LMFDB - Embedded newform 2208.2.n.b.367.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2208,2,Mod(367,2208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2208, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2208.367");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2208 = 2^{5} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2208.n (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.6309687663$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			367.16
Character	$\chi$	$=$	2208.367
Dual form			2208.2.n.b.367.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.969269 q^{5} -4.55308 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.969269 q^{5} -4.55308 q^{7} +1.00000 q^{9} +0.915699i q^{11} -4.49735i q^{13} +0.969269 q^{15} +5.37023i q^{17} +0.688174i q^{19} -4.55308 q^{21} +(-4.70330 + 0.937530i) q^{23} -4.06052 q^{25} +1.00000 q^{27} +6.35610i q^{29} -5.64064i q^{31} +0.915699i q^{33} -4.41316 q^{35} +3.25973 q^{37} -4.49735i q^{39} -8.87354 q^{41} +9.54349i q^{43} +0.969269 q^{45} -0.156006i q^{47} +13.7305 q^{49} +5.37023i q^{51} -2.42613 q^{53} +0.887559i q^{55} +0.688174i q^{57} -3.81302 q^{59} -12.2676 q^{61} -4.55308 q^{63} -4.35914i q^{65} +10.6451i q^{67} +(-4.70330 + 0.937530i) q^{69} +14.9045i q^{71} -8.18221 q^{73} -4.06052 q^{75} -4.16925i q^{77} -12.0667 q^{79} +1.00000 q^{81} -11.1608i q^{83} +5.20519i q^{85} +6.35610i q^{87} -10.9680i q^{89} +20.4768i q^{91} -5.64064i q^{93} +0.667026i q^{95} -17.8948i q^{97} +0.915699i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 24 * q^9	$ 24 q + 24 q^{3} + 24 q^{9} + 24 q^{25} + 24 q^{27} + 56 q^{49} + 32 q^{73} + 24 q^{75} + 24 q^{81}+O(q^{100}) $ 24 * q + 24 * q^3 + 24 * q^9 + 24 * q^25 + 24 * q^27 + 56 * q^49 + 32 * q^73 + 24 * q^75 + 24 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2208\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$737$	$1381$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	0.969269			0.433470			0.216735	−	0.976230i	$-0.430459\pi$
$5$	0.969269			0.433470			0.216735	+	0.976230i	$0.430459\pi$
$6$		0			0
$7$	−4.55308			−1.72090			−0.860451	−	0.509533i	$-0.829818\pi$
$7$	−4.55308			−1.72090			−0.860451	+	0.509533i	$0.829818\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			0.915699i			0.276094i	0.990426	+	0.138047i	$0.0440825\pi$
$11$			0.915699i			0.276094i	−0.990426	+	0.138047i	$0.955918\pi$
$12$		0			0
$13$		−	4.49735i		−	1.24734i	−0.781687	−	0.623671i	$-0.785641\pi$
$13$		−	4.49735i		−	1.24734i	0.781687	−	0.623671i	$-0.214359\pi$
$14$		0			0
$15$	0.969269			0.250264
$16$		0			0
$17$			5.37023i			1.30247i	0.758875	+	0.651236i	$0.225749\pi$
$17$			5.37023i			1.30247i	−0.758875	+	0.651236i	$0.774251\pi$
$18$		0			0
$19$			0.688174i			0.157878i	0.996879	+	0.0789390i	$0.0251532\pi$
$19$			0.688174i			0.157878i	−0.996879	+	0.0789390i	$0.974847\pi$
$20$		0			0
$21$	−4.55308			−0.993564
$22$		0			0
$23$	−4.70330	+	0.937530i	−0.980706	+	0.195489i
$23$	−4.70330	+	0.937530i	−0.980706	+	0.195489i
$24$		0			0
$25$	−4.06052			−0.812104
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			6.35610i			1.18030i	0.807294	+	0.590149i	$0.200931\pi$
$29$			6.35610i			1.18030i	−0.807294	+	0.590149i	$0.799069\pi$
$30$		0			0
$31$		−	5.64064i		−	1.01309i	−0.862214	−	0.506544i	$-0.830923\pi$
$31$		−	5.64064i		−	1.01309i	0.862214	−	0.506544i	$-0.169077\pi$
$32$		0			0
$33$			0.915699i			0.159403i
$34$		0			0
$35$	−4.41316			−0.745960
$36$		0			0
$37$	3.25973			0.535896			0.267948	−	0.963433i	$-0.413654\pi$
$37$	3.25973			0.535896			0.267948	+	0.963433i	$0.413654\pi$
$38$		0			0
$39$		−	4.49735i		−	0.720153i
$40$		0			0
$41$	−8.87354			−1.38581			−0.692907	−	0.721027i	$-0.743670\pi$
$41$	−8.87354			−1.38581			−0.692907	+	0.721027i	$0.743670\pi$
$42$		0			0
$43$			9.54349i			1.45537i	0.685912	+	0.727684i	$0.259403\pi$
$43$			9.54349i			1.45537i	−0.685912	+	0.727684i	$0.740597\pi$
$44$		0			0
$45$	0.969269			0.144490
$46$		0			0
$47$		−	0.156006i		−	0.0227558i	−0.999935	−	0.0113779i	$-0.996378\pi$
$47$		−	0.156006i		−	0.0227558i	0.999935	−	0.0113779i	$-0.00362177\pi$
$48$		0			0
$49$	13.7305			1.96151
$50$		0			0
$51$			5.37023i			0.751982i
$52$		0			0
$53$	−2.42613			−0.333255			−0.166628	−	0.986020i	$-0.553288\pi$
$53$	−2.42613			−0.333255			−0.166628	+	0.986020i	$0.553288\pi$
$54$		0			0
$55$			0.887559i			0.119678i
$56$		0			0
$57$			0.688174i			0.0911509i
$58$		0			0
$59$	−3.81302			−0.496413			−0.248207	−	0.968707i	$-0.579841\pi$
$59$	−3.81302			−0.496413			−0.248207	+	0.968707i	$0.579841\pi$
$60$		0			0
$61$	−12.2676			−1.57070			−0.785352	−	0.619049i	$-0.787518\pi$
$61$	−12.2676			−1.57070			−0.785352	+	0.619049i	$0.787518\pi$
$62$		0			0
$63$	−4.55308			−0.573634
$64$		0			0
$65$		−	4.35914i		−	0.540685i
$66$		0			0
$67$			10.6451i			1.30050i	0.759719	+	0.650251i	$0.225336\pi$
$67$			10.6451i			1.30050i	−0.759719	+	0.650251i	$0.774664\pi$
$68$		0			0
$69$	−4.70330	+	0.937530i	−0.566211	+	0.112865i
$70$		0			0
$71$			14.9045i			1.76884i	0.466689	+	0.884422i	$0.345447\pi$
$71$			14.9045i			1.76884i	−0.466689	+	0.884422i	$0.654553\pi$
$72$		0			0
$73$	−8.18221			−0.957655			−0.478828	−	0.877909i	$-0.658938\pi$
$73$	−8.18221			−0.957655			−0.478828	+	0.877909i	$0.658938\pi$
$74$		0			0
