LMFDB - Embedded newform 2208.2.j.c.47.37

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2208,2,Mod(47,2208)]

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, 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("2208.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2208 = 2^{5} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2208.j (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:	$42$
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.37
Character	$\chi$	$=$	2208.47
Dual form			2208.2.j.c.47.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.59465 - 0.676097i) q^{3} +1.67234 q^{5} -2.28745i q^{7} +(2.08579 - 2.15627i) q^{9} +O(q^{10})$	\(q+(1.59465 - 0.676097i) q^{3} +1.67234 q^{5} -2.28745i q^{7} +(2.08579 - 2.15627i) q^{9} +2.91064i q^{11} -4.63743i q^{13} +(2.66679 - 1.13066i) q^{15} -3.10443i q^{17} -7.70658 q^{19} +(-1.54654 - 3.64767i) q^{21} +1.00000 q^{23} -2.20328 q^{25} +(1.86824 - 4.84868i) q^{27} +8.53054 q^{29} -7.88372i q^{31} +(1.96787 + 4.64143i) q^{33} -3.82539i q^{35} +3.97637i q^{37} +(-3.13535 - 7.39505i) q^{39} -5.01274i q^{41} -7.62705 q^{43} +(3.48814 - 3.60601i) q^{45} +4.61765 q^{47} +1.76758 q^{49} +(-2.09889 - 4.95046i) q^{51} +0.427900 q^{53} +4.86757i q^{55} +(-12.2893 + 5.21039i) q^{57} +9.10118i q^{59} +9.06577i q^{61} +(-4.93235 - 4.77113i) q^{63} -7.75536i q^{65} +8.29515 q^{67} +(1.59465 - 0.676097i) q^{69} -4.33432 q^{71} +6.91010 q^{73} +(-3.51345 + 1.48963i) q^{75} +6.65793 q^{77} -6.46025i q^{79} +(-0.298990 - 8.99503i) q^{81} +12.7870i q^{83} -5.19165i q^{85} +(13.6032 - 5.76747i) q^{87} +4.52095i q^{89} -10.6079 q^{91} +(-5.33016 - 12.5717i) q^{93} -12.8880 q^{95} -8.58440 q^{97} +(6.27612 + 6.07097i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 2 q^{3} - 8 q^{5} + 2 q^{9}+O(q^{10}) $ 42 * q - 2 * q^3 - 8 * q^5 + 2 * q^9	$ 42 q - 2 q^{3} - 8 q^{5} + 2 q^{9} - 8 q^{15} - 4 q^{19} - 8 q^{21} + 42 q^{23} + 22 q^{25} + 16 q^{27} + 12 q^{33} + 8 q^{39} - 28 q^{43} + 8 q^{45} - 50 q^{49} - 28 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63} + 4 q^{67} - 2 q^{69} + 4 q^{73} + 6 q^{75} - 32 q^{77} + 18 q^{81} - 48 q^{87} + 8 q^{91} + 22 q^{93} - 16 q^{95} + 20 q^{97}+O(q^{100}) $ 42 * q - 2 * q^3 - 8 * q^5 + 2 * q^9 - 8 * q^15 - 4 * q^19 - 8 * q^21 + 42 * q^23 + 22 * q^25 + 16 * q^27 + 12 * q^33 + 8 * q^39 - 28 * q^43 + 8 * q^45 - 50 * q^49 - 28 * q^51 - 24 * q^53 - 8 * q^57 - 16 * q^63 + 4 * q^67 - 2 * q^69 + 4 * q^73 + 6 * q^75 - 32 * q^77 + 18 * q^81 - 48 * q^87 + 8 * q^91 + 22 * q^93 - 16 * q^95 + 20 * q^97

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
$3$	1.59465	−	0.676097i	0.920669	−	0.390345i
$3$	1.59465	−	0.676097i	0.920669	−	0.390345i
$4$		0			0
$5$	1.67234			0.747893			0.373947	−	0.927450i	$-0.378004\pi$
$5$	1.67234			0.747893			0.373947	+	0.927450i	$0.378004\pi$
$6$		0			0
$7$		−	2.28745i		−	0.864574i	−0.901736	−	0.432287i	$-0.857707\pi$
$7$		−	2.28745i		−	0.864574i	0.901736	−	0.432287i	$-0.142293\pi$
$8$		0			0
$9$	2.08579	−	2.15627i	0.695262	−	0.718756i
$10$		0			0
$11$			2.91064i			0.877590i	0.898587	+	0.438795i	$0.144595\pi$
$11$			2.91064i			0.877590i	−0.898587	+	0.438795i	$0.855405\pi$
$12$		0			0
$13$		−	4.63743i		−	1.28619i	−0.765786	−	0.643096i	$-0.777650\pi$
$13$		−	4.63743i		−	1.28619i	0.765786	−	0.643096i	$-0.222350\pi$
$14$		0			0
$15$	2.66679	−	1.13066i	0.688562	−	0.291936i
$16$		0			0
$17$		−	3.10443i		−	0.752934i	−0.926430	−	0.376467i	$-0.877139\pi$
$17$		−	3.10443i		−	0.752934i	0.926430	−	0.376467i	$-0.122861\pi$
$18$		0			0
$19$	−7.70658			−1.76801			−0.884005	−	0.467477i	$-0.845163\pi$
$19$	−7.70658			−1.76801			−0.884005	+	0.467477i	$0.845163\pi$
$20$		0			0
$21$	−1.54654	−	3.64767i	−0.337482	−	0.795986i
$22$		0			0
$23$	1.00000			0.208514
$23$	1.00000			0.208514
$24$		0			0
$25$	−2.20328			−0.440656
$26$		0			0
$27$	1.86824	−	4.84868i	0.359544	−	0.933128i
$28$		0			0
$29$	8.53054			1.58408			0.792041	−	0.610468i	$-0.209018\pi$
$29$	8.53054			1.58408			0.792041	+	0.610468i	$0.209018\pi$
$30$		0			0
$31$		−	7.88372i		−	1.41596i	−0.706234	−	0.707979i	$-0.749607\pi$
$31$		−	7.88372i		−	1.41596i	0.706234	−	0.707979i	$-0.250393\pi$
$32$		0			0
$33$	1.96787	+	4.64143i	0.342563	+	0.807970i
$34$		0			0
$35$		−	3.82539i		−	0.646609i
$36$		0			0
$37$			3.97637i			0.653711i	0.945074	+	0.326856i	$0.105989\pi$
$37$			3.97637i			0.653711i	−0.945074	+	0.326856i	$0.894011\pi$
$38$		0			0
$39$	−3.13535	−	7.39505i	−0.502058	−	1.18416i
$40$		0			0
$41$		−	5.01274i		−	0.782859i	−0.920208	−	0.391429i	$-0.871981\pi$
$41$		−	5.01274i		−	0.782859i	0.920208	−	0.391429i	$-0.128019\pi$
$42$		0			0
$43$	−7.62705			−1.16311			−0.581557	−	0.813505i	$-0.697556\pi$
$43$	−7.62705			−1.16311			−0.581557	+	0.813505i	$0.697556\pi$
$44$		0			0
$45$	3.48814	−	3.60601i	0.519982	−	0.537553i
$46$		0			0
$47$	4.61765			0.673554			0.336777	−	0.941584i	$-0.390663\pi$
$47$	4.61765			0.673554			0.336777	+	0.941584i	$0.390663\pi$
$48$		0			0
$49$	1.76758			0.252512
$50$		0			0
$51$	−2.09889	−	4.95046i	−0.293904	−	0.693203i
$52$		0			0
$53$	0.427900			0.0587766			0.0293883	−	0.999568i	$-0.490644\pi$
$53$	0.427900			0.0587766			0.0293883	+	0.999568i	$0.490644\pi$
$54$		0			0
$55$			4.86757i			0.656344i
$56$		0			0
$57$	−12.2893	+	5.21039i	−1.62775	+	0.690133i
$58$		0			0
$59$			9.10118i			1.18487i	0.805617	+	0.592436i	$0.201834\pi$
$59$			9.10118i			1.18487i	−0.805617	+	0.592436i	$0.798166\pi$
$60$		0			0
$61$			9.06577i			1.16075i	0.814348	+	0.580376i	$0.197095\pi$
