pi$
$499$		−	5.58305i		−	0.249932i	0.992161	−	0.124966i	$-0.0398821\pi$
$500$		0			0
$501$	−5.08340	−	23.0662i	−0.227110	−	1.03052i
$502$		0			0
$503$	17.6877			0.788657			0.394328	−	0.918970i	$-0.370977\pi$
$503$	17.6877			0.788657			0.394328	+	0.918970i	$0.370977\pi$
$504$		0			0
$505$	48.8153			2.17225
$506$		0			0
$507$	4.17488	+	18.9437i	0.185413	+	0.841321i
$508$		0			0
$509$		−	16.2328i		−	0.719506i	−0.933048	−	0.359753i	$-0.882861\pi$
$509$		−	16.2328i		−	0.719506i	0.933048	−	0.359753i	$-0.117139\pi$
$510$		0			0
$511$			8.27969i			0.366272i
$512$		0			0
$513$	−4.13422	+	3.14773i	−0.182530	+	0.138976i
$514$		0			0
$515$	74.1149			3.26589
$516$		0			0
$517$	4.79483			0.210876
$518$		0			0
$519$	−0.322299	+	0.0710293i	−0.0141473	+	0.00311784i
$520$		0			0
$521$		−	33.5735i		−	1.47088i	−0.677590	−	0.735440i	$-0.736976\pi$
$521$		−	33.5735i		−	1.47088i	0.677590	−	0.735440i	$-0.263024\pi$
$522$		0			0
$523$			25.2490i			1.10406i	0.833824	+	0.552030i	$0.186147\pi$
$523$			25.2490i			1.10406i	−0.833824	+	0.552030i	$0.813853\pi$
$524$		0			0
$525$	−54.6498	+	12.0439i	−2.38511	+	0.525639i
$526$		0			0
$527$	13.4091			0.584109
$528$		0			0
$529$	12.7317			0.553551
$530$		0			0
$531$	−27.7275	+	12.8452i	−1.20327	+	0.557436i
$532$		0			0
$533$			23.4097i			1.01398i
$534$		0			0
$535$		−	28.7949i		−	1.24491i
$536$		0			0
$537$	6.84702	+	31.0687i	0.295471	+	1.34071i
$538$		0			0
$539$	6.48977			0.279534
$540$		0			0
$541$	44.3432			1.90646			0.953232	−	0.302241i	$-0.0977347\pi$
$541$	44.3432			1.90646			0.953232	+	0.302241i	$0.0977347\pi$
$542$		0			0
$543$	3.79738	+	17.2308i	0.162961	+	0.739444i
$544$		0			0
$545$			36.8128i			1.57689i
$546$		0			0
$547$		−	18.1643i		−	0.776648i	−0.921523	−	0.388324i	$-0.873054\pi$
$547$		−	18.1643i		−	0.776648i	0.921523	−	0.388324i	$-0.126946\pi$
$548$		0			0
$549$	4.69482	−	2.17495i	0.200370	−	0.0928248i
$550$		0			0
$551$	−7.58678			−0.323208
$552$		0			0
$553$	−2.48367			−0.105616
$554$		0			0
$555$	6.97387	−	1.53693i	0.296024	−	0.0652389i
$556$		0			0
$557$		−	10.5657i		−	0.447681i	−0.974626	−	0.223841i	$-0.928140\pi$
$557$		−	10.5657i		−	0.447681i	0.974626	−	0.223841i	$-0.0718595\pi$
$558$		0			0
$559$		−	38.7427i		−	1.63864i
$560$		0			0
$561$	−9.77306	+	2.15382i	−0.412619	+	0.0909344i
$562$		0			0
$563$	22.0038			0.927349			0.463675	−	0.886006i	$-0.346531\pi$
$563$	22.0038			0.927349			0.463675	+	0.886006i	$0.346531\pi$
$564$		0			0
$565$	−70.4608			−2.96431
$566$		0			0
$567$	−14.8625	−	12.5983i	−0.624165	−	0.529078i
$568$		0			0
$569$			39.0018i			1.63504i	0.575898	+	0.817521i	$0.304652\pi$
$569$			39.0018i			1.63504i	−0.575898	+	0.817521i	$0.695348\pi$