$75$	−4.06052			−0.468868
$76$		0			0
$77$		−	4.16925i		−	0.475130i
$78$		0			0
$79$	−12.0667			−1.35761			−0.678805	−	0.734318i	$-0.737502\pi$
$79$	−12.0667			−1.35761			−0.678805	+	0.734318i	$0.737502\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	11.1608i		−	1.22506i	−0.790447	−	0.612531i	$-0.790152\pi$
$83$		−	11.1608i		−	1.22506i	0.790447	−	0.612531i	$-0.209848\pi$
$84$		0			0
$85$			5.20519i			0.564582i
$86$		0			0
$87$			6.35610i			0.681446i
$88$		0			0
$89$		−	10.9680i		−	1.16260i	−0.813688	−	0.581302i	$-0.802544\pi$
$89$		−	10.9680i		−	1.16260i	0.813688	−	0.581302i	$-0.197456\pi$
$90$		0			0
$91$			20.4768i			2.14655i
$92$		0			0
$93$		−	5.64064i		−	0.584907i
$94$		0			0
$95$			0.667026i			0.0684354i
$96$		0			0
$97$		−	17.8948i		−	1.81695i	−0.417944	−	0.908473i	$-0.637249\pi$
$97$		−	17.8948i		−	1.81695i	0.417944	−	0.908473i	$-0.362751\pi$
$98$		0			0
$99$			0.915699i			0.0920312i
$100$		0			0
$101$		−	2.18685i		−	0.217600i	−0.994064	−	0.108800i	$-0.965299\pi$
$101$		−	2.18685i		−	0.217600i	0.994064	−	0.108800i	$-0.0347007\pi$
$102$		0			0
$103$	−1.82776			−0.180095			−0.0900475	−	0.995937i	$-0.528702\pi$
$103$	−1.82776			−0.180095			−0.0900475	+	0.995937i	$0.528702\pi$
$104$		0			0
$105$	−4.41316			−0.430680
$106$		0			0
$107$			2.63011i			0.254262i	0.991886	+	0.127131i	$0.0405769\pi$
$107$			2.63011i			0.254262i	−0.991886	+	0.127131i	$0.959423\pi$
$108$		0			0
$109$	−6.14687			−0.588764			−0.294382	−	0.955688i	$-0.595114\pi$
$109$	−6.14687			−0.588764			−0.294382	+	0.955688i	$0.595114\pi$
$110$		0			0
$111$	3.25973			0.309400
$112$		0			0
$113$			18.8052i			1.76905i	0.466495	+	0.884524i	$0.345517\pi$
$113$			18.8052i			1.76905i	−0.466495	+	0.884524i	$0.654483\pi$
$114$		0			0
$115$	−4.55876	+	0.908719i	−0.425107	+	0.0847385i
$116$		0			0
$117$		−	4.49735i		−	0.415780i
$118$		0			0
$119$		−	24.4511i		−	2.24143i
$120$		0			0
$121$	10.1615			0.923772
$122$		0			0
$123$	−8.87354			−0.800100
$124$		0			0
$125$	−8.78208			−0.785493
$126$		0			0
$127$			0.261572i			0.0232107i	0.999933	+	0.0116054i	$0.00369418\pi$
$127$			0.261572i			0.0232107i	−0.999933	+	0.0116054i	$0.996306\pi$
$128$		0			0
$129$			9.54349i			0.840258i
$130$		0			0
$131$	8.22618			0.718725			0.359362	−	0.933198i	$-0.382994\pi$
$131$	8.22618			0.718725			0.359362	+	0.933198i	$0.382994\pi$
$132$		0			0
$133$		−	3.13331i		−	0.271693i
$134$		0			0
$135$	0.969269			0.0834214
$136$		0			0
$137$			6.70631i			0.572959i	0.958086	+	0.286479i	$0.0924850\pi$
$137$			6.70631i			0.572959i	−0.958086	+	0.286479i	$0.907515\pi$
$138$		0			0
$139$	7.42604			0.629868			0.314934	−	0.949114i	$-0.398018\pi$
$139$	7.42604			0.629868			0.314934	+	0.949114i	$0.398018\pi$
$140$		0			0
$141$		−	0.156006i		−	0.0131381i
$142$		0			0
$143$	4.11822			0.344383
$144$		0			0
$145$			6.16077i			0.511624i
$146$		0			0
$147$	13.7305			1.13248
$148$		0			0
$149$	16.0149			1.31199			0.655997	−	0.754764i	$-0.272249\pi$
$149$	16.0149			1.31199			0.655997	+	0.754764i	$0.272249\pi$
$150$		0			0
$151$		−	8.36872i		−	0.681037i	−0.940238	−	0.340518i	$-0.889397\pi$
$151$		−	8.36872i		−	0.681037i	0.940238	−	0.340518i	$-0.110603\pi$
$152$		0			0
$153$			5.37023i			0.434157i
$154$		0			0
$155$		−	5.46729i		−	0.439143i
$156$		0			0
$157$	−0.188409			−0.0150366			−0.00751832	−	0.999972i	$-0.502393\pi$
$157$	−0.188409			−0.0150366			−0.00751832	+	0.999972i	$0.502393\pi$
$158$		0			0
$159$	−2.42613			−0.192405
$160$		0			0
$161$	21.4145	−	4.26865i	1.68770	−	0.336417i
$162$		0			0
$163$	−6.04337			−0.473354			−0.236677	−	0.971588i	$-0.576058\pi$
$163$	−6.04337			−0.473354			−0.236677	+	0.971588i	$0.576058\pi$
$164$		0			0
$165$			0.887559i			0.0690963i
$166$		0			0
$167$		−	3.28126i		−	0.253912i	−0.991908	−	0.126956i	$-0.959479\pi$
$167$		−	3.28126i		−	0.253912i	0.991908	−	0.126956i	$-0.0405207\pi$
$168$		0			0
$169$	−7.22618			−0.555860
$170$		0			0
$171$			0.688174i			0.0526260i
$172$		0			0
$173$			4.28422i			0.325723i	0.986649	+	0.162862i	$0.0520724\pi$
$173$			4.28422i			0.325723i	−0.986649	+	0.162862i	$0.947928\pi$
$174$		0			0
$175$	18.4879			1.39755
$176$		0			0
$177$	−3.81302			−0.286604
$178$		0			0
$179$	21.7569			1.62618			0.813092	−	0.582135i	$-0.197783\pi$
$179$	21.7569			1.62618			0.813092	+	0.582135i	$0.197783\pi$
$180$		0			0
$181$	2.42739			0.180426			0.0902131	−	0.995922i	$-0.471245\pi$
$181$	2.42739			0.180426			0.0902131	+	0.995922i	$0.471245\pi$
$182$		0			0
$183$	−12.2676			−0.906847
$184$		0			0
$185$	3.15955			0.232295
$186$		0			0
$187$	−4.91751			−0.359604
$188$		0			0
$189$	−4.55308			−0.331188
$190$		0			0
$191$	8.94391			0.647158			0.323579	−	0.946201i	$-0.395114\pi$
$191$	8.94391			0.647158			0.323579	+	0.946201i	$0.395114\pi$
$192$		0			0
$193$	−12.2220			−0.879760			−0.439880	−	0.898057i	$-0.644979\pi$
$193$	−12.2220			−0.879760			−0.439880	+	0.898057i	$0.644979\pi$
$194$		0			0
$195$		−	4.35914i		−	0.312165i
$196$		0			0
$197$		−	10.5254i		−	0.749900i	−0.927045	−	0.374950i	$-0.877660\pi$
$197$		−	10.5254i		−	0.749900i	0.927045	−	0.374950i	$-0.122340\pi$
$198$		0			0
$199$	−16.2033			−1.14862			−0.574312	−	0.818637i	$-0.694730\pi$
$199$	−16.2033			−1.14862			−0.574312	+	0.818637i	$0.694730\pi$
$200$		0			0
$201$			10.6451i			0.750845i
$202$		0			0