$61$			9.06577i			1.16075i	−0.814348	+	0.580376i	$0.802905\pi$
$62$		0			0
$63$	−4.93235	−	4.77113i	−0.621418	−	0.601106i
$64$		0			0
$65$		−	7.75536i		−	0.961933i
$66$		0			0
$67$	8.29515			1.01341			0.506707	−	0.862118i	$-0.330863\pi$
$67$	8.29515			1.01341			0.506707	+	0.862118i	$0.330863\pi$
$68$		0			0
$69$	1.59465	−	0.676097i	0.191973	−	0.0813925i
$70$		0			0
$71$	−4.33432			−0.514390			−0.257195	−	0.966360i	$-0.582798\pi$
$71$	−4.33432			−0.514390			−0.257195	+	0.966360i	$0.582798\pi$
$72$		0			0
$73$	6.91010			0.808766			0.404383	−	0.914590i	$-0.367486\pi$
$73$	6.91010			0.808766			0.404383	+	0.914590i	$0.367486\pi$
$74$		0			0
$75$	−3.51345	+	1.48963i	−0.405698	+	0.172008i
$76$		0			0
$77$	6.65793			0.758742
$78$		0			0
$79$		−	6.46025i		−	0.726835i	−0.931626	−	0.363418i	$-0.881610\pi$
$79$		−	6.46025i		−	0.726835i	0.931626	−	0.363418i	$-0.118390\pi$
$80$		0			0
$81$	−0.298990	−	8.99503i	−0.0332212	−	0.999448i
$82$		0			0
$83$			12.7870i			1.40355i	0.712396	+	0.701777i	$0.247610\pi$
$83$			12.7870i			1.40355i	−0.712396	+	0.701777i	$0.752390\pi$
$84$		0			0
$85$		−	5.19165i		−	0.563114i
$86$		0			0
$87$	13.6032	−	5.76747i	1.45842	−	0.618338i
$88$		0			0
$89$			4.52095i			0.479220i	0.970869	+	0.239610i	$0.0770195\pi$
$89$			4.52095i			0.479220i	−0.970869	+	0.239610i	$0.922980\pi$
$90$		0			0
$91$	−10.6079			−1.11201
$92$		0			0
$93$	−5.33016	−	12.5717i	−0.552712	−	1.30363i
$94$		0			0
$95$	−12.8880			−1.32228
$96$		0			0
$97$	−8.58440			−0.871613			−0.435807	−	0.900040i	$-0.643537\pi$
$97$	−8.58440			−0.871613			−0.435807	+	0.900040i	$0.643537\pi$
$98$		0			0
$99$	6.27612	+	6.07097i	0.630774	+	0.610155i
$100$		0			0
$101$	19.6859			1.95882			0.979412	−	0.201874i	$-0.0647030\pi$
$101$	19.6859			1.95882			0.979412	+	0.201874i	$0.0647030\pi$
$102$		0			0
$103$		−	2.89731i		−	0.285481i	−0.989760	−	0.142740i	$-0.954409\pi$
$103$		−	2.89731i		−	0.285481i	0.989760	−	0.142740i	$-0.0455914\pi$
$104$		0			0
$105$	−2.58633	−	6.10014i	−0.252400	−	0.595313i
$106$		0			0
$107$			2.73473i			0.264376i	0.991225	+	0.132188i	$0.0422003\pi$
$107$			2.73473i			0.264376i	−0.991225	+	0.132188i	$0.957800\pi$
$108$		0			0
$109$		−	18.5012i		−	1.77210i	−0.463592	−	0.886049i	$-0.653440\pi$
$109$		−	18.5012i		−	1.77210i	0.463592	−	0.886049i	$-0.346560\pi$
$110$		0			0
$111$	2.68841	+	6.34090i	0.255173	+	0.601852i
$112$		0			0
$113$		−	3.69006i		−	0.347131i	−0.984822	−	0.173566i	$-0.944471\pi$
$113$		−	3.69006i		−	0.347131i	0.984822	−	0.173566i	$-0.0555289\pi$
$114$		0			0
$115$	1.67234			0.155946
$116$		0			0
$117$	−9.99954	−	9.67269i	−0.924458	−	0.894240i
$118$		0			0
$119$	−7.10121			−0.650967
$120$		0			0
$121$	2.52819			0.229835
$122$		0			0
$123$	−3.38910	−	7.99354i	−0.305585	−	0.720753i
$124$		0			0
$125$	−12.0463			−1.07746
$126$		0			0
$127$			3.49738i			0.310342i	0.987888	+	0.155171i	$0.0495929\pi$
$127$			3.49738i			0.310342i	−0.987888	+	0.155171i	$0.950407\pi$
$128$		0			0
$129$	−12.1624	+	5.15663i	−1.07084	+	0.454016i
$130$		0			0
$131$		−	2.59666i		−	0.226871i	−0.993545	−	0.113436i	$-0.963814\pi$
$131$		−	2.59666i		−	0.226871i	0.993545	−	0.113436i	$-0.0361856\pi$
$132$		0			0
$133$			17.6284i			1.52858i
$134$		0			0
$135$	3.12434	−	8.10863i	0.268900	−	0.697880i
$136$		0			0
$137$			18.1669i			1.55210i	0.630668	+	0.776052i	$0.282781\pi$
$137$			18.1669i			1.55210i	−0.630668	+	0.776052i	$0.717219\pi$
$138$		0			0
$139$	4.65828			0.395110			0.197555	−	0.980292i	$-0.436700\pi$
$139$	4.65828			0.395110			0.197555	+	0.980292i	$0.436700\pi$
$140$		0			0
$141$	7.36352	−	3.12198i	0.620120	−	0.262918i
$142$		0			0
$143$	13.4979			1.12875
$144$		0			0
$145$	14.2660			1.18472
$146$		0			0
$147$	2.81867	−	1.19506i	0.232480	−	0.0985666i
$148$		0			0
$149$	14.7435			1.20784			0.603919	−	0.797045i	$-0.293605\pi$
$149$	14.7435			1.20784			0.603919	+	0.797045i	$0.293605\pi$
$150$		0			0
$151$			8.88943i			0.723412i	0.932292	+	0.361706i	$0.117806\pi$
$151$			8.88943i			0.723412i	−0.932292	+	0.361706i	$0.882194\pi$
$152$		0			0
$153$	−6.69398	−	6.47517i	−0.541176	−	0.523486i
$154$		0			0
$155$		−	13.1843i		−	1.05899i
$156$		0			0
$157$		−	8.04774i		−	0.642279i	−0.947032	−	0.321140i	$-0.895934\pi$
$157$		−	8.04774i		−	0.642279i	0.947032	−	0.321140i	$-0.104066\pi$
$158$		0			0
$159$	0.682349	−	0.289302i	0.0541138	−	0.0229431i
$160$		0			0
$161$		−	2.28745i		−	0.180276i
$162$		0			0
$163$	8.22815			0.644479			0.322239	−	0.946658i	$-0.395564\pi$
$163$	8.22815			0.644479			0.322239	+	0.946658i	$0.395564\pi$
$164$		0			0
$165$	3.29095	+	7.76205i	0.256200	+	0.604275i
$166$		0			0
$167$	−17.1392			−1.32627			−0.663134	−	0.748501i	$-0.730774\pi$
$167$	−17.1392			−1.32627			−0.663134	+	0.748501i	$0.730774\pi$
$168$		0			0
$169$	−8.50574			−0.654288
$170$		0			0
$171$	−16.0743	+	16.6175i	−1.22923	+	1.27077i
$172$		0			0
$173$	−20.0458			−1.52405			−0.762027	−	0.647546i	$-0.775796\pi$
$173$	−20.0458			−1.52405			−0.762027	+	0.647546i	$0.775796\pi$
$174$		0			0
$175$			5.03989i			0.380980i
$176$		0			0
$177$	6.15328	+	14.5131i	0.462509	+	1.09088i
$178$		0			0
$179$			13.0220i			0.973309i	0.873594	+	0.486655i	$0.161783\pi$
$179$			13.0220i			0.973309i	−0.873594	+	0.486655i	$0.838217\pi$
$180$		0			0
$181$			21.4856i			1.59701i	0.601987	+	0.798506i	$0.294376\pi$
$181$			21.4856i			1.59701i	−0.601987	+	0.798506i	$0.705624\pi$
$182$		0			0
$183$	6.12934	+	14.4567i	0.453094	+	1.06867i
$184$		0			0
$185$			6.64984i			0.488906i
$186$		0			0
$187$	9.03586			0.660767