$570$		0			0
$571$		−	19.8494i		−	0.830671i	−0.909668	−	0.415335i	$-0.863664\pi$
$571$		−	19.8494i		−	0.830671i	0.909668	−	0.415335i	$-0.136336\pi$
$572$		0			0
$573$	−5.92572	−	26.8882i	−0.247551	−	1.12327i
$574$		0			0
$575$	89.2127			3.72043
$576$		0			0
$577$	23.5745			0.981420			0.490710	−	0.871323i	$-0.336737\pi$
$577$	23.5745			0.981420			0.490710	+	0.871323i	$0.336737\pi$
$578$		0			0
$579$	1.65367	+	7.50360i	0.0687241	+	0.311839i
$580$		0			0
$581$		−	36.0244i		−	1.49455i
$582$		0			0
$583$			3.96615i			0.164261i
$584$		0			0
$585$	−27.6905	−	59.7723i	−1.14486	−	2.47128i
$586$		0			0
$587$	4.02667			0.166199			0.0830993	−	0.996541i	$-0.473518\pi$
$587$	4.02667			0.166199			0.0830993	+	0.996541i	$0.473518\pi$
$588$		0			0
$589$	−6.51027			−0.268251
$590$		0			0
$591$	−28.3264	+	6.24266i	−1.16519	+	0.256789i
$592$		0			0
$593$		−	13.6372i		−	0.560012i	−0.959998	−	0.280006i	$-0.909664\pi$
$593$		−	13.6372i		−	0.560012i	0.959998	−	0.280006i	$-0.0903364\pi$
$594$		0			0
$595$		−	19.9031i		−	0.815948i
$596$		0			0
$597$	−17.4719	+	3.85053i	−0.715079	+	0.157592i
$598$		0			0
$599$	39.7181			1.62284			0.811418	−	0.584466i	$-0.198696\pi$
$599$	39.7181			1.62284			0.811418	+	0.584466i	$0.198696\pi$
$600$		0			0
$601$	−24.7654			−1.01020			−0.505102	−	0.863060i	$-0.668545\pi$
$601$	−24.7654			−1.01020			−0.505102	+	0.863060i	$0.668545\pi$
$602$		0			0
$603$	7.29337	+	15.7433i	0.297009	+	0.641118i
$604$		0			0
$605$			13.9745i			0.568145i
$606$		0			0
$607$		−	20.8822i		−	0.847582i	−0.905760	−	0.423791i	$-0.860699\pi$
$607$		−	20.8822i		−	0.847582i	0.905760	−	0.423791i	$-0.139301\pi$
$608$		0			0
$609$	−6.12245	−	27.7809i	−0.248094	−	1.12574i
$610$		0			0
$611$	−8.40833			−0.340165
$612$		0			0
$613$	38.1009			1.53888			0.769441	−	0.638718i	$-0.220535\pi$
$613$	38.1009			1.53888			0.769441	+	0.638718i	$0.220535\pi$
$614$		0			0
$615$	−7.91818	−	35.9291i	−0.319292	−	1.44880i
$616$		0			0
$617$			6.53753i			0.263191i	0.991303	+	0.131596i	$0.0420100\pi$
$617$			6.53753i			0.263191i	−0.991303	+	0.131596i	$0.957990\pi$
$618$		0			0
$619$		−	48.8059i		−	1.96168i	−0.194827	−	0.980838i	$-0.562415\pi$
$619$		−	48.8059i		−	1.96168i	0.194827	−	0.980838i	$-0.437585\pi$
$620$		0			0
$621$	18.8159	+	24.7127i	0.755056	+	0.991687i
$622$		0			0
$623$	−23.1462			−0.927331
$624$		0			0
$625$	123.118			4.92473
$626$		0			0
$627$	4.74493	−	1.04570i	0.189494	−	0.0417614i
$628$		0			0
$629$			1.90247i			0.0758566i
$630$		0			0
$631$		−	33.5750i		−	1.33660i	−0.743892	−	0.668300i	$-0.767022\pi$
$631$		−	33.5750i		−	1.33660i	0.743892	−	0.668300i	$-0.232978\pi$
$632$		0			0
$633$	23.9667	−	5.28187i	0.952591	−	0.209935i