$203$		−	28.9398i		−	2.03118i
$204$		0			0
$205$	−8.60085			−0.600709
$206$		0			0
$207$	−4.70330	+	0.937530i	−0.326902	+	0.0651629i
$208$		0			0
$209$	−0.630161			−0.0435891
$210$		0			0
$211$	−6.20898			−0.427444			−0.213722	−	0.976895i	$-0.568559\pi$
$211$	−6.20898			−0.427444			−0.213722	+	0.976895i	$0.568559\pi$
$212$		0			0
$213$			14.9045i			1.02124i
$214$		0			0
$215$			9.25021i			0.630859i
$216$		0			0
$217$			25.6823i			1.74343i
$218$		0			0
$219$	−8.18221			−0.552902
$220$		0			0
$221$	24.1518			1.62463
$222$		0			0
$223$			22.7592i			1.52407i	0.647538	+	0.762033i	$0.275799\pi$
$223$			22.7592i			1.52407i	−0.647538	+	0.762033i	$0.724201\pi$
$224$		0			0
$225$	−4.06052			−0.270701
$226$		0			0
$227$			5.19084i			0.344528i	0.985051	+	0.172264i	$0.0551083\pi$
$227$			5.19084i			0.344528i	−0.985051	+	0.172264i	$0.944892\pi$
$228$		0			0
$229$	−16.5481			−1.09353			−0.546764	−	0.837287i	$-0.684140\pi$
$229$	−16.5481			−1.09353			−0.546764	+	0.837287i	$0.684140\pi$
$230$		0			0
$231$		−	4.16925i		−	0.274317i
$232$		0			0
$233$	−15.1175			−0.990382			−0.495191	−	0.868784i	$-0.664902\pi$
$233$	−15.1175			−0.990382			−0.495191	+	0.868784i	$0.664902\pi$
$234$		0			0
$235$		−	0.151211i		−	0.00986395i
$236$		0			0
$237$	−12.0667			−0.783817
$238$		0			0
$239$		−	4.20687i		−	0.272120i	−0.990701	−	0.136060i	$-0.956556\pi$
$239$		−	4.20687i		−	0.272120i	0.990701	−	0.136060i	$-0.0434439\pi$
$240$		0			0
$241$		−	11.2163i		−	0.722503i	−0.932469	−	0.361251i	$-0.882350\pi$
$241$		−	11.2163i		−	0.722503i	0.932469	−	0.361251i	$-0.117650\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	13.3086			0.850254
$246$		0			0
$247$	3.09496			0.196928
$248$		0			0
$249$		−	11.1608i		−	0.707289i
$250$		0			0
$251$		−	19.6483i		−	1.24019i	−0.784526	−	0.620096i	$-0.787094\pi$
$251$		−	19.6483i		−	1.24019i	0.784526	−	0.620096i	$-0.212906\pi$
$252$		0			0
$253$	−0.858496	−	4.30681i	−0.0539732	−	0.270767i
$254$		0			0
$255$			5.20519i			0.325962i
$256$		0			0
$257$	12.3772			0.772071			0.386036	−	0.922484i	$-0.373844\pi$
$257$	12.3772			0.772071			0.386036	+	0.922484i	$0.373844\pi$
$258$		0			0
$259$	−14.8418			−0.922224
$260$		0			0
$261$			6.35610i			0.393433i
$262$		0			0
$263$	−0.300441			−0.0185260			−0.00926300	−	0.999957i	$-0.502949\pi$
$263$	−0.300441			−0.0185260			−0.00926300	+	0.999957i	$0.502949\pi$
$264$		0			0
$265$	−2.35158			−0.144456
$266$		0			0
$267$		−	10.9680i		−	0.671230i
$268$		0			0
$269$		−	1.53065i		−	0.0933252i	−0.998911	−	0.0466626i	$-0.985141\pi$
$269$		−	1.53065i		−	0.0933252i	0.998911	−	0.0466626i	$-0.0148586\pi$
$270$		0			0
$271$			6.52820i			0.396560i	0.980145	+	0.198280i	$0.0635355\pi$
$271$			6.52820i			0.396560i	−0.980145	+	0.198280i	$0.936465\pi$
$272$		0			0
$273$			20.4768i			1.23931i
$274$		0			0
$275$		−	3.71821i		−	0.224217i
$276$		0			0
$277$			7.38599i			0.443781i	0.975072	+	0.221891i	$0.0712228\pi$
$277$			7.38599i			0.443781i	−0.975072	+	0.221891i	$0.928777\pi$
$278$		0			0
$279$		−	5.64064i		−	0.337696i
$280$		0			0
$281$		−	7.79441i		−	0.464976i	−0.972599	−	0.232488i	$-0.925313\pi$
$281$		−	7.79441i		−	0.464976i	0.972599	−	0.232488i	$-0.0746866\pi$
$282$		0			0
$283$		−	27.8464i		−	1.65530i	−0.561246	−	0.827649i	$-0.689678\pi$
$283$		−	27.8464i		−	1.65530i	0.561246	−	0.827649i	$-0.310322\pi$
$284$		0			0
$285$			0.667026i			0.0395112i
$286$		0			0
$287$	40.4019			2.38485
$288$		0			0
$289$	−11.8393			−0.696432
$290$		0			0
$291$		−	17.8948i		−	1.04901i
$292$		0			0
$293$	−12.2770			−0.717232			−0.358616	−	0.933485i	$-0.616751\pi$
$293$	−12.2770			−0.717232			−0.358616	+	0.933485i	$0.616751\pi$
$294$		0			0
$295$	−3.69584			−0.215180
$296$		0			0
$297$			0.915699i			0.0531343i
$298$		0			0
$299$	4.21640	+	21.1524i	0.243841	+	1.22327i
$300$		0			0
$301$		−	43.4523i		−	2.50455i
$302$		0			0
$303$		−	2.18685i		−	0.125631i
$304$		0			0
$305$	−11.8906			−0.680853
$306$		0			0
$307$	0.0911995			0.00520503			0.00260251	−	0.999997i	$-0.499172\pi$
$307$	0.0911995			0.00520503			0.00260251	+	0.999997i	$0.499172\pi$
$308$		0			0
$309$	−1.82776			−0.103978
$310$		0			0
$311$			2.75985i			0.156497i	0.996934	+	0.0782483i	$0.0249327\pi$
$311$			2.75985i			0.156497i	−0.996934	+	0.0782483i	$0.975067\pi$
$312$		0			0
$313$		−	11.2930i		−	0.638317i	−0.947701	−	0.319158i	$-0.896600\pi$
$313$		−	11.2930i		−	0.638317i	0.947701	−	0.319158i	$-0.103400\pi$
$314$		0			0
$315$	−4.41316			−0.248653
$316$		0			0
$317$			23.3082i			1.30912i	0.756010	+	0.654560i	$0.227146\pi$
$317$			23.3082i			1.30912i	−0.756010	+	0.654560i	$0.772854\pi$
$318$		0			0
$319$	−5.82028			−0.325873
$320$		0			0
$321$			2.63011i			0.146798i
$322$		0			0
$323$	−3.69565			−0.205632
$324$		0			0
$325$			18.2616i			1.01297i
$326$		0			0
$327$	−6.14687			−0.339923
$328$		0			0
$329$			0.710307i			0.0391605i
$330$		0			0
$331$	14.7506			0.810766			0.405383	−	0.914147i	$-0.367138\pi$
$331$	14.7506			0.810766			0.405383	+	0.914147i	$0.367138\pi$
$332$		0			0
$333$	3.25973			0.178632
$334$		0			0
$335$			10.3179i			0.563729i
$336$		0			0
$337$			22.4118i			1.22085i	0.792075	+	0.610423i	$0.209001\pi$
$337$			22.4118i			1.22085i	−0.792075	+	0.610423i	$0.790999\pi$
$338$		0			0
$339$			18.8052i			1.02136i
$340$		0			0
$341$	5.16513			0.279707
$342$		0			0