$188$		0			0
$189$	−11.0911	−	4.27351i	−0.806759	−	0.310852i
$190$		0			0
$191$	22.2870			1.61263			0.806314	−	0.591488i	$-0.201459\pi$
$191$	22.2870			1.61263			0.806314	+	0.591488i	$0.201459\pi$
$192$		0			0
$193$	5.72840			0.412339			0.206170	−	0.978516i	$-0.433900\pi$
$193$	5.72840			0.412339			0.206170	+	0.978516i	$0.433900\pi$
$194$		0			0
$195$	−5.24337	−	12.3670i	−0.375486	−	0.885622i
$196$		0			0
$197$	−12.4216			−0.884999			−0.442499	−	0.896769i	$-0.645908\pi$
$197$	−12.4216			−0.884999			−0.442499	+	0.896769i	$0.645908\pi$
$198$		0			0
$199$		−	14.2662i		−	1.01130i	−0.862738	−	0.505650i	$-0.831253\pi$
$199$		−	14.2662i		−	1.01130i	0.862738	−	0.505650i	$-0.168747\pi$
$200$		0			0
$201$	13.2278	−	5.60832i	0.933019	−	0.395581i
$202$		0			0
$203$		−	19.5132i		−	1.36956i
$204$		0			0
$205$		−	8.38300i		−	0.585494i
$206$		0			0
$207$	2.08579	−	2.15627i	0.144972	−	0.149871i
$208$		0			0
$209$		−	22.4311i		−	1.55159i
$210$		0			0
$211$	25.3506			1.74521			0.872604	−	0.488428i	$-0.162429\pi$
$211$	25.3506			1.74521			0.872604	+	0.488428i	$0.162429\pi$
$212$		0			0
$213$	−6.91171	+	2.93042i	−0.473582	+	0.200789i
$214$		0			0
$215$	−12.7550			−0.869885
$216$		0			0
$217$	−18.0336			−1.22420
$218$		0			0
$219$	11.0192	−	4.67189i	0.744605	−	0.315697i
$220$		0			0
$221$	−14.3966			−0.968417
$222$		0			0
$223$		−	4.13720i		−	0.277048i	−0.990359	−	0.138524i	$-0.955764\pi$
$223$		−	4.13720i		−	0.277048i	0.990359	−	0.138524i	$-0.0442358\pi$
$224$		0			0
$225$	−4.59557	+	4.75086i	−0.306371	+	0.316724i
$226$		0			0
$227$		−	11.1121i		−	0.737536i	−0.929522	−	0.368768i	$-0.879780\pi$
$227$		−	11.1121i		−	0.737536i	0.929522	−	0.368768i	$-0.120220\pi$
$228$		0			0
$229$		−	8.33587i		−	0.550850i	−0.961323	−	0.275425i	$-0.911182\pi$
$229$		−	8.33587i		−	0.550850i	0.961323	−	0.275425i	$-0.0888185\pi$
$230$		0			0
$231$	10.6170	−	4.50141i	0.698550	−	0.296171i
$232$		0			0
$233$			1.94318i			0.127302i	0.997972	+	0.0636510i	$0.0202744\pi$
$233$			1.94318i			0.127302i	−0.997972	+	0.0636510i	$0.979726\pi$
$234$		0			0
$235$	7.72228			0.503746
$236$		0			0
$237$	−4.36776	−	10.3018i	−0.283716	−	0.669175i
$238$		0			0
$239$	16.9988			1.09956			0.549781	−	0.835309i	$-0.314711\pi$
$239$	16.9988			1.09956			0.549781	+	0.835309i	$0.314711\pi$
$240$		0			0
$241$	4.17570			0.268981			0.134490	−	0.990915i	$-0.457060\pi$
$241$	4.17570			0.268981			0.134490	+	0.990915i	$0.457060\pi$
$242$		0			0
$243$	−6.55830	−	14.1417i	−0.420715	−	0.907193i
$244$		0			0
$245$	2.95600			0.188852
$246$		0			0
$247$			35.7387i			2.27400i
$248$		0			0
$249$	8.64524	+	20.3907i	0.547870	+	1.29221i
$250$		0			0
$251$		−	2.42594i		−	0.153124i	−0.997065	−	0.0765619i	$-0.975606\pi$
$251$		−	2.42594i		−	0.153124i	0.997065	−	0.0765619i	$-0.0243943\pi$
$252$		0			0
$253$			2.91064i			0.182990i
$254$		0			0
$255$	−3.51006	−	8.27885i	−0.219809	−	0.518442i
$256$		0			0
$257$		−	6.11395i		−	0.381378i	−0.981651	−	0.190689i	$-0.938928\pi$
$257$		−	6.11395i		−	0.381378i	0.981651	−	0.190689i	$-0.0610722\pi$
$258$		0			0
$259$	9.09574			0.565182
$260$		0			0
$261$	17.7929	−	18.3941i	1.10135	−	1.13857i
$262$		0			0
$263$	−6.15397			−0.379470			−0.189735	−	0.981835i	$-0.560763\pi$
$263$	−6.15397			−0.379470			−0.189735	+	0.981835i	$0.560763\pi$
$264$		0			0
$265$	0.715594			0.0439586
$266$		0			0
$267$	3.05660	+	7.20931i	0.187061	+	0.441203i
$268$		0			0
$269$	−20.4694			−1.24804			−0.624021	−	0.781407i	$-0.714502\pi$
$269$	−20.4694			−1.24804			−0.624021	+	0.781407i	$0.714502\pi$
$270$		0			0
$271$			5.78075i			0.351156i	0.984466	+	0.175578i	$0.0561794\pi$
$271$			5.78075i			0.351156i	−0.984466	+	0.175578i	$0.943821\pi$
$272$		0			0
$273$	−16.9158	+	7.17195i	−1.02379	+	0.434066i
$274$		0			0
$275$		−	6.41295i		−	0.386715i
$276$		0			0
$277$			6.92990i			0.416377i	0.978089	+	0.208189i	$0.0667568\pi$
$277$			6.92990i			0.416377i	−0.978089	+	0.208189i	$0.933243\pi$
$278$		0			0
$279$	−16.9994	−	16.4438i	−1.01773	−	0.984462i
$280$		0			0
$281$			6.80878i			0.406178i	0.979160	+	0.203089i	$0.0650980\pi$
$281$			6.80878i			0.406178i	−0.979160	+	0.203089i	$0.934902\pi$
$282$		0			0
$283$	−23.2198			−1.38027			−0.690136	−	0.723679i	$-0.742449\pi$
$283$	−23.2198			−1.38027			−0.690136	+	0.723679i	$0.742449\pi$
$284$		0			0
$285$	−20.5518	+	8.71355i	−1.21738	+	0.516146i
$286$		0			0
$287$	−11.4664			−0.676839
$288$		0			0
$289$	7.36254			0.433091
$290$		0			0
$291$	−13.6891	+	5.80388i	−0.802467	+	0.340230i
$292$		0			0
$293$	4.43092			0.258857			0.129429	−	0.991589i	$-0.458686\pi$
$293$	4.43092			0.258857			0.129429	+	0.991589i	$0.458686\pi$
$294$		0			0
$295$			15.2203i			0.886158i
$296$		0			0
$297$	14.1127	+	5.43778i	0.818904	+	0.315532i
$298$		0			0
$299$		−	4.63743i		−	0.268189i
$300$		0			0
$301$			17.4465i			1.00560i
$302$		0			0
$303$	31.3921	−	13.3096i	1.80343	−	0.764616i
$304$		0			0
$305$			15.1610i			0.868119i
$306$		0			0
$307$	−24.5271			−1.39983			−0.699917	−	0.714224i	$-0.746780\pi$
$307$	−24.5271			−1.39983			−0.699917	+	0.714224i	$0.746780\pi$
$308$		0			0
$309$	−1.95886	−	4.62019i	−0.111436	−	0.262833i
$310$		0			0
$311$	−6.45799			−0.366199			−0.183100	−	0.983094i	$-0.558613\pi$
$311$	−6.45799			−0.366199			−0.183100	+	0.983094i	$0.558613\pi$
$312$		0			0
$313$	−1.06905			−0.0604265			−0.0302132	−	0.999543i	$-0.509619\pi$
$313$	−1.06905			−0.0604265			−0.0302132	+	0.999543i	$0.509619\pi$
$314$		0			0
$315$	−8.24857	−	7.97895i	−0.464754	−	0.449563i
$316$		0			0