$634$		0			0
$635$	−49.2694			−1.95520
$636$		0			0
$637$	−11.3806			−0.450916
$638$		0			0
$639$	−12.1996	+	5.65169i	−0.482610	+	0.223577i
$640$		0			0
$641$		−	34.6489i		−	1.36855i	−0.729224	−	0.684275i	$-0.760119\pi$
$641$		−	34.6489i		−	1.36855i	0.729224	−	0.684275i	$-0.239881\pi$
$642$		0			0
$643$			0.619071i			0.0244138i	0.999925	+	0.0122069i	$0.00388567\pi$
$643$			0.619071i			0.0244138i	−0.999925	+	0.0122069i	$0.996114\pi$
$644$		0			0
$645$	13.1045	+	59.4622i	0.515989	+	2.34132i
$646$		0			0
$647$	−50.0696			−1.96844			−0.984219	−	0.176954i	$-0.943376\pi$
$647$	−50.0696			−1.96844			−0.984219	+	0.176954i	$0.943376\pi$
$648$		0			0
$649$	28.5744			1.12164
$650$		0			0
$651$	−5.25371	−	23.8390i	−0.205909	−	0.934323i
$652$		0			0
$653$		−	39.5827i		−	1.54899i	−0.632579	−	0.774496i	$-0.718003\pi$
$653$		−	39.5827i		−	1.54899i	0.632579	−	0.774496i	$-0.281997\pi$
$654$		0			0
$655$		−	7.64932i		−	0.298884i
$656$		0			0
$657$	10.4109	−	4.82304i	0.406169	−	0.188165i
$658$		0			0
$659$	−31.6525			−1.23301			−0.616503	−	0.787352i	$-0.711451\pi$
$659$	−31.6525			−1.23301			−0.616503	+	0.787352i	$0.711451\pi$
$660$		0			0
$661$	2.73626			0.106428			0.0532141	−	0.998583i	$-0.483053\pi$
$661$	2.73626			0.106428			0.0532141	+	0.998583i	$0.483053\pi$
$662$		0			0
$663$	17.1383	−	3.77699i	0.665596	−	0.146686i
$664$		0			0
$665$			9.66318i			0.374722i
$666$		0			0
$667$			45.3507i			1.75599i
$668$		0			0
$669$	−18.0607	+	3.98029i	−0.698269	+	0.153887i
$670$		0			0
$671$	−4.83821			−0.186777
$672$		0			0
$673$	13.1782			0.507982			0.253991	−	0.967207i	$-0.418257\pi$
$673$	13.1782			0.507982			0.253991	+	0.967207i	$0.418257\pi$
$674$		0			0
$675$	46.9784	+	61.7012i	1.80820	+	2.37488i
$676$		0			0
$677$			3.15582i			0.121288i	0.998159	+	0.0606441i	$0.0193155\pi$
$677$			3.15582i			0.121288i	−0.998159	+	0.0606441i	$0.980685\pi$
$678$		0			0
$679$			10.5362i			0.404342i
$680$		0			0
$681$	9.53575	+	43.2689i	0.365411	+	1.65807i
$682$		0			0
$683$	−8.92376			−0.341458			−0.170729	−	0.985318i	$-0.554612\pi$
$683$	−8.92376			−0.341458			−0.170729	+	0.985318i	$0.554612\pi$
$684$		0			0
$685$	−63.2475			−2.41656
$686$		0			0
$687$	0.175240	+	0.795158i	0.00668581	+	0.0303372i
$688$		0			0
$689$		−	6.95515i		−	0.264970i
$690$		0			0
$691$		−	5.33135i		−	0.202814i	−0.994845	−	0.101407i	$-0.967666\pi$
$691$		−	5.33135i		−	0.202814i	0.994845	−	0.101407i	$-0.0323344\pi$
$692$		0			0
$693$	7.65821	+	16.5309i	0.290911	+	0.627956i
$694$		0			0
$695$	−47.8902			−1.81658
$696$		0			0
$697$	9.80147			0.371257
$698$		0			0
$699$	20.9474	−	4.61646i	0.792303	−	0.174610i
$700$		0			0