$343$	−30.6447			−1.65466
$344$		0			0
$345$	−4.55876	+	0.908719i	−0.245435	+	0.0489238i
$346$		0			0
$347$	−8.93888			−0.479864			−0.239932	−	0.970790i	$-0.577125\pi$
$347$	−8.93888			−0.479864			−0.239932	+	0.970790i	$0.577125\pi$
$348$		0			0
$349$			33.9944i			1.81968i	0.414960	+	0.909840i	$0.363796\pi$
$349$			33.9944i			1.81968i	−0.414960	+	0.909840i	$0.636204\pi$
$350$		0			0
$351$		−	4.49735i		−	0.240051i
$352$		0			0
$353$	−14.3436			−0.763430			−0.381715	−	0.924280i	$-0.624666\pi$
$353$	−14.3436			−0.763430			−0.381715	+	0.924280i	$0.624666\pi$
$354$		0			0
$355$			14.4465i			0.766741i
$356$		0			0
$357$		−	24.4511i		−	1.29409i
$358$		0			0
$359$	17.4705			0.922059			0.461029	−	0.887385i	$-0.347480\pi$
$359$	17.4705			0.922059			0.461029	+	0.887385i	$0.347480\pi$
$360$		0			0
$361$	18.5264			0.975075
$362$		0			0
$363$	10.1615			0.533340
$364$		0			0
$365$	−7.93076			−0.415115
$366$		0			0
$367$	4.20241			0.219364			0.109682	−	0.993967i	$-0.465017\pi$
$367$	4.20241			0.219364			0.109682	+	0.993967i	$0.465017\pi$
$368$		0			0
$369$	−8.87354			−0.461938
$370$		0			0
$371$	11.0464			0.573500
$372$		0			0
$373$	37.2998			1.93131			0.965655	−	0.259826i	$-0.0836652\pi$
$373$	37.2998			1.93131			0.965655	+	0.259826i	$0.0836652\pi$
$374$		0			0
$375$	−8.78208			−0.453504
$376$		0			0
$377$	28.5856			1.47223
$378$		0			0
$379$		−	12.0640i		−	0.619684i	−0.950788	−	0.309842i	$-0.899724\pi$
$379$		−	12.0640i		−	0.619684i	0.950788	−	0.309842i	$-0.100276\pi$
$380$		0			0
$381$			0.261572i			0.0134007i
$382$		0			0
$383$	15.8402			0.809395			0.404698	−	0.914451i	$-0.367377\pi$
$383$	15.8402			0.809395			0.404698	+	0.914451i	$0.367377\pi$
$384$		0			0
$385$		−	4.04113i		−	0.205955i
$386$		0			0
$387$			9.54349i			0.485123i
$388$		0			0
$389$	2.04276			0.103572			0.0517860	−	0.998658i	$-0.483509\pi$
$389$	2.04276			0.103572			0.0517860	+	0.998658i	$0.483509\pi$
$390$		0			0
$391$	−5.03475	−	25.2578i	−0.254618	−	1.27734i
$392$		0			0
$393$	8.22618			0.414956
$394$		0			0
$395$	−11.6959			−0.588484
$396$		0			0
$397$			34.7911i			1.74612i	0.487617	+	0.873058i	$0.337866\pi$
$397$			34.7911i			1.74612i	−0.487617	+	0.873058i	$0.662134\pi$
$398$		0			0
$399$		−	3.13331i		−	0.156862i
$400$		0			0
$401$		−	27.9532i		−	1.39592i	−0.716138	−	0.697958i	$-0.754092\pi$
$401$		−	27.9532i		−	1.39592i	0.716138	−	0.697958i	$-0.245908\pi$
$402$		0			0
$403$	−25.3679			−1.26367
$404$		0			0
$405$	0.969269			0.0481633
$406$		0			0
$407$			2.98493i			0.147957i
$408$		0			0
$409$	−10.4393			−0.516192			−0.258096	−	0.966119i	$-0.583095\pi$
$409$	−10.4393			−0.516192			−0.258096	+	0.966119i	$0.583095\pi$
$410$		0			0
$411$			6.70631i			0.330798i
$412$		0			0
$413$	17.3610			0.854279
$414$		0			0
$415$		−	10.8179i		−	0.531027i
$416$		0			0
$417$	7.42604			0.363655
$418$		0			0
$419$		−	27.9406i		−	1.36499i	−0.730891	−	0.682494i	$-0.760895\pi$
$419$		−	27.9406i		−	1.36499i	0.730891	−	0.682494i	$-0.239105\pi$
$420$		0			0
$421$	32.6815			1.59280			0.796400	−	0.604770i	$-0.206735\pi$
$421$	32.6815			1.59280			0.796400	+	0.604770i	$0.206735\pi$
$422$		0			0
$423$		−	0.156006i		−	0.00758526i
$424$		0			0
$425$		−	21.8059i		−	1.05774i
$426$		0			0
$427$	55.8553			2.70303
$428$		0			0
$429$	4.11822			0.198830
$430$		0			0
$431$	−16.1528			−0.778055			−0.389028	−	0.921226i	$-0.627189\pi$
$431$	−16.1528			−0.778055			−0.389028	+	0.921226i	$0.627189\pi$
$432$		0			0
$433$			19.9138i			0.956996i	0.878089	+	0.478498i	$0.158819\pi$
$433$			19.9138i			0.956996i	−0.878089	+	0.478498i	$0.841181\pi$
$434$		0			0
$435$			6.16077i			0.295386i
$436$		0			0
$437$	−0.645184	−	3.23669i	−0.0308633	−	0.154832i
$438$		0			0
$439$			29.3002i			1.39842i	0.714915	+	0.699211i	$0.246465\pi$
$439$			29.3002i			1.39842i	−0.714915	+	0.699211i	$0.753535\pi$
$440$		0			0
$441$	13.7305			0.653835
$442$		0			0
$443$	−32.9596			−1.56596			−0.782978	−	0.622049i	$-0.786301\pi$
$443$	−32.9596			−1.56596			−0.782978	+	0.622049i	$0.786301\pi$
$444$		0			0
$445$		−	10.6309i		−	0.503954i
$446$		0			0
$447$	16.0149			0.757480
$448$		0			0
$449$	16.6963			0.787949			0.393975	−	0.919121i	$-0.371100\pi$
$449$	16.6963			0.787949			0.393975	+	0.919121i	$0.371100\pi$
$450$		0			0
$451$		−	8.12550i		−	0.382615i
$452$		0			0
$453$		−	8.36872i		−	0.393197i
$454$		0			0
$455$			19.8475i			0.930466i
$456$		0			0
$457$			7.54667i			0.353018i	0.984299	+	0.176509i	$0.0564805\pi$
$457$			7.54667i			0.353018i	−0.984299	+	0.176509i	$0.943519\pi$
$458$		0			0
$459$			5.37023i			0.250661i
$460$		0			0
$461$		−	22.0330i		−	1.02618i	−0.858335	−	0.513090i	$-0.828501\pi$
$461$		−	22.0330i		−	1.02618i	0.858335	−	0.513090i	$-0.171499\pi$
$462$		0			0
$463$		−	10.5899i		−	0.492153i	−0.969250	−	0.246077i	$-0.920858\pi$
$463$		−	10.5899i		−	0.492153i	0.969250	−	0.246077i	$-0.0791415\pi$
$464$		0			0
$465$		−	5.46729i		−	0.253540i
$466$		0			0
$467$			23.8323i			1.10283i	0.834231	+	0.551414i	$0.185912\pi$
$467$			23.8323i			1.10283i	−0.834231	+	0.551414i	$0.814088\pi$
$468$		0			0
$469$		−	48.4679i		−	2.23804i
$470$		0			0
$471$	−0.188409			−0.00868141
$472$		0			0
$473$	−8.73897			−0.401818
$474$		0			0
$475$		−	2.79434i		−	0.128213i
$476$		0			0
$477$	−2.42613			−0.111085
$478$		0			0
$479$	18.4098			0.841165			0.420583	−	0.907254i	$-0.361826\pi$