$317$	5.60014			0.314535			0.157268	−	0.987556i	$-0.449732\pi$
$317$	5.60014			0.314535			0.157268	+	0.987556i	$0.449732\pi$
$318$		0			0
$319$			24.8293i			1.39018i
$320$		0			0
$321$	1.84894	+	4.36092i	0.103198	+	0.243403i
$322$		0			0
$323$			23.9245i			1.33120i
$324$		0			0
$325$			10.2176i			0.566768i
$326$		0			0
$327$	−12.5086	−	29.5029i	−0.691729	−	1.63151i
$328$		0			0
$329$		−	10.5626i		−	0.582337i
$330$		0			0
$331$	3.95776			0.217538			0.108769	−	0.994067i	$-0.465309\pi$
$331$	3.95776			0.217538			0.108769	+	0.994067i	$0.465309\pi$
$332$		0			0
$333$	8.57413	+	8.29386i	0.469859	+	0.454501i
$334$		0			0
$335$	13.8723			0.757925
$336$		0			0
$337$	−15.1916			−0.827540			−0.413770	−	0.910381i	$-0.635788\pi$
$337$	−15.1916			−0.827540			−0.413770	+	0.910381i	$0.635788\pi$
$338$		0			0
$339$	−2.49483	−	5.88433i	−0.135501	−	0.319593i
$340$		0			0
$341$	22.9467			1.24263
$342$		0			0
$343$		−	20.0554i		−	1.08289i
$344$		0			0
$345$	2.66679	−	1.13066i	0.143575	−	0.0608729i
$346$		0			0
$347$		−	17.8879i		−	0.960272i	−0.877194	−	0.480136i	$-0.840587\pi$
$347$		−	17.8879i		−	0.960272i	0.877194	−	0.480136i	$-0.159413\pi$
$348$		0			0
$349$			6.58198i			0.352325i	0.984361	+	0.176163i	$0.0563685\pi$
$349$			6.58198i			0.352325i	−0.984361	+	0.176163i	$0.943632\pi$
$350$		0			0
$351$	−22.4854	−	8.66384i	−1.20018	−	0.462442i
$352$		0			0
$353$			19.9404i			1.06132i	0.847584	+	0.530661i	$0.178056\pi$
$353$			19.9404i			1.06132i	−0.847584	+	0.530661i	$0.821944\pi$
$354$		0			0
$355$	−7.24846			−0.384708
$356$		0			0
$357$	−11.3239	+	4.80111i	−0.599325	+	0.254102i
$358$		0			0
$359$	20.8705			1.10150			0.550752	−	0.834669i	$-0.314341\pi$
$359$	20.8705			1.10150			0.550752	+	0.834669i	$0.314341\pi$
$360$		0			0
$361$	40.3914			2.12586
$362$		0			0
$363$	4.03156	−	1.70930i	0.211602	−	0.0897150i
$364$		0			0
$365$	11.5560			0.604870
$366$		0			0
$367$			28.7454i			1.50050i	0.661155	+	0.750249i	$0.270066\pi$
$367$			28.7454i			1.50050i	−0.661155	+	0.750249i	$0.729934\pi$
$368$		0			0
$369$	−10.8088	−	10.4555i	−0.562684	−	0.544292i
$370$		0			0
$371$		−	0.978799i		−	0.0508167i
$372$		0			0
$373$		−	2.12022i		−	0.109781i	−0.998492	−	0.0548903i	$-0.982519\pi$
$373$		−	2.12022i		−	0.109781i	0.998492	−	0.0548903i	$-0.0174809\pi$
$374$		0			0
$375$	−19.2096	+	8.14449i	−0.991981	+	0.420579i
$376$		0			0
$377$		−	39.5598i		−	2.03743i
$378$		0			0
$379$	17.5425			0.901099			0.450550	−	0.892751i	$-0.351228\pi$
$379$	17.5425			0.901099			0.450550	+	0.892751i	$0.351228\pi$
$380$		0			0
$381$	2.36457	+	5.57708i	0.121141	+	0.285723i
$382$		0			0
$383$	13.6105			0.695464			0.347732	−	0.937594i	$-0.386952\pi$
$383$	13.6105			0.695464			0.347732	+	0.937594i	$0.386952\pi$
$384$		0			0
$385$	11.1343			0.567458
$386$		0			0
$387$	−15.9084	+	16.4460i	−0.808670	+	0.835996i
$388$		0			0
$389$	9.47011			0.480154			0.240077	−	0.970754i	$-0.422827\pi$
$389$	9.47011			0.480154			0.240077	+	0.970754i	$0.422827\pi$
$390$		0			0
$391$		−	3.10443i		−	0.156998i
$392$		0			0
$393$	−1.75559	−	4.14075i	−0.0885580	−	0.208873i
$394$		0			0
$395$		−	10.8037i		−	0.543595i
$396$		0			0
$397$			22.0808i			1.10821i	0.832448	+	0.554103i	$0.186939\pi$
$397$			22.0808i			1.10821i	−0.832448	+	0.554103i	$0.813061\pi$
$398$		0			0
$399$	11.9185	+	28.1110i	0.596671	+	1.40731i
$400$		0			0
$401$		−	15.5379i		−	0.775927i	−0.921675	−	0.387963i	$-0.873179\pi$
$401$		−	15.5379i		−	0.775927i	0.921675	−	0.387963i	$-0.126821\pi$
$402$		0			0
$403$	−36.5602			−1.82119
$404$		0			0
$405$	−0.500013	−	15.0427i	−0.0248459	−	0.747480i
$406$		0			0
$407$	−11.5738			−0.573691
$408$		0			0
$409$	−27.3129			−1.35054			−0.675268	−	0.737572i	$-0.735972\pi$
$409$	−27.3129			−1.35054			−0.675268	+	0.737572i	$0.735972\pi$
$410$		0			0
$411$	12.2826	+	28.9698i	0.605856	+	1.42897i
$412$		0			0
$413$	20.8185			1.02441
$414$		0			0
$415$			21.3842i			1.04971i
$416$		0			0
$417$	7.42831	−	3.14945i	0.363766	−	0.154229i
$418$		0			0
$419$			26.1832i			1.27913i	0.768736	+	0.639566i	$0.220886\pi$
$419$			26.1832i			1.27913i	−0.768736	+	0.639566i	$0.779114\pi$
$420$		0			0
$421$			13.1612i			0.641436i	0.947175	+	0.320718i	$0.103924\pi$
$421$			13.1612i			0.641436i	−0.947175	+	0.320718i	$0.896076\pi$
$422$		0			0
$423$	9.63144	−	9.95690i	0.468296	−	0.484121i
$424$		0			0
$425$			6.83992i			0.331785i
$426$		0			0
$427$	20.7375			1.00356
$428$		0			0
$429$	21.5243	−	9.12587i	1.03920	−	0.440601i
$430$		0			0
$431$	9.94800			0.479178			0.239589	−	0.970874i	$-0.422987\pi$
$431$	9.94800			0.479178			0.239589	+	0.970874i	$0.422987\pi$
$432$		0			0
$433$	−2.05253			−0.0986383			−0.0493192	−	0.998783i	$-0.515705\pi$
$433$	−2.05253			−0.0986383			−0.0493192	+	0.998783i	$0.515705\pi$
$434$		0			0
$435$	22.7492	−	9.64517i	1.09074	−	0.462451i
$436$		0			0
$437$	−7.70658			−0.368656
$438$		0			0
$439$		−	32.5669i		−	1.55433i	−0.629296	−	0.777166i	$-0.716657\pi$
$439$		−	32.5669i		−	1.55433i	0.629296	−	0.777166i	$-0.283343\pi$
$440$		0			0
$441$	3.68680	−	3.81138i	0.175562	−	0.181494i
$442$		0			0
$443$		−	16.4142i		−	0.779861i	−0.920844	−	0.389931i	$-0.872499\pi$
$443$		−	16.4142i		−	0.779861i	0.920844	−	0.389931i	$-0.127501\pi$
$444$		0			0
$445$			7.56056i			0.358405i
$446$		0			0
$447$	23.5107	−	9.96807i	1.11202	−	0.471473i
$448$		0			0
$449$			30.6773i			1.44775i	0.689929	+	0.723877i	$0.257642\pi$
$449$			30.6773i			1.44775i	−0.689929	+	0.723877i	$0.742358\pi$
$450$		0			0
$451$	14.5903			0.687029
$452$		0			0
$453$	6.01012	+	14.1755i	0.282380	+	0.666023i