$701$			24.3162i			0.918412i	0.888330	+	0.459206i	$0.151866\pi$
$701$			24.3162i			0.918412i	−0.888330	+	0.459206i	$0.848134\pi$
$702$		0			0
$703$		−	0.923673i		−	0.0348370i
$704$		0			0
$705$	12.9051	−	2.84407i	0.486034	−	0.107114i
$706$		0			0
$707$	23.6749			0.890387
$708$		0			0
$709$	−16.3457			−0.613876			−0.306938	−	0.951730i	$-0.599304\pi$
$709$	−16.3457			−0.613876			−0.306938	+	0.951730i	$0.599304\pi$
$710$		0			0
$711$	1.44678	+	3.12298i	0.0542583	+	0.117121i
$712$		0			0
$713$			38.9158i			1.45741i
$714$		0			0
$715$			61.5979i			2.30363i
$716$		0			0
$717$	−5.30775	−	24.0842i	−0.198222	−	0.899439i
$718$		0			0
$719$	0.498558			0.0185931			0.00929654	−	0.999957i	$-0.497041\pi$
$719$	0.498558			0.0185931			0.00929654	+	0.999957i	$0.497041\pi$
$720$		0			0
$721$	35.9450			1.33866
$722$		0			0
$723$	3.75139	+	17.0221i	0.139516	+	0.633059i
$724$		0			0
$725$			113.229i			4.20522i
$726$		0			0
$727$		−	2.74527i		−	0.101817i	−0.998703	−	0.0509083i	$-0.983788\pi$
$727$		−	2.74527i		−	0.101817i	0.998703	−	0.0509083i	$-0.0162116\pi$
$728$		0			0
$729$	−7.18354	+	26.0268i	−0.266057	+	0.963957i
$730$		0			0
$731$	−16.2213			−0.599967
$732$		0			0
$733$	33.4648			1.23605			0.618025	−	0.786159i	$-0.287933\pi$
$733$	33.4648			1.23605			0.618025	+	0.786159i	$0.287933\pi$
$734$		0			0
$735$	17.4670	−	3.84943i	0.644278	−	0.141988i
$736$		0			0
$737$		−	16.2242i		−	0.597625i
$738$		0			0
$739$		−	26.9919i		−	0.992915i	−0.868061	−	0.496457i	$-0.834634\pi$
$739$		−	26.9919i		−	0.992915i	0.868061	−	0.496457i	$-0.165366\pi$
$740$		0			0
$741$	−8.32083	+	1.83377i	−0.305673	+	0.0673653i
$742$		0			0
$743$	−22.1647			−0.813143			−0.406571	−	0.913619i	$-0.633276\pi$
$743$	−22.1647			−0.813143			−0.406571	+	0.913619i	$0.633276\pi$
$744$		0			0
$745$	−30.9122			−1.13254
$746$		0			0
$747$	−45.2973	+	20.9848i	−1.65734	+	0.767792i
$748$		0			0
$749$		−	13.9652i		−	0.510279i
$750$		0			0
$751$		−	34.8139i		−	1.27038i	−0.772357	−	0.635189i	$-0.780922\pi$
$751$		−	34.8139i		−	1.27038i	0.772357	−	0.635189i	$-0.219078\pi$
$752$		0			0
$753$	2.03434	+	9.23090i	0.0741354	+	0.336393i
$754$		0			0
$755$	7.72105			0.280998
$756$		0			0
$757$	−14.8163			−0.538508			−0.269254	−	0.963069i	$-0.586777\pi$
$757$	−14.8163			−0.538508			−0.269254	+	0.963069i	$0.586777\pi$
$758$		0			0
$759$	−6.25080	−	28.3633i	−0.226890	−	1.02952i
$760$		0			0
$761$			35.3771i			1.28242i	0.767366	+	0.641209i	$0.221567\pi$
$761$			35.3771i			1.28242i	−0.767366	+	0.641209i	$0.778433\pi$
$762$		0			0
$763$			17.8538i			0.646352i
$764$		0			0
$765$	−25.0263	+	11.5939i	−0.904827	+	0.419177i
$766$		0			0
$767$	−50.1087			−1.80932
$768$		0			0