$479$	18.4098			0.841165			0.420583	+	0.907254i	$0.361826\pi$
$480$		0			0
$481$		−	14.6601i		−	0.668445i
$482$		0			0
$483$	21.4145	−	4.26865i	0.974394	−	0.194230i
$484$		0			0
$485$		−	17.3449i		−	0.787591i
$486$		0			0
$487$		−	39.6459i		−	1.79653i	−0.439458	−	0.898263i	$-0.644829\pi$
$487$		−	39.6459i		−	1.79653i	0.439458	−	0.898263i	$-0.355171\pi$
$488$		0			0
$489$	−6.04337			−0.273291
$490$		0			0
$491$	30.1004			1.35841			0.679206	−	0.733947i	$-0.262324\pi$
$491$	30.1004			1.35841			0.679206	+	0.733947i	$0.262324\pi$
$492$		0			0
$493$	−34.1337			−1.53730
$494$		0			0
$495$			0.887559i			0.0398928i
$496$		0			0
$497$		−	67.8616i		−	3.04401i
$498$		0			0
$499$	21.5335			0.963974			0.481987	−	0.876178i	$-0.339915\pi$
$499$	21.5335			0.963974			0.481987	+	0.876178i	$0.339915\pi$
$500$		0			0
$501$		−	3.28126i		−	0.146596i
$502$		0			0
$503$	−22.2489			−0.992030			−0.496015	−	0.868314i	$-0.665204\pi$
$503$	−22.2489			−0.992030			−0.496015	+	0.868314i	$0.665204\pi$
$504$		0			0
$505$		−	2.11964i		−	0.0943229i
$506$		0			0
$507$	−7.22618			−0.320926
$508$		0			0
$509$		−	30.3648i		−	1.34590i	−0.739689	−	0.672949i	$-0.765028\pi$
$509$		−	30.3648i		−	1.34590i	0.739689	−	0.672949i	$-0.234972\pi$
$510$		0			0
$511$	37.2543			1.64803
$512$		0			0
$513$			0.688174i			0.0303836i
$514$		0			0
$515$	−1.77159			−0.0780658
$516$		0			0
$517$	0.142854			0.00628273
$518$		0			0
$519$			4.28422i			0.188056i
$520$		0			0
$521$		−	35.7402i		−	1.56580i	−0.622145	−	0.782902i	$-0.713738\pi$
$521$		−	35.7402i		−	1.56580i	0.622145	−	0.782902i	$-0.286262\pi$
$522$		0			0
$523$			9.30899i			0.407054i	0.979069	+	0.203527i	$0.0652404\pi$
$523$			9.30899i			0.407054i	−0.979069	+	0.203527i	$0.934760\pi$
$524$		0			0
$525$	18.4879			0.806877
$526$		0			0
$527$	30.2915			1.31952
$528$		0			0
$529$	21.2421	−	8.81897i	0.923568	−	0.383434i
$530$		0			0
$531$	−3.81302			−0.165471
$532$		0			0
$533$			39.9074i			1.72858i
$534$		0			0
$535$			2.54928i			0.110215i
$536$		0			0
$537$	21.7569			0.938878
$538$		0			0
$539$			12.5730i			0.541559i
$540$		0			0
$541$			20.3348i			0.874263i	0.899398	+	0.437131i	$0.144006\pi$
$541$			20.3348i			0.874263i	−0.899398	+	0.437131i	$0.855994\pi$
$542$		0			0
$543$	2.42739			0.104169
$544$		0			0
$545$	−5.95797			−0.255211
$546$		0			0
$547$	−39.4942			−1.68865			−0.844324	−	0.535832i	$-0.819998\pi$
$547$	−39.4942			−1.68865			−0.844324	+	0.535832i	$0.819998\pi$
$548$		0			0
$549$	−12.2676			−0.523568
$550$		0			0
$551$	−4.37410			−0.186343
$552$		0			0
$553$	54.9407			2.33632
$554$		0			0
$555$	3.15955			0.134115
$556$		0			0
$557$	0.909976			0.0385569			0.0192785	−	0.999814i	$-0.493863\pi$
$557$	0.909976			0.0385569			0.0192785	+	0.999814i	$0.493863\pi$
$558$		0			0
$559$	42.9205			1.81534
$560$		0			0
$561$	−4.91751			−0.207618
$562$		0			0
$563$		−	24.7038i		−	1.04114i	−0.853819	−	0.520571i	$-0.825719\pi$
$563$		−	24.7038i		−	1.04114i	0.853819	−	0.520571i	$-0.174281\pi$
$564$		0			0
$565$			18.2273i			0.766829i
$566$		0			0
$567$	−4.55308			−0.191211
$568$		0			0
$569$			4.12434i			0.172901i	0.996256	+	0.0864507i	$0.0275525\pi$
$569$			4.12434i			0.172901i	−0.996256	+	0.0864507i	$0.972448\pi$
$570$		0			0
$571$			27.5717i			1.15384i	0.816801	+	0.576919i	$0.195745\pi$
$571$			27.5717i			1.15384i	−0.816801	+	0.576919i	$0.804255\pi$
$572$		0			0
$573$	8.94391			0.373637
$574$		0			0
$575$	19.0978	−	3.80686i	0.796435	−	0.158757i
$576$		0			0
$577$	−30.3113			−1.26188			−0.630938	−	0.775833i	$-0.717330\pi$
$577$	−30.3113			−1.26188			−0.630938	+	0.775833i	$0.717330\pi$
$578$		0			0
$579$	−12.2220			−0.507930
$580$		0			0
$581$			50.8162i			2.10821i
$582$		0			0
$583$		−	2.22161i		−	0.0920097i
$584$		0			0
$585$		−	4.35914i		−	0.180228i
$586$		0			0
$587$	−14.3608			−0.592732			−0.296366	−	0.955074i	$-0.595775\pi$
$587$	−14.3608			−0.592732			−0.296366	+	0.955074i	$0.595775\pi$
$588$		0			0
$589$	3.88174			0.159944
$590$		0			0
$591$		−	10.5254i		−	0.432955i
$592$		0			0
$593$	−25.1240			−1.03172			−0.515859	−	0.856674i	$-0.672527\pi$
$593$	−25.1240			−1.03172			−0.515859	+	0.856674i	$0.672527\pi$
$594$		0			0
$595$		−	23.6997i		−	0.971591i
$596$		0			0
$597$	−16.2033			−0.663158
$598$		0			0
$599$		−	3.10960i		−	0.127055i	−0.997980	−	0.0635274i	$-0.979765\pi$
$599$		−	3.10960i		−	0.127055i	0.997980	−	0.0635274i	$-0.0202350\pi$
$600$		0			0
$601$	−14.1864			−0.578674			−0.289337	−	0.957227i	$-0.593435\pi$
$601$	−14.1864			−0.578674			−0.289337	+	0.957227i	$0.593435\pi$
$602$		0			0
$603$			10.6451i			0.433501i
$604$		0			0
$605$	9.84922			0.400428
$606$		0			0
$607$		−	6.65785i		−	0.270234i	−0.990830	−	0.135117i	$-0.956859\pi$
$607$		−	6.65785i		−	0.270234i	0.990830	−	0.135117i	$-0.0431410\pi$
$608$		0			0
$609$		−	28.9398i		−	1.17270i
$610$		0			0
$611$	−0.701613			−0.0283842
$612$		0			0
$613$	−8.35314			−0.337380			−0.168690	−	0.985669i	$-0.553954\pi$
$613$	−8.35314			−0.337380			−0.168690	+	0.985669i	$0.553954\pi$
$614$		0			0
$615$	−8.60085			−0.346820
$616$		0			0
$617$			8.98374i			0.361672i	0.983513	+	0.180836i	$0.0578803\pi$
$617$			8.98374i			0.361672i	−0.983513	+	0.180836i	$0.942120\pi$
$618$		0			0
$619$			25.1148i			1.00945i	0.863280	+	0.504725i	$0.168406\pi$