$454$		0			0
$455$	−17.7400			−0.831663
$456$		0			0
$457$	−21.4345			−1.00266			−0.501331	−	0.865255i	$-0.667156\pi$
$457$	−21.4345			−1.00266			−0.501331	+	0.865255i	$0.667156\pi$
$458$		0			0
$459$	−15.0524	−	5.79982i	−0.702584	−	0.270712i
$460$		0			0
$461$	9.57418			0.445914			0.222957	−	0.974828i	$-0.428429\pi$
$461$	9.57418			0.445914			0.222957	+	0.974828i	$0.428429\pi$
$462$		0			0
$463$			23.0254i			1.07008i	0.844826	+	0.535042i	$0.179704\pi$
$463$			23.0254i			1.07008i	−0.844826	+	0.535042i	$0.820296\pi$
$464$		0			0
$465$	−8.91383	−	21.0242i	−0.413369	−	0.974975i
$466$		0			0
$467$			31.8460i			1.47366i	0.676080	+	0.736828i	$0.263677\pi$
$467$			31.8460i			1.47366i	−0.676080	+	0.736828i	$0.736323\pi$
$468$		0			0
$469$		−	18.9747i		−	0.876171i
$470$		0			0
$471$	−5.44105	−	12.8333i	−0.250710	−	0.591326i
$472$		0			0
$473$		−	22.1996i		−	1.02074i
$474$		0			0
$475$	16.9798			0.779084
$476$		0			0
$477$	0.892508	−	0.922668i	0.0408652	−	0.0422461i
$478$		0			0
$479$	−8.50882			−0.388778			−0.194389	−	0.980925i	$-0.562272\pi$
$479$	−8.50882			−0.388778			−0.194389	+	0.980925i	$0.562272\pi$
$480$		0			0
$481$	18.4401			0.840798
$482$		0			0
$483$	−1.54654	−	3.64767i	−0.0703698	−	0.165975i
$484$		0			0
$485$	−14.3560			−0.651874
$486$		0			0
$487$			10.2419i			0.464104i	0.972703	+	0.232052i	$0.0745440\pi$
$487$			10.2419i			0.464104i	−0.972703	+	0.232052i	$0.925456\pi$
$488$		0			0
$489$	13.1210	−	5.56303i	0.593351	−	0.251569i
$490$		0			0
$491$			32.3151i			1.45836i	0.684321	+	0.729181i	$0.260099\pi$
$491$			32.3151i			1.45836i	−0.684321	+	0.729181i	$0.739901\pi$
$492$		0			0
$493$		−	26.4824i		−	1.19271i
$494$		0			0
$495$	10.4958	+	10.1527i	0.471751	+	0.456331i
$496$		0			0
$497$			9.91454i			0.444728i
$498$		0			0
$499$	42.5250			1.90368			0.951841	−	0.306593i	$-0.0991890\pi$
$499$	42.5250			1.90368			0.951841	+	0.306593i	$0.0991890\pi$
$500$		0			0
$501$	−27.3309	+	11.5877i	−1.22105	+	0.517702i
$502$		0			0
$503$	6.73586			0.300337			0.150168	−	0.988660i	$-0.452018\pi$
$503$	6.73586			0.300337			0.150168	+	0.988660i	$0.452018\pi$
$504$		0			0
$505$	32.9216			1.46499
$506$		0			0
$507$	−13.5636	+	5.75070i	−0.602382	+	0.255398i
$508$		0			0
$509$	−41.3469			−1.83267			−0.916334	−	0.400416i	$-0.868866\pi$
$509$	−41.3469			−1.83267			−0.916334	+	0.400416i	$0.868866\pi$
$510$		0			0
$511$		−	15.8065i		−	0.699238i
$512$		0			0
$513$	−14.3978	+	37.3667i	−0.635677	+	1.64978i
$514$		0			0
$515$		−	4.84529i		−	0.213509i
$516$		0			0
$517$			13.4403i			0.591104i
$518$		0			0
$519$	−31.9659	+	13.5529i	−1.40315	+	0.594906i
$520$		0			0
$521$			24.8813i			1.09007i	0.838413	+	0.545035i	$0.183484\pi$
$521$			24.8813i			1.09007i	−0.838413	+	0.545035i	$0.816516\pi$
$522$		0			0
$523$	6.99915			0.306052			0.153026	−	0.988222i	$-0.451098\pi$
$523$	6.99915			0.306052			0.153026	+	0.988222i	$0.451098\pi$
$524$		0			0
$525$	3.40745	+	8.03683i	0.148713	+	0.350756i
$526$		0			0
$527$	−24.4744			−1.06612
$528$		0			0
$529$	1.00000			0.0434783
$530$		0			0
$531$	19.6246	+	18.9831i	0.851635	+	0.823797i
$532$		0			0
$533$	−23.2462			−1.00691
$534$		0			0
$535$			4.57339i			0.197725i
$536$		0			0
$537$	8.80412	+	20.7655i	0.379926	+	0.896095i
$538$		0			0
$539$			5.14479i			0.221602i
$540$		0			0
$541$		−	0.299027i		−	0.0128562i	−0.999979	−	0.00642808i	$-0.997954\pi$
$541$		−	0.299027i		−	0.0128562i	0.999979	−	0.00642808i	$-0.00204614\pi$
$542$		0			0
$543$	14.5263	+	34.2619i	0.623385	+	1.47032i
$544$		0			0
$545$		−	30.9404i		−	1.32534i
$546$		0			0
$547$	22.3488			0.955564			0.477782	−	0.878478i	$-0.341441\pi$
$547$	22.3488			0.955564			0.477782	+	0.878478i	$0.341441\pi$
$548$		0			0
$549$	19.5482	+	18.9093i	0.834298	+	0.807028i
$550$		0			0
$551$	−65.7413			−2.80067
$552$		0			0
$553$	−14.7775			−0.628403
$554$		0			0
$555$	4.49594	+	10.6041i	0.190842	+	0.450121i
$556$		0			0
$557$	−16.6755			−0.706562			−0.353281	−	0.935517i	$-0.614934\pi$
$557$	−16.6755			−0.706562			−0.353281	+	0.935517i	$0.614934\pi$
$558$		0			0
$559$			35.3699i			1.49599i
$560$		0			0
$561$	14.4090	−	6.10912i	0.608348	−	0.257927i
$562$		0			0
$563$		−	12.6843i		−	0.534578i	−0.963616	−	0.267289i	$-0.913872\pi$
$563$		−	12.6843i		−	0.534578i	0.963616	−	0.267289i	$-0.0861279\pi$
$564$		0			0
$565$		−	6.17103i		−	0.259617i
$566$		0			0
$567$	−20.5757	+	0.683925i	−0.864097	+	0.0287221i
$568$		0			0
$569$		−	23.7045i		−	0.993743i	−0.867824	−	0.496872i	$-0.834482\pi$
$569$		−	23.7045i		−	0.993743i	0.867824	−	0.496872i	$-0.165518\pi$
$570$		0			0
$571$	14.6959			0.615004			0.307502	−	0.951548i	$-0.400507\pi$
$571$	14.6959			0.615004			0.307502	+	0.951548i	$0.400507\pi$
$572$		0			0
$573$	35.5398	−	15.0681i	1.48470	−	0.629481i
$574$		0			0
$575$	−2.20328			−0.0918831
$576$		0			0
$577$	10.0148			0.416920			0.208460	−	0.978031i	$-0.433155\pi$
$577$	10.0148			0.416920			0.208460	+	0.978031i	$0.433155\pi$
$578$		0			0
$579$	9.13476	−	3.87295i	0.379628	−	0.160954i
$580$		0			0
$581$	29.2496			1.21348
$582$		0			0
$583$			1.24546i			0.0515818i
$584$		0			0
$585$	−16.7226	−	16.1760i	−0.691396	−	0.668796i
$586$		0			0
$587$			31.6906i			1.30801i	0.756489	+	0.654006i	$0.226913\pi$
$587$			31.6906i			1.30801i	−0.756489	+	0.654006i	$0.773087\pi$
$588$		0			0
$589$			60.7565i			2.50343i
$590$		0			0
$591$	−19.8080	+	8.39817i	−0.814791	+	0.345455i
$592$		0			0
$593$			23.0186i			0.945262i	0.881261	+	0.472631i	$0.156696\pi$
$593$			23.0186i			0.945262i	−0.881261	+	0.472631i	$0.843304\pi$