$769$	22.8083			0.822490			0.411245	−	0.911525i	$-0.365094\pi$
$769$	22.8083			0.822490			0.411245	+	0.911525i	$0.365094\pi$
$770$		0			0
$771$	25.1967	−	5.55294i	0.907438	−	0.199984i
$772$		0			0
$773$		−	0.905503i		−	0.0325687i	−0.999867	−	0.0162843i	$-0.994816\pi$
$773$		−	0.905503i		−	0.0325687i	0.999867	−	0.0162843i	$-0.00518370\pi$
$774$		0			0
$775$			97.1625i			3.49018i
$776$		0			0
$777$	3.38226	−	0.745393i	0.121338	−	0.0267408i
$778$		0			0
$779$	−4.75873			−0.170499
$780$		0			0
$781$	12.5722			0.449870
$782$		0			0
$783$	−31.3654	+	23.8812i	−1.12091	+	0.853443i
$784$		0			0
$785$		−	68.4038i		−	2.44144i
$786$		0			0
$787$			25.3507i			0.903656i	0.892105	+	0.451828i	$0.149228\pi$
$787$			25.3507i			0.903656i	−0.892105	+	0.451828i	$0.850772\pi$
$788$		0			0
$789$	4.00618	+	18.1782i	0.142624	+	0.647162i
$790$		0			0
$791$	−34.1728			−1.21504
$792$		0			0
$793$	8.48440			0.301290
$794$		0			0
$795$	2.35254	+	10.6748i	0.0834359	+	0.378594i
$796$		0			0
$797$		−	37.6954i		−	1.33524i	−0.744502	−	0.667620i	$-0.767313\pi$
$797$		−	37.6954i		−	1.33524i	0.744502	−	0.667620i	$-0.232687\pi$
$798$		0			0
$799$			3.52051i			0.124547i
$800$		0			0
$801$	13.4830	+	29.1041i	0.476397	+	1.02834i
$802$		0			0
$803$	−10.7289			−0.378615
$804$		0			0
$805$	57.7626			2.03586
$806$		0			0
$807$	26.8782	−	5.92351i	0.946158	−	0.208517i
$808$		0			0
$809$		−	39.4859i		−	1.38825i	−0.719855	−	0.694124i	$-0.755792\pi$
$809$		−	39.4859i		−	1.38825i	0.719855	−	0.694124i	$-0.244208\pi$
$810$		0			0
$811$		−	4.16418i		−	0.146224i	−0.997324	−	0.0731121i	$-0.976707\pi$
$811$		−	4.16418i		−	0.146224i	0.997324	−	0.0731121i	$-0.0232931\pi$
$812$		0			0
$813$	−27.1703	+	5.98789i	−0.952904	+	0.210004i
$814$		0			0
$815$	36.6462			1.28366
$816$		0			0
$817$	7.87563			0.275533
$818$		0			0
$819$	−13.4296	−	28.9889i	−0.469269	−	1.01296i
$820$		0			0
$821$		−	13.4613i		−	0.469802i	−0.972019	−	0.234901i	$-0.924523\pi$
$821$		−	13.4613i		−	0.469802i	0.972019	−	0.234901i	$-0.0754766\pi$
$822$		0			0
$823$		−	0.833750i		−	0.0290627i	−0.999894	−	0.0145313i	$-0.995374\pi$
$823$		−	0.833750i		−	0.0290627i	0.999894	−	0.0145313i	$-0.00462563\pi$
$824$		0			0
$825$	−15.6066	−	70.8157i	−0.543352	−	2.46549i
$826$		0			0
$827$	−24.3703			−0.847439			−0.423719	−	0.905794i	$-0.639276\pi$
$827$	−24.3703			−0.847439			−0.423719	+	0.905794i	$0.639276\pi$
$828$		0			0
$829$	−13.0706			−0.453960			−0.226980	−	0.973899i	$-0.572885\pi$
$829$	−13.0706			−0.453960			−0.226980	+	0.973899i	$0.572885\pi$
$830$		0			0
$831$	9.39832	+	42.6453i	0.326024	+	1.47935i
$832$		0			0
$833$			4.76499i			0.165097i
$834$		0			0
$835$			60.8705i			2.10651i
$836$		0			0