$619$			25.1148i			1.00945i	−0.863280	+	0.504725i	$0.831594\pi$
$620$		0			0
$621$	−4.70330	+	0.937530i	−0.188737	+	0.0376218i
$622$		0			0
$623$			49.9381i			2.00073i
$624$		0			0
$625$	11.7904			0.471616
$626$		0			0
$627$	−0.630161			−0.0251662
$628$		0			0
$629$			17.5055i			0.697989i
$630$		0			0
$631$	−16.1347			−0.642312			−0.321156	−	0.947026i	$-0.604071\pi$
$631$	−16.1347			−0.642312			−0.321156	+	0.947026i	$0.604071\pi$
$632$		0			0
$633$	−6.20898			−0.246785
$634$		0			0
$635$			0.253533i			0.0100612i
$636$		0			0
$637$		−	61.7511i		−	2.44667i
$638$		0			0
$639$			14.9045i			0.589614i
$640$		0			0
$641$		−	17.9723i		−	0.709865i	−0.934892	−	0.354932i	$-0.884504\pi$
$641$		−	17.9723i		−	0.709865i	0.934892	−	0.354932i	$-0.115496\pi$
$642$		0			0
$643$		−	37.3857i		−	1.47435i	−0.675704	−	0.737173i	$-0.736160\pi$
$643$		−	37.3857i		−	1.47435i	0.675704	−	0.737173i	$-0.263840\pi$
$644$		0			0
$645$			9.25021i			0.364227i
$646$		0			0
$647$			0.412854i			0.0162310i	0.999967	+	0.00811548i	$0.00258327\pi$
$647$			0.412854i			0.0162310i	−0.999967	+	0.00811548i	$0.997417\pi$
$648$		0			0
$649$		−	3.49158i		−	0.137057i
$650$		0			0
$651$			25.6823i			1.00657i
$652$		0			0
$653$			47.0964i			1.84302i	0.388350	+	0.921512i	$0.373045\pi$
$653$			47.0964i			1.84302i	−0.388350	+	0.921512i	$0.626955\pi$
$654$		0			0
$655$	7.97338			0.311546
$656$		0			0
$657$	−8.18221			−0.319218
$658$		0			0
$659$			35.5752i			1.38581i	0.721027	+	0.692907i	$0.243670\pi$
$659$			35.5752i			1.38581i	−0.721027	+	0.692907i	$0.756330\pi$
$660$		0			0
$661$	24.5240			0.953872			0.476936	−	0.878938i	$-0.341747\pi$
$661$	24.5240			0.953872			0.476936	+	0.878938i	$0.341747\pi$
$662$		0			0
$663$	24.1518			0.937979
$664$		0			0
$665$		−	3.03702i		−	0.117771i
$666$		0			0
$667$	−5.95904	−	29.8947i	−0.230735	−	1.15753i
$668$		0			0
$669$			22.7592i			0.879920i
$670$		0			0
$671$		−	11.2334i		−	0.433662i
$672$		0			0
$673$	−49.3893			−1.90382			−0.951910	−	0.306379i	$-0.900882\pi$
$673$	−49.3893			−1.90382			−0.951910	+	0.306379i	$0.900882\pi$
$674$		0			0
$675$	−4.06052			−0.156289
$676$		0			0
$677$	38.7298			1.48851			0.744254	−	0.667896i	$-0.232805\pi$
$677$	38.7298			1.48851			0.744254	+	0.667896i	$0.232805\pi$
$678$		0			0
$679$			81.4766i			3.12679i
$680$		0			0
$681$			5.19084i			0.198914i
$682$		0			0
$683$	11.5331			0.441303			0.220651	−	0.975353i	$-0.429182\pi$
$683$	11.5331			0.441303			0.220651	+	0.975353i	$0.429182\pi$
$684$		0			0
$685$			6.50022i			0.248361i
$686$		0			0
$687$	−16.5481			−0.631348
$688$		0			0
$689$			10.9112i			0.415683i
$690$		0			0
$691$	19.3937			0.737771			0.368885	−	0.929475i	$-0.379739\pi$
$691$	19.3937			0.737771			0.368885	+	0.929475i	$0.379739\pi$
$692$		0			0
$693$		−	4.16925i		−	0.158377i
$694$		0			0
$695$	7.19783			0.273029
$696$		0			0
$697$		−	47.6529i		−	1.80498i
$698$		0			0
$699$	−15.1175			−0.571797
$700$		0			0
$701$	42.0341			1.58761			0.793803	−	0.608174i	$-0.208098\pi$
$701$	42.0341			1.58761			0.793803	+	0.608174i	$0.208098\pi$
$702$		0			0
$703$			2.24326i			0.0846061i
$704$		0			0
$705$		−	0.151211i		−	0.00569495i
$706$		0			0
$707$			9.95690i			0.374468i
$708$		0			0
$709$	−7.00113			−0.262933			−0.131467	−	0.991321i	$-0.541969\pi$
$709$	−7.00113			−0.262933			−0.131467	+	0.991321i	$0.541969\pi$
$710$		0			0
$711$	−12.0667			−0.452537
$712$		0			0
$713$	5.28827	+	26.5296i	0.198047	+	0.993542i
$714$		0			0
$715$	3.99166			0.149280
$716$		0			0
$717$		−	4.20687i		−	0.157108i
$718$		0			0
$719$		−	32.5628i		−	1.21439i	−0.794554	−	0.607193i	$-0.792295\pi$
$719$		−	32.5628i		−	1.21439i	0.794554	−	0.607193i	$-0.207705\pi$
$720$		0			0
$721$	8.32196			0.309926
$722$		0			0
$723$		−	11.2163i		−	0.417137i
$724$		0			0
$725$		−	25.8091i		−	0.958525i
$726$		0			0
$727$	−8.30822			−0.308135			−0.154067	−	0.988060i	$-0.549237\pi$
$727$	−8.30822			−0.308135			−0.154067	+	0.988060i	$0.549237\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−51.2507			−1.89558
$732$		0			0
$733$	5.54133			0.204674			0.102337	−	0.994750i	$-0.467368\pi$
$733$	5.54133			0.204674			0.102337	+	0.994750i	$0.467368\pi$
$734$		0			0
$735$	13.3086			0.490894
$736$		0			0
$737$	−9.74768			−0.359061
$738$		0			0
$739$	−16.1333			−0.593472			−0.296736	−	0.954960i	$-0.595898\pi$
$739$	−16.1333			−0.593472			−0.296736	+	0.954960i	$0.595898\pi$
$740$		0			0
$741$	3.09496			0.113696
$742$		0			0
$743$	−46.0973			−1.69115			−0.845574	−	0.533859i	$-0.820741\pi$
$743$	−46.0973			−1.69115			−0.845574	+	0.533859i	$0.820741\pi$
$744$		0			0
$745$	15.5228			0.568710
$746$		0			0
$747$		−	11.1608i		−	0.408354i
$748$		0			0
$749$		−	11.9751i		−	0.437561i
$750$		0			0
$751$	2.29470			0.0837349			0.0418675	−	0.999123i	$-0.486669\pi$
$751$	2.29470			0.0837349			0.0418675	+	0.999123i	$0.486669\pi$
$752$		0			0
$753$		−	19.6483i		−	0.716025i
$754$		0			0
$755$		−	8.11154i		−	0.295209i
$756$		0			0
$757$	−25.5612			−0.929037			−0.464518	−	0.885563i	$-0.653773\pi$
$757$	−25.5612			−0.929037			−0.464518	+	0.885563i	$0.653773\pi$
$758$		0			0
$759$	−0.858496	−	4.30681i	−0.0311614	−	0.156327i
$760$		0			0
$761$	16.0610			0.582210			0.291105	−	0.956691i	$-0.405977\pi$
$761$	16.0610			0.582210			0.291105	+	0.956691i	$0.405977\pi$
$762$		0			0
$763$	27.9872			1.01321
$764$		0			0