$594$		0			0
$595$	−11.8756			−0.486854
$596$		0			0
$597$	−9.64530	−	22.7495i	−0.394756	−	0.931073i
$598$		0			0
$599$	11.4463			0.467681			0.233841	−	0.972275i	$-0.424871\pi$
$599$	11.4463			0.467681			0.233841	+	0.972275i	$0.424871\pi$
$600$		0			0
$601$	−31.9168			−1.30191			−0.650957	−	0.759114i	$-0.725632\pi$
$601$	−31.9168			−1.30191			−0.650957	+	0.759114i	$0.725632\pi$
$602$		0			0
$603$	17.3019	−	17.8866i	0.704588	−	0.728398i
$604$		0			0
$605$	4.22799			0.171892
$606$		0			0
$607$			8.79097i			0.356815i	0.983957	+	0.178407i	$0.0570945\pi$
$607$			8.79097i			0.356815i	−0.983957	+	0.178407i	$0.942906\pi$
$608$		0			0
$609$	−13.1928	−	31.1166i	−0.534599	−	1.26091i
$610$		0			0
$611$		−	21.4140i		−	0.866319i
$612$		0			0
$613$			23.9647i			0.967925i	0.875089	+	0.483963i	$0.160803\pi$
$613$			23.9647i			0.967925i	−0.875089	+	0.483963i	$0.839197\pi$
$614$		0			0
$615$	−5.66772	−	13.3679i	−0.228545	−	0.539046i
$616$		0			0
$617$			20.0313i			0.806431i	0.915105	+	0.403215i	$0.132107\pi$
$617$			20.0313i			0.806431i	−0.915105	+	0.403215i	$0.867893\pi$
$618$		0			0
$619$	−36.0899			−1.45057			−0.725287	−	0.688446i	$-0.758293\pi$
$619$	−36.0899			−1.45057			−0.725287	+	0.688446i	$0.758293\pi$
$620$		0			0
$621$	1.86824	−	4.84868i	0.0749700	−	0.194571i
$622$		0			0
$623$	10.3414			0.414321
$624$		0			0
$625$	−9.12916			−0.365166
$626$		0			0
$627$	−15.1656	−	35.7696i	−0.605654	−	1.42850i
$628$		0			0
$629$	12.3444			0.492202
$630$		0			0
$631$			23.6810i			0.942725i	0.881940	+	0.471362i	$0.156238\pi$
$631$			23.6810i			0.942725i	−0.881940	+	0.471362i	$0.843762\pi$
$632$		0			0
$633$	40.4252	−	17.1395i	1.60676	−	0.681233i
$634$		0			0
$635$			5.84881i			0.232103i
$636$		0			0
$637$		−	8.19704i		−	0.324778i
$638$		0			0
$639$	−9.04047	+	9.34597i	−0.357636	+	0.369721i
$640$		0			0
$641$			20.8766i			0.824577i	0.911053	+	0.412288i	$0.135270\pi$
$641$			20.8766i			0.824577i	−0.911053	+	0.412288i	$0.864730\pi$
$642$		0			0
$643$	−33.5342			−1.32246			−0.661230	−	0.750183i	$-0.729965\pi$
$643$	−33.5342			−1.32246			−0.661230	+	0.750183i	$0.729965\pi$
$644$		0			0
$645$	−20.3397	+	8.62363i	−0.800876	+	0.339555i
$646$		0			0
$647$	18.4765			0.726385			0.363193	−	0.931714i	$-0.381687\pi$
$647$	18.4765			0.726385			0.363193	+	0.931714i	$0.381687\pi$
$648$		0			0
$649$	−26.4902			−1.03983
$650$		0			0
$651$	−28.7572	+	12.1925i	−1.12708	+	0.477860i
$652$		0			0
$653$	−23.6502			−0.925504			−0.462752	−	0.886488i	$-0.653138\pi$
$653$	−23.6502			−0.925504			−0.462752	+	0.886488i	$0.653138\pi$
$654$		0			0
$655$		−	4.34250i		−	0.169676i
$656$		0			0
$657$	14.4130	−	14.9000i	0.562304	−	0.581305i
$658$		0			0
$659$		−	22.5920i		−	0.880059i	−0.897983	−	0.440029i	$-0.854968\pi$
$659$		−	22.5920i		−	0.880059i	0.897983	−	0.440029i	$-0.145032\pi$
$660$		0			0
$661$			40.2172i			1.56427i	0.623110	+	0.782135i	$0.285869\pi$
$661$			40.2172i			1.56427i	−0.623110	+	0.782135i	$0.714131\pi$
$662$		0			0
$663$	−22.9574	+	9.73346i	−0.891591	+	0.378016i
$664$		0			0
$665$			29.4807i			1.14321i
$666$		0			0
$667$	8.53054			0.330304
$668$		0			0
$669$	−2.79715	−	6.59737i	−0.108144	−	0.255069i
$670$		0			0
$671$	−26.3872			−1.01867
$672$		0			0
$673$	9.31167			0.358939			0.179469	−	0.983764i	$-0.442562\pi$
$673$	9.31167			0.358939			0.179469	+	0.983764i	$0.442562\pi$
$674$		0			0
$675$	−4.11626	+	10.6830i	−0.158435	+	0.411189i
$676$		0			0
$677$	−15.7282			−0.604482			−0.302241	−	0.953231i	$-0.597735\pi$
$677$	−15.7282			−0.604482			−0.302241	+	0.953231i	$0.597735\pi$
$678$		0			0
$679$			19.6364i			0.753574i
$680$		0			0
$681$	−7.51285	−	17.7199i	−0.287893	−	0.679026i
$682$		0			0
$683$		−	29.9693i		−	1.14674i	−0.819295	−	0.573372i	$-0.805635\pi$
$683$		−	29.9693i		−	1.14674i	0.819295	−	0.573372i	$-0.194365\pi$
$684$		0			0
$685$			30.3813i			1.16081i
$686$		0			0
$687$	−5.63585	−	13.2927i	−0.215021	−	0.507150i
$688$		0			0
$689$		−	1.98436i		−	0.0755980i
$690$		0			0
$691$	−10.7673			−0.409606			−0.204803	−	0.978803i	$-0.565655\pi$
$691$	−10.7673			−0.409606			−0.204803	+	0.978803i	$0.565655\pi$
$692$		0			0
$693$	13.8870	−	14.3563i	0.527524	−	0.545350i
$694$		0			0
$695$	7.79023			0.295500
$696$		0			0
$697$	−15.5617			−0.589441
$698$		0			0
$699$	1.31378	+	3.09868i	0.0496916	+	0.117203i
$700$		0			0
$701$	−39.5604			−1.49417			−0.747087	−	0.664726i	$-0.768548\pi$
$701$	−39.5604			−1.49417			−0.747087	+	0.664726i	$0.768548\pi$
$702$		0			0
$703$		−	30.6442i		−	1.15577i
$704$		0			0
$705$	12.3143	−	5.22101i	0.463783	−	0.196635i
$706$		0			0
$707$		−	45.0305i		−	1.69355i
$708$		0			0
$709$		−	31.7211i		−	1.19131i	−0.803240	−	0.595655i	$-0.796892\pi$
$709$		−	31.7211i		−	1.19131i	0.803240	−	0.595655i	$-0.203108\pi$
$710$		0			0
$711$	−13.9300	−	13.4747i	−0.522417	−	0.505341i
$712$		0			0
$713$		−	7.88372i		−	0.295248i
$714$		0			0
$715$	22.5730			0.844183
$716$		0			0
$717$	27.1071	−	11.4928i	1.01233	−	0.429208i
$718$		0			0
$719$	16.6643			0.621475			0.310737	−	0.950496i	$-0.399424\pi$
$719$	16.6643			0.621475			0.310737	+	0.950496i	$0.399424\pi$
$720$		0			0
$721$	−6.62746			−0.246819
$722$		0			0
$723$	6.65876	−	2.82318i	0.247642	−	0.104995i
$724$		0			0
$725$	−18.7952			−0.698035
$726$		0			0
$727$		−	28.4645i		−	1.05569i	−0.849341	−	0.527845i	$-0.823000\pi$
$727$		−	28.4645i		−	1.05569i	0.849341	−	0.527845i	$-0.177000\pi$
$728$		0			0
$729$	−20.0193	−	18.1170i	−0.741457	−	0.671000i
$730$		0			0
$731$			23.6776i			0.875748i
$732$		0			0