$837$	−26.9149	+	20.4926i	−0.930314	+	0.708328i
$838$		0			0
$839$	48.2217			1.66480			0.832399	−	0.554176i	$-0.186967\pi$
$839$	48.2217			1.66480			0.832399	+	0.554176i	$0.186967\pi$
$840$		0			0
$841$	−28.5593			−0.984802
$842$		0			0
$843$	−27.9020	+	6.14914i	−0.960996	+	0.211787i
$844$		0			0
$845$		−	49.9916i		−	1.71976i
$846$		0			0
$847$			6.77750i			0.232878i
$848$		0			0
$849$	−20.6409	+	4.54892i	−0.708394	+	0.156118i
$850$		0			0
$851$	−5.52134			−0.189269
$852$		0			0
$853$	42.0086			1.43835			0.719174	−	0.694830i	$-0.244521\pi$
$853$	42.0086			1.43835			0.719174	+	0.694830i	$0.244521\pi$
$854$		0			0
$855$	12.1505	−	5.62894i	0.415540	−	0.192506i
$856$		0			0
$857$		−	43.5704i		−	1.48834i	−0.667992	−	0.744168i	$-0.732846\pi$
$857$		−	43.5704i		−	1.48834i	0.667992	−	0.744168i	$-0.267154\pi$
$858$		0			0
$859$			7.94535i			0.271092i	0.990771	+	0.135546i	$0.0432788\pi$
$859$			7.94535i			0.271092i	−0.990771	+	0.135546i	$0.956721\pi$
$860$		0			0
$861$	−3.84024	−	17.4253i	−0.130875	−	0.593851i
$862$		0			0
$863$	−18.8058			−0.640158			−0.320079	−	0.947391i	$-0.603709\pi$
$863$	−18.8058			−0.640158			−0.320079	+	0.947391i	$0.603709\pi$
$864$		0			0
$865$	0.850531			0.0289189
$866$		0			0
$867$	4.75569	+	21.5792i	0.161512	+	0.732867i
$868$		0			0
$869$		−	3.21837i		−	0.109176i
$870$		0			0
$871$			28.4511i			0.964029i
$872$		0			0
$873$	13.2483	−	6.13749i	0.448386	−	0.207723i
$874$		0			0
$875$	95.9022			3.24209
$876$		0			0
$877$	−56.7762			−1.91719			−0.958597	−	0.284765i	$-0.908085\pi$
$877$	−56.7762			−1.91719			−0.958597	+	0.284765i	$0.908085\pi$
$878$		0			0
$879$	42.6961	−	9.40951i	1.44010	−	0.317375i
$880$		0			0
$881$			1.57381i			0.0530229i	0.999649	+	0.0265114i	$0.00843984\pi$
$881$			1.57381i			0.0530229i	−0.999649	+	0.0265114i	$0.991560\pi$
$882$		0			0
$883$			12.4647i			0.419472i	0.977758	+	0.209736i	$0.0672604\pi$
$883$			12.4647i			0.419472i	−0.977758	+	0.209736i	$0.932740\pi$
$884$		0			0
$885$	76.9068	−	16.9490i	2.58519	−	0.569734i
$886$		0			0
$887$	52.1556			1.75122			0.875608	−	0.483023i	$-0.160461\pi$
$887$	52.1556			1.75122			0.875608	+	0.483023i	$0.160461\pi$
$888$		0			0
$889$	−23.8952			−0.801419
$890$		0			0
$891$	16.3250	−	19.2589i	0.546908	−	0.645199i
$892$		0			0
$893$		−	1.70925i		−	0.0571979i
$894$		0			0
$895$		−	81.9888i		−	2.74058i
$896$		0			0
$897$	10.9616	+	49.7386i	0.365996	+	1.66072i
$898$		0			0
$899$	−49.3920			−1.64731
$900$		0			0
$901$	−2.91208			−0.0970153
$902$		0			0
$903$	6.35555	+	28.8386i	0.211499	+	0.959688i
$904$		0			0
$905$		−	45.4713i		−	1.51152i
$906$		0			0
$907$		−	11.3056i		−	0.375396i	−0.982227	−	0.187698i	$-0.939897\pi$