$765$			5.20519i			0.188194i
$766$		0			0
$767$			17.1485i			0.619197i
$768$		0			0
$769$			28.1563i			1.01534i	0.861551	+	0.507671i	$0.169493\pi$
$769$			28.1563i			1.01534i	−0.861551	+	0.507671i	$0.830507\pi$
$770$		0			0
$771$	12.3772			0.445756
$772$		0			0
$773$	−24.8284			−0.893015			−0.446508	−	0.894780i	$-0.647332\pi$
$773$	−24.8284			−0.893015			−0.446508	+	0.894780i	$0.647332\pi$
$774$		0			0
$775$			22.9039i			0.822733i
$776$		0			0
$777$	−14.8418			−0.532447
$778$		0			0
$779$		−	6.10654i		−	0.218790i
$780$		0			0
$781$	−13.6481			−0.488367
$782$		0			0
$783$			6.35610i			0.227149i
$784$		0			0
$785$	−0.182618			−0.00651793
$786$		0			0
$787$		−	30.2006i		−	1.07653i	−0.842774	−	0.538267i	$-0.819079\pi$
$787$		−	30.2006i		−	1.07653i	0.842774	−	0.538267i	$-0.180921\pi$
$788$		0			0
$789$	−0.300441			−0.0106960
$790$		0			0
$791$		−	85.6217i		−	3.04436i
$792$		0			0
$793$			55.1717i			1.95920i
$794$		0			0
$795$	−2.35158			−0.0834018
$796$		0			0
$797$	−0.413948			−0.0146628			−0.00733139	−	0.999973i	$-0.502334\pi$
$797$	−0.413948			−0.0146628			−0.00733139	+	0.999973i	$0.502334\pi$
$798$		0			0
$799$	0.837786			0.0296387
$800$		0			0
$801$		−	10.9680i		−	0.387535i
$802$		0			0
$803$		−	7.49244i		−	0.264403i
$804$		0			0
$805$	20.7564	−	4.13747i	0.731567	−	0.145827i
$806$		0			0
$807$		−	1.53065i		−	0.0538814i
$808$		0			0
$809$	3.27386			0.115103			0.0575513	−	0.998343i	$-0.481671\pi$
$809$	3.27386			0.115103			0.0575513	+	0.998343i	$0.481671\pi$
$810$		0			0
$811$	−5.52427			−0.193983			−0.0969916	−	0.995285i	$-0.530922\pi$
$811$	−5.52427			−0.193983			−0.0969916	+	0.995285i	$0.530922\pi$
$812$		0			0
$813$			6.52820i			0.228954i
$814$		0			0
$815$	−5.85765			−0.205185
$816$		0			0
$817$	−6.56758			−0.229771
$818$		0			0
$819$			20.4768i			0.715518i
$820$		0			0
$821$		−	40.6786i		−	1.41969i	−0.704356	−	0.709847i	$-0.748764\pi$
$821$		−	40.6786i		−	1.41969i	0.704356	−	0.709847i	$-0.251236\pi$
$822$		0			0
$823$		−	31.2641i		−	1.08980i	−0.838502	−	0.544898i	$-0.816568\pi$
$823$		−	31.2641i		−	1.08980i	0.838502	−	0.544898i	$-0.183432\pi$
$824$		0			0
$825$		−	3.71821i		−	0.129452i
$826$		0			0
$827$		−	20.8630i		−	0.725478i	−0.931891	−	0.362739i	$-0.881842\pi$
$827$		−	20.8630i		−	0.725478i	0.931891	−	0.362739i	$-0.118158\pi$
$828$		0			0
$829$		−	11.6152i		−	0.403414i	−0.979446	−	0.201707i	$-0.935351\pi$
$829$		−	11.6152i		−	0.403414i	0.979446	−	0.201707i	$-0.0646489\pi$
$830$		0			0
$831$			7.38599i			0.256217i
$832$		0			0
$833$			73.7361i			2.55481i
$834$		0			0
$835$		−	3.18042i		−	0.110063i
$836$		0			0
$837$		−	5.64064i		−	0.194969i
$838$		0			0
$839$	−35.2227			−1.21602			−0.608011	−	0.793929i	$-0.708032\pi$
$839$	−35.2227			−1.21602			−0.608011	+	0.793929i	$0.708032\pi$
$840$		0			0
$841$	−11.4000			−0.393104
$842$		0			0
$843$		−	7.79441i		−	0.268454i
$844$		0			0
$845$	−7.00411			−0.240949
$846$		0			0
$847$	−46.2661			−1.58972
$848$		0			0
$849$		−	27.8464i		−	0.955687i
$850$		0			0
$851$	−15.3315	+	3.05609i	−0.525556	+	0.104762i
$852$		0			0
$853$			11.0149i			0.377145i	0.982059	+	0.188572i	$0.0603860\pi$
$853$			11.0149i			0.377145i	−0.982059	+	0.188572i	$0.939614\pi$
$854$		0			0
$855$			0.667026i			0.0228118i
$856$		0			0
$857$	4.44133			0.151713			0.0758565	−	0.997119i	$-0.475831\pi$
$857$	4.44133			0.151713			0.0758565	+	0.997119i	$0.475831\pi$
$858$		0			0
$859$	−55.9465			−1.90887			−0.954436	−	0.298417i	$-0.903541\pi$
$859$	−55.9465			−1.90887			−0.954436	+	0.298417i	$0.903541\pi$
$860$		0			0
$861$	40.4019			1.37689
$862$		0			0
$863$			19.0075i			0.647022i	0.946224	+	0.323511i	$0.104863\pi$
$863$			19.0075i			0.647022i	−0.946224	+	0.323511i	$0.895137\pi$
$864$		0			0
$865$			4.15256i			0.141191i
$866$		0			0
$867$	−11.8393			−0.402085
$868$		0			0
$869$		−	11.0495i		−	0.374828i
$870$		0			0
$871$	47.8746			1.62217
$872$		0			0
$873$		−	17.8948i		−	0.605648i
$874$		0			0
$875$	39.9855			1.35176
$876$		0			0
$877$			39.4573i			1.33238i	0.745782	+	0.666190i	$0.232076\pi$
$877$			39.4573i			1.33238i	−0.745782	+	0.666190i	$0.767924\pi$
$878$		0			0
$879$	−12.2770			−0.414094
$880$		0			0
$881$			17.9584i			0.605035i	0.953144	+	0.302517i	$0.0978270\pi$
$881$			17.9584i			0.605035i	−0.953144	+	0.302517i	$0.902173\pi$
$882$		0			0
$883$	39.4681			1.32821			0.664104	−	0.747640i	$-0.268813\pi$
$883$	39.4681			1.32821			0.664104	+	0.747640i	$0.268813\pi$
$884$		0			0
$885$	−3.69584			−0.124234
$886$		0			0
$887$			15.0370i			0.504892i	0.967611	+	0.252446i	$0.0812350\pi$
$887$			15.0370i			0.504892i	−0.967611	+	0.252446i	$0.918765\pi$
$888$		0			0
$889$		−	1.19096i		−	0.0399434i
$890$		0			0
$891$			0.915699i			0.0306771i
$892$		0			0
$893$	0.107359			0.00359264
$894$		0			0
$895$	21.0882			0.704902
$896$		0			0
$897$	4.21640	+	21.1524i	0.140782	+	0.706258i
$898$		0			0
$899$	35.8525			1.19575
$900$		0			0
$901$		−	13.0289i		−	0.434055i
$902$		0			0
$903$		−	43.4523i		−	1.44600i
$904$		0			0
$905$	2.35279			0.0782094
$906$		0			0
$907$		−	9.18521i		−	0.304990i	−0.988304	−	0.152495i	$-0.951269\pi$
$907$		−	9.18521i		−	0.304990i	0.988304	−	0.152495i	$-0.0487308\pi$
$908$		0			0
$909$		−	2.18685i		−	0.0725332i
$910$		0			0
$911$	−20.7570			−0.687710			−0.343855	−	0.939023i	$-0.611733\pi$