$733$		−	19.2342i		−	0.710431i	−0.934785	−	0.355215i	$-0.884408\pi$
$733$		−	19.2342i		−	0.710431i	0.934785	−	0.355215i	$-0.115592\pi$
$734$		0			0
$735$	4.71377	−	1.99854i	0.173870	−	0.0737173i
$736$		0			0
$737$			24.1442i			0.889362i
$738$		0			0
$739$	21.7895			0.801538			0.400769	−	0.916179i	$-0.368743\pi$
$739$	21.7895			0.801538			0.400769	+	0.916179i	$0.368743\pi$
$740$		0			0
$741$	24.1628	+	56.9906i	0.887644	+	2.09360i
$742$		0			0
$743$	24.4226			0.895980			0.447990	−	0.894039i	$-0.352140\pi$
$743$	24.4226			0.895980			0.447990	+	0.894039i	$0.352140\pi$
$744$		0			0
$745$	24.6562			0.903334
$746$		0			0
$747$	27.5722	+	26.6709i	1.00881	+	0.975838i
$748$		0			0
$749$	6.25555			0.228573
$750$		0			0
$751$			13.8921i			0.506931i	0.967344	+	0.253465i	$0.0815704\pi$
$751$			13.8921i			0.506931i	−0.967344	+	0.253465i	$0.918430\pi$
$752$		0			0
$753$	−1.64017	−	3.86851i	−0.0597710	−	0.140976i
$754$		0			0
$755$			14.8662i			0.541035i
$756$		0			0
$757$			8.05974i			0.292936i	0.989215	+	0.146468i	$0.0467906\pi$
$757$			8.05974i			0.292936i	−0.989215	+	0.146468i	$0.953209\pi$
$758$		0			0
$759$	1.96787	+	4.64143i	0.0714293	+	0.168473i
$760$		0			0
$761$		−	18.9559i		−	0.687151i	−0.939125	−	0.343576i	$-0.888362\pi$
$761$		−	18.9559i		−	0.687151i	0.939125	−	0.343576i	$-0.111638\pi$
$762$		0			0
$763$	−42.3206			−1.53211
$764$		0			0
$765$	−11.1946	−	10.8287i	−0.404742	−	0.391512i
$766$		0			0
$767$	42.2061			1.52397
$768$		0			0
$769$	33.9584			1.22457			0.612286	−	0.790636i	$-0.290250\pi$
$769$	33.9584			1.22457			0.612286	+	0.790636i	$0.290250\pi$
$770$		0			0
$771$	−4.13362	−	9.74959i	−0.148869	−	0.351123i
$772$		0			0
$773$	2.59044			0.0931716			0.0465858	−	0.998914i	$-0.485166\pi$
$773$	2.59044			0.0931716			0.0465858	+	0.998914i	$0.485166\pi$
$774$		0			0
$775$			17.3700i			0.623950i
$776$		0			0
$777$	14.5045	−	6.14960i	0.520345	−	0.220616i
$778$		0			0
$779$			38.6311i			1.38410i
$780$		0			0
$781$		−	12.6156i		−	0.451423i
$782$		0			0
$783$	15.9371	−	41.3619i	0.569546	−	1.47815i
$784$		0			0
$785$		−	13.4585i		−	0.480356i
$786$		0			0
$787$	−22.7336			−0.810365			−0.405182	−	0.914236i	$-0.632792\pi$
$787$	−22.7336			−0.810365			−0.405182	+	0.914236i	$0.632792\pi$
$788$		0			0
$789$	−9.81340	+	4.16068i	−0.349366	+	0.148124i
$790$		0			0
$791$	−8.44081			−0.300121
$792$		0			0
$793$	42.0419			1.49295
$794$		0			0
$795$	1.14112	−	0.483811i	0.0404713	−	0.0171590i
$796$		0			0
$797$	44.9272			1.59140			0.795702	−	0.605689i	$-0.207102\pi$
$797$	44.9272			1.59140			0.795702	+	0.605689i	$0.207102\pi$
$798$		0			0
$799$		−	14.3352i		−	0.507141i
$800$		0			0
$801$	9.74838	+	9.42973i	0.344442	+	0.333183i
$802$		0			0
$803$			20.1128i			0.709765i
$804$		0			0
$805$		−	3.82539i		−	0.134827i
$806$		0			0
$807$	−32.6415	+	13.8393i	−1.14903	+	0.487167i
$808$		0			0
$809$		−	37.6032i		−	1.32206i	−0.750361	−	0.661028i	$-0.770120\pi$
$809$		−	37.6032i		−	1.32206i	0.750361	−	0.661028i	$-0.229880\pi$
$810$		0			0
$811$	−9.33068			−0.327644			−0.163822	−	0.986490i	$-0.552382\pi$
$811$	−9.33068			−0.327644			−0.163822	+	0.986490i	$0.552382\pi$
$812$		0			0
$813$	3.90835	+	9.21825i	0.137072	+	0.323298i
$814$		0			0
$815$	13.7603			0.482001
$816$		0			0
$817$	58.7785			2.05640
$818$		0			0
$819$	−22.1258	+	22.8734i	−0.773137	+	0.799262i
$820$		0			0
$821$	32.4718			1.13327			0.566636	−	0.823968i	$-0.308244\pi$
$821$	32.4718			1.13327			0.566636	+	0.823968i	$0.308244\pi$
$822$		0			0
$823$		−	32.3017i		−	1.12597i	−0.826468	−	0.562983i	$-0.809653\pi$
$823$		−	32.3017i		−	1.12597i	0.826468	−	0.562983i	$-0.190347\pi$
$824$		0			0
$825$	−4.33577	−	10.2264i	−0.150952	−	0.356037i
$826$		0			0
$827$		−	47.3767i		−	1.64745i	−0.566989	−	0.823725i	$-0.691892\pi$
$827$		−	47.3767i		−	1.64745i	0.566989	−	0.823725i	$-0.308108\pi$
$828$		0			0
$829$		−	10.6496i		−	0.369875i	−0.982750	−	0.184937i	$-0.940792\pi$
$829$		−	10.6496i		−	0.369875i	0.982750	−	0.184937i	$-0.0592082\pi$
$830$		0			0
$831$	4.68528	+	11.0507i	0.162531	+	0.383346i
$832$		0			0
$833$		−	5.48733i		−	0.190125i
$834$		0			0
$835$	−28.6625			−0.991906
$836$		0			0
$837$	−38.2256	−	14.7287i	−1.32127	−	0.509098i
$838$		0			0
$839$	−48.9856			−1.69117			−0.845585	−	0.533841i	$-0.820748\pi$
$839$	−48.9856			−1.69117			−0.845585	+	0.533841i	$0.820748\pi$
$840$		0			0
$841$	43.7702			1.50932
$842$		0			0
$843$	4.60339	+	10.8576i	0.158549	+	0.373955i
$844$		0			0
$845$	−14.2245			−0.489337
$846$		0			0
$847$		−	5.78310i		−	0.198710i
$848$		0			0
$849$	−37.0273	+	15.6988i	−1.27077	+	0.538782i
$850$		0			0
$851$			3.97637i			0.136308i
$852$		0			0
$853$			7.48762i			0.256371i	0.991750	+	0.128186i	$0.0409153\pi$
$853$			7.48762i			0.256371i	−0.991750	+	0.128186i	$0.959085\pi$
$854$		0			0
$855$	−26.8817	+	27.7900i	−0.919333	+	0.950399i
$856$		0			0
$857$			21.3312i			0.728659i	0.931270	+	0.364330i	$0.118702\pi$
$857$			21.3312i			0.728659i	−0.931270	+	0.364330i	$0.881298\pi$
$858$		0			0
$859$	−35.6085			−1.21495			−0.607474	−	0.794340i	$-0.707817\pi$
$859$	−35.6085			−1.21495			−0.607474	+	0.794340i	$0.707817\pi$
$860$		0			0
$861$	−18.2848	+	7.75238i	−0.623145	+	0.264201i
$862$		0			0
$863$	16.0883			0.547653			0.273827	−	0.961779i	$-0.411711\pi$
$863$	16.0883			0.547653			0.273827	+	0.961779i	$0.411711\pi$
$864$		0			0
$865$	−33.5234			−1.13983
$866$		0			0
$867$	11.7406	−	4.97779i	0.398733	−	0.169055i
$868$		0			0
$869$	18.8035			0.637864
$870$		0			0
$871$		−	38.4682i		−	1.30344i
$872$		0			0