$907$		−	11.3056i		−	0.375396i	0.982227	−	0.187698i	$-0.0601027\pi$
$908$		0			0
$909$	−13.7910	−	29.7690i	−0.457418	−	0.987374i
$910$		0			0
$911$	−6.71121			−0.222352			−0.111176	−	0.993801i	$-0.535462\pi$
$911$	−6.71121			−0.222352			−0.111176	+	0.993801i	$0.535462\pi$
$912$		0			0
$913$	46.6808			1.54491
$914$		0			0
$915$	−13.0219	+	2.86980i	−0.430489	+	0.0948727i
$916$		0			0
$917$		−	3.70984i		−	0.122510i
$918$		0			0
$919$			42.0708i			1.38779i	0.720077	+	0.693894i	$0.244107\pi$
$919$			42.0708i			1.38779i	−0.720077	+	0.693894i	$0.755893\pi$
$920$		0			0
$921$	−47.7351	+	10.5200i	−1.57293	+	0.346647i
$922$		0			0
$923$	−22.0470			−0.725686
$924$		0			0
$925$	−13.7854			−0.453260
$926$		0			0
$927$	−20.9385	−	45.1974i	−0.687710	−	1.48448i
$928$		0			0
$929$			9.30385i			0.305250i	0.988284	+	0.152625i	$0.0487726\pi$
$929$			9.30385i			0.305250i	−0.988284	+	0.152625i	$0.951227\pi$
$930$		0			0
$931$		−	2.31346i		−	0.0758205i
$932$		0			0
$933$	6.43643	+	29.2056i	0.210719	+	0.956149i
$934$		0			0
$935$	25.7906			0.843444
$936$		0			0
$937$	−58.8896			−1.92384			−0.961920	−	0.273330i	$-0.911875\pi$
$937$	−58.8896			−1.92384			−0.961920	+	0.273330i	$0.911875\pi$
$938$		0			0
$939$	−8.50731	−	38.6023i	−0.277626	−	1.25974i
$940$		0			0
$941$			52.4642i			1.71028i	0.518393	+	0.855142i	$0.326530\pi$
$941$			52.4642i			1.71028i	−0.518393	+	0.855142i	$0.673470\pi$
$942$		0			0
$943$			28.4458i			0.926321i
$944$		0			0
$945$	30.4171	+	39.9497i	0.989469	+	1.29956i
$946$		0			0
$947$	14.4365			0.469123			0.234561	−	0.972101i	$-0.424635\pi$
$947$	14.4365			0.469123			0.234561	+	0.972101i	$0.424635\pi$
$948$		0			0
$949$	18.8145			0.610744
$950$		0			0
$951$	9.82753	−	2.16582i	0.318679	−	0.0702317i
$952$		0			0
$953$		−	17.1564i		−	0.555750i	−0.960617	−	0.277875i	$-0.910370\pi$
$953$		−	17.1564i		−	0.555750i	0.960617	−	0.277875i	$-0.0896301\pi$
$954$		0			0
$955$			70.9568i			2.29611i
$956$		0			0
$957$	35.9987	−	7.93353i	1.16367	−	0.256455i
$958$		0			0
$959$	−30.6744			−0.990529
$960$		0			0
$961$	−11.3836			−0.367212
$962$		0			0
$963$	−17.5600	+	8.13496i	−0.565862	+	0.262146i
$964$		0			0
$965$		−	19.8016i		−	0.637437i
$966$		0			0
$967$			4.16814i			0.134038i	0.997752	+	0.0670192i	$0.0213489\pi$
$967$			4.16814i			0.134038i	−0.997752	+	0.0670192i	$0.978651\pi$
$968$		0			0
$969$	0.767789	+	3.48388i	0.0246649	+	0.111918i
$970$		0			0
$971$	5.92051			0.189998			0.0949991	−	0.995477i	$-0.469715\pi$
$971$	5.92051			0.189998			0.0949991	+	0.995477i	$0.469715\pi$
$972$		0			0
$973$	−23.2262			−0.744599
$974$		0			0
$975$	27.3681	+	124.184i	0.876482	+	3.97708i
$976$		0			0
$977$			23.2708i			0.744498i	0.928133	+	0.372249i	$0.121413\pi$