$911$	−20.7570			−0.687710			−0.343855	+	0.939023i	$0.611733\pi$
$912$		0			0
$913$	10.2200			0.338232
$914$		0			0
$915$	−11.8906			−0.393091
$916$		0			0
$917$	−37.4545			−1.23686
$918$		0			0
$919$	−51.6210			−1.70282			−0.851409	−	0.524502i	$-0.824251\pi$
$919$	−51.6210			−1.70282			−0.851409	+	0.524502i	$0.824251\pi$
$920$		0			0
$921$	0.0911995			0.00300513
$922$		0			0
$923$	67.0310			2.20635
$924$		0			0
$925$	−13.2362			−0.435203
$926$		0			0
$927$	−1.82776			−0.0600317
$928$		0			0
$929$	36.6120			1.20120			0.600600	−	0.799549i	$-0.294928\pi$
$929$	36.6120			1.20120			0.600600	+	0.799549i	$0.294928\pi$
$930$		0			0
$931$			9.44900i			0.309679i
$932$		0			0
$933$			2.75985i			0.0903534i
$934$		0			0
$935$	−4.76639			−0.155878
$936$		0			0
$937$		−	6.16375i		−	0.201361i	−0.994919	−	0.100680i	$-0.967898\pi$
$937$		−	6.16375i		−	0.201361i	0.994919	−	0.100680i	$-0.0321020\pi$
$938$		0			0
$939$		−	11.2930i		−	0.368532i
$940$		0			0
$941$	12.0779			0.393729			0.196865	−	0.980431i	$-0.436924\pi$
$941$	12.0779			0.393729			0.196865	+	0.980431i	$0.436924\pi$
$942$		0			0
$943$	41.7349	−	8.31921i	1.35908	−	0.270911i
$944$		0			0
$945$	−4.41316			−0.143560
$946$		0			0
$947$	51.7679			1.68223			0.841115	−	0.540856i	$-0.181900\pi$
$947$	51.7679			1.68223			0.841115	+	0.540856i	$0.181900\pi$
$948$		0			0
$949$			36.7983i			1.19452i
$950$		0			0
$951$			23.3082i			0.755821i
$952$		0			0
$953$			18.0036i			0.583193i	0.956541	+	0.291597i	$0.0941865\pi$
$953$			18.0036i			0.583193i	−0.956541	+	0.291597i	$0.905813\pi$
$954$		0			0
$955$	8.66905			0.280524
$956$		0			0
$957$	−5.82028			−0.188143
$958$		0			0
$959$		−	30.5344i		−	0.986006i
$960$		0			0
$961$	−0.816783			−0.0263478
$962$		0			0
$963$			2.63011i			0.0847541i
$964$		0			0
$965$	−11.8464			−0.381350
$966$		0			0
$967$		−	51.6314i		−	1.66035i	−0.557500	−	0.830177i	$-0.688240\pi$
$967$		−	51.6314i		−	1.66035i	0.557500	−	0.830177i	$-0.311760\pi$
$968$		0			0
$969$	−3.69565			−0.118721
$970$		0			0
$971$			6.84543i			0.219680i	0.993949	+	0.109840i	$0.0350339\pi$
$971$			6.84543i			0.219680i	−0.993949	+	0.109840i	$0.964966\pi$
$972$		0			0
$973$	−33.8113			−1.08394
$974$		0			0
$975$			18.2616i			0.584839i
$976$		0			0
$977$			18.2136i			0.582706i	0.956616	+	0.291353i	$0.0941054\pi$
$977$			18.2136i			0.582706i	−0.956616	+	0.291353i	$0.905895\pi$
$978$		0			0
$979$	10.0434			0.320988
$980$		0			0
$981$	−6.14687			−0.196255
$982$		0			0
$983$	9.93131			0.316760			0.158380	−	0.987378i	$-0.449373\pi$
$983$	9.93131			0.316760			0.158380	+	0.987378i	$0.449373\pi$
$984$		0			0
$985$		−	10.2019i		−	0.325059i
$986$		0			0
$987$			0.710307i			0.0226093i
$988$		0			0
$989$	−8.94731	−	44.8859i	−0.284508	−	1.42729i
$990$		0			0
$991$			26.5845i			0.844485i	0.906483	+	0.422242i	$0.138757\pi$
$991$			26.5845i			0.844485i	−0.906483	+	0.422242i	$0.861243\pi$
$992$		0			0
$993$	14.7506			0.468096
$994$		0			0
$995$	−15.7054			−0.497894
$996$		0			0
$997$			9.89264i			0.313303i	0.987654	+	0.156652i	$0.0500700\pi$
$997$			9.89264i			0.313303i	−0.987654	+	0.156652i	$0.949930\pi$
$998$		0			0
$999$	3.25973			0.103133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.n.b.367.16		24
4.3	odd	2		552.2.n.b.91.20	yes	24
8.3	odd	2	inner	2208.2.n.b.367.10		24
8.5	even	2		552.2.n.b.91.17	✓	24
23.22	odd	2	inner	2208.2.n.b.367.9		24
92.91	even	2		552.2.n.b.91.19	yes	24
184.45	odd	2		552.2.n.b.91.18	yes	24
184.91	even	2	inner	2208.2.n.b.367.15		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.n.b.91.17	✓	24	8.5	even	2
552.2.n.b.91.18	yes	24	184.45	odd	2
552.2.n.b.91.19	yes	24	92.91	even	2
552.2.n.b.91.20	yes	24	4.3	odd	2
2208.2.n.b.367.9		24	23.22	odd	2	inner
2208.2.n.b.367.10		24	8.3	odd	2	inner
2208.2.n.b.367.15		24	184.91	even	2	inner
2208.2.n.b.367.16		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:	\( N \)	\(=\)	\( 2208 = 2^{5} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2208.n (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.969269 q^{5} -4.55308 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.969269 q^{5} -4.55308 q^{7} +1.00000 q^{9} +0.915699i q^{11} -4.49735i q^{13} +0.969269 q^{15} +5.37023i q^{17} +0.688174i q^{19} -4.55308 q^{21} +(-4.70330 + 0.937530i) q^{23} -4.06052 q^{25} +1.00000 q^{27} +6.35610i q^{29} -5.64064i q^{31} +0.915699i q^{33} -4.41316 q^{35} +3.25973 q^{37} -4.49735i q^{39} -8.87354 q^{41} +9.54349i q^{43} +0.969269 q^{45} -0.156006i q^{47} +13.7305 q^{49} +5.37023i q^{51} -2.42613 q^{53} +0.887559i q^{55} +0.688174i q^{57} -3.81302 q^{59} -12.2676 q^{61} -4.55308 q^{63} -4.35914i q^{65} +10.6451i q^{67} +(-4.70330 + 0.937530i) q^{69} +14.9045i q^{71} -8.18221 q^{73} -4.06052 q^{75} -4.16925i q^{77} -12.0667 q^{79} +1.00000 q^{81} -11.1608i q^{83} +5.20519i q^{85} +6.35610i q^{87} -10.9680i q^{89} +20.4768i q^{91} -5.64064i q^{93} +0.667026i q^{95} -17.8948i q^{97} +0.915699i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 24 * q^9	\( 24 q + 24 q^{3} + 24 q^{9} + 24 q^{25} + 24 q^{27} + 56 q^{49} + 32 q^{73} + 24 q^{75} + 24 q^{81}+O(q^{100}) \) 24 * q + 24 * q^3 + 24 * q^9 + 24 * q^25 + 24 * q^27 + 56 * q^49 + 32 * q^73 + 24 * q^75 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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