$977$			23.2708i			0.744498i	−0.928133	+	0.372249i	$0.878587\pi$
$978$		0			0
$979$		−	29.9930i		−	0.958581i
$980$		0			0
$981$	22.4495	−	10.4001i	0.716757	−	0.332050i
$982$		0			0
$983$	4.59799			0.146653			0.0733266	−	0.997308i	$-0.476638\pi$
$983$	4.59799			0.146653			0.0733266	+	0.997308i	$0.476638\pi$
$984$		0			0
$985$	74.7519			2.38179
$986$		0			0
$987$	6.25884	−	1.37934i	0.199221	−	0.0439050i
$988$		0			0
$989$		−	47.0774i		−	1.49697i
$990$		0			0
$991$			19.3619i			0.615050i	0.951540	+	0.307525i	$0.0995007\pi$
$991$			19.3619i			0.615050i	−0.951540	+	0.307525i	$0.900499\pi$
$992$		0			0
$993$	−26.4963	+	5.83934i	−0.840834	+	0.185306i
$994$		0			0
$995$	46.1076			1.46171
$996$		0			0
$997$	−45.8378			−1.45170			−0.725848	−	0.687855i	$-0.758553\pi$
$997$	−45.8378			−1.45170			−0.725848	+	0.687855i	$0.758553\pi$
$998$		0			0
$999$	−2.90748	−	3.81867i	−0.0919885	−	0.120817i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.b.191.14	yes	24
3.2	odd	2	inner	912.2.d.b.191.12	yes	24
4.3	odd	2	inner	912.2.d.b.191.11	✓	24
12.11	even	2	inner	912.2.d.b.191.13	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.b.191.11	✓	24	4.3	odd	2	inner
912.2.d.b.191.12	yes	24	3.2	odd	2	inner
912.2.d.b.191.13	yes	24	12.11	even	2	inner
912.2.d.b.191.14	yes	24	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(0.372770 + 1.69146i) q^{3} -4.46369i q^{5} -2.16484i q^{7} +(-2.72208 + 1.26105i) q^{9} +O(q^{10})\)	\(q+(0.372770 + 1.69146i) q^{3} -4.46369i q^{5} -2.16484i q^{7} +(-2.72208 + 1.26105i) q^{9} +2.80522 q^{11} -4.91931 q^{13} +(7.55016 - 1.66393i) q^{15} +2.05968i q^{17} -1.00000i q^{19} +(3.66175 - 0.806989i) q^{21} -5.97760 q^{23} -14.9245 q^{25} +(-3.14773 - 4.13422i) q^{27} -7.58678i q^{29} -6.51027i q^{31} +(1.04570 + 4.74493i) q^{33} -9.66318 q^{35} +0.923673 q^{37} +(-1.83377 - 8.32083i) q^{39} -4.75873i q^{41} +7.87563i q^{43} +(5.62894 + 12.1505i) q^{45} +1.70925 q^{47} +2.31346 q^{49} +(-3.48388 + 0.767789i) q^{51} +1.41385i q^{53} -12.5216i q^{55} +(1.69146 - 0.372770i) q^{57} +10.1861 q^{59} -1.72471 q^{61} +(2.72998 + 5.89289i) q^{63} +21.9583i q^{65} -5.78356i q^{67} +(-2.22827 - 10.1109i) q^{69} +4.48172 q^{71} -3.82461 q^{73} +(-5.56341 - 25.2442i) q^{75} -6.07287i q^{77} -1.14728i q^{79} +(5.81949 - 6.86538i) q^{81} +16.6407 q^{83} +9.19379 q^{85} +(12.8328 - 2.82813i) q^{87} -10.6918i q^{89} +10.6495i q^{91} +(11.0119 - 2.42683i) q^{93} -4.46369 q^{95} -4.86696 q^{97} +(-7.63606 + 3.53754i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 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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