$873$	−17.9052	+	18.5103i	−0.606000	+	0.626478i
$874$		0			0
$875$			27.5554i			0.931541i
$876$		0			0
$877$			30.9575i			1.04536i	0.852529	+	0.522680i	$0.175068\pi$
$877$			30.9575i			1.04536i	−0.852529	+	0.522680i	$0.824932\pi$
$878$		0			0
$879$	7.06575	−	2.99573i	0.238322	−	0.101044i
$880$		0			0
$881$			35.0906i			1.18223i	0.806586	+	0.591116i	$0.201313\pi$
$881$			35.0906i			1.18223i	−0.806586	+	0.591116i	$0.798687\pi$
$882$		0			0
$883$	30.4343			1.02420			0.512098	−	0.858927i	$-0.328869\pi$
$883$	30.4343			1.02420			0.512098	+	0.858927i	$0.328869\pi$
$884$		0			0
$885$	10.2904	+	24.2709i	0.345907	+	0.815858i
$886$		0			0
$887$	35.1003			1.17855			0.589277	−	0.807931i	$-0.299413\pi$
$887$	35.1003			1.17855			0.589277	+	0.807931i	$0.299413\pi$
$888$		0			0
$889$	8.00008			0.268314
$890$		0			0
$891$	26.1813	−	0.870253i	0.877106	−	0.0291546i
$892$		0			0
$893$	−35.5863			−1.19085
$894$		0			0
$895$			21.7772i			0.727931i
$896$		0			0
$897$	−3.13535	−	7.39505i	−0.104686	−	0.246914i
$898$		0			0
$899$		−	67.2524i		−	2.24299i
$900$		0			0
$901$		−	1.32838i		−	0.0442549i
$902$		0			0
$903$	11.7955	+	27.8210i	0.392530	+	0.925823i
$904$		0			0
$905$			35.9312i			1.19439i
$906$		0			0
$907$	6.22500			0.206698			0.103349	−	0.994645i	$-0.467044\pi$
$907$	6.22500			0.206698			0.103349	+	0.994645i	$0.467044\pi$
$908$		0			0
$909$	41.0606	−	42.4482i	1.36190	−	1.40792i
$910$		0			0
$911$	12.2456			0.405714			0.202857	−	0.979208i	$-0.434977\pi$
$911$	12.2456			0.405714			0.202857	+	0.979208i	$0.434977\pi$
$912$		0			0
$913$	−37.2183			−1.23175
$914$		0			0
$915$	10.2503	+	24.1765i	0.338866	+	0.799250i
$916$		0			0
$917$	−5.93973			−0.196147
$918$		0			0
$919$		−	23.4762i		−	0.774407i	−0.921994	−	0.387203i	$-0.873441\pi$
$919$		−	23.4762i		−	0.774407i	0.921994	−	0.387203i	$-0.126559\pi$
$920$		0			0
$921$	−39.1120	+	16.5827i	−1.28878	+	0.546418i
$922$		0			0
$923$			20.1001i			0.661603i
$924$		0			0
$925$		−	8.76106i		−	0.288062i
$926$		0			0
$927$	−6.24739	−	6.04318i	−0.205191	−	0.198484i
$928$		0			0
$929$			8.59904i			0.282126i	0.990001	+	0.141063i	$0.0450519\pi$
$929$			8.59904i			0.282126i	−0.990001	+	0.141063i	$0.954948\pi$
$930$		0			0
$931$	−13.6220			−0.446443
$932$		0			0
$933$	−10.2982	+	4.36623i	−0.337148	+	0.142944i
$934$		0			0
$935$	15.1110			0.494183
$936$		0			0
$937$	40.6559			1.32817			0.664085	−	0.747657i	$-0.268821\pi$
$937$	40.6559			1.32817			0.664085	+	0.747657i	$0.268821\pi$
$938$		0			0
$939$	−1.70476	+	0.722784i	−0.0556328	+	0.0235872i
$940$		0			0
$941$	11.6231			0.378901			0.189450	−	0.981890i	$-0.439329\pi$
$941$	11.6231			0.378901			0.189450	+	0.981890i	$0.439329\pi$
$942$		0			0
$943$		−	5.01274i		−	0.163237i
$944$		0			0
$945$	−18.5481	−	7.14676i	−0.603369	−	0.232484i
$946$		0			0
$947$			30.2719i			0.983705i	0.870678	+	0.491853i	$0.163680\pi$
$947$			30.2719i			0.983705i	−0.870678	+	0.491853i	$0.836320\pi$
$948$		0			0
$949$		−	32.0451i		−	1.04023i
$950$		0			0
$951$	8.93023	−	3.78623i	0.289583	−	0.122777i
$952$		0			0
$953$		−	2.66840i		−	0.0864379i	−0.999066	−	0.0432189i	$-0.986239\pi$
$953$		−	2.66840i		−	0.0864379i	0.999066	−	0.0432189i	$-0.0137613\pi$
$954$		0			0
$955$	37.2714			1.20607
$956$		0			0
$957$	16.7870	+	39.5940i	0.542647	+	1.27989i
$958$		0			0
$959$	41.5559			1.34191
$960$		0			0
$961$	−31.1530			−1.00494
$962$		0			0
$963$	5.89681	+	5.70406i	0.190022	+	0.183811i
$964$		0			0
$965$	9.57983			0.308386
$966$		0			0
$967$		−	41.6296i		−	1.33872i	−0.742939	−	0.669359i	$-0.766569\pi$
$967$		−	41.6296i		−	1.33872i	0.742939	−	0.669359i	$-0.233431\pi$
$968$		0			0
$969$	16.1753	+	38.1511i	0.519625	+	1.22559i
$970$		0			0
$971$		−	31.1228i		−	0.998778i	−0.866378	−	0.499389i	$-0.833558\pi$
$971$		−	31.1228i		−	0.998778i	0.866378	−	0.499389i	$-0.166442\pi$
$972$		0			0
$973$		−	10.6556i		−	0.341602i
$974$		0			0
$975$	6.90805	+	16.2934i	0.221235	+	0.521806i
$976$		0			0
$977$			20.3893i			0.652312i	0.945316	+	0.326156i	$0.105753\pi$
$977$			20.3893i			0.652312i	−0.945316	+	0.326156i	$0.894247\pi$
$978$		0			0
$979$	−13.1588			−0.420558
$980$		0			0
$981$	−39.8937	−	38.5896i	−1.27371	−	1.23207i
$982$		0			0
$983$	−54.2769			−1.73116			−0.865582	−	0.500766i	$-0.833052\pi$
$983$	−54.2769			−1.73116			−0.865582	+	0.500766i	$0.833052\pi$
$984$		0			0
$985$	−20.7731			−0.661884
$986$		0			0
$987$	−7.14137	−	16.8437i	−0.227312	−	0.536140i
$988$		0			0
$989$	−7.62705			−0.242526
$990$		0			0
$991$		−	50.6391i		−	1.60861i	−0.594220	−	0.804303i	$-0.702539\pi$
$991$		−	50.6391i		−	1.60861i	0.594220	−	0.804303i	$-0.297461\pi$
$992$		0			0
$993$	6.31122	−	2.67583i	0.200281	−	0.0849148i
$994$		0			0
$995$		−	23.8579i		−	0.756345i
$996$		0			0
$997$		−	42.5149i		−	1.34646i	−0.739433	−	0.673230i	$-0.764906\pi$
$997$		−	42.5149i		−	1.34646i	0.739433	−	0.673230i	$-0.235094\pi$
$998$		0			0
$999$	19.2801	+	7.42883i	0.609997	+	0.235038i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.j.c.47.37		42
3.2	odd	2		2208.2.j.d.47.38		42
4.3	odd	2		552.2.j.c.323.20	yes	42
8.3	odd	2		2208.2.j.d.47.37		42
8.5	even	2		552.2.j.d.323.24	yes	42
12.11	even	2		552.2.j.d.323.23	yes	42
24.5	odd	2		552.2.j.c.323.19	✓	42
24.11	even	2	inner	2208.2.j.c.47.38		42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.j.c.323.19	✓	42	24.5	odd	2
552.2.j.c.323.20	yes	42	4.3	odd	2
552.2.j.d.323.23	yes	42	12.11	even	2
552.2.j.d.323.24	yes	42	8.5	even	2
2208.2.j.c.47.37		42	1.1	even	1	trivial
2208.2.j.c.47.38		42	24.11	even	2	inner
2208.2.j.d.47.37		42	8.3	odd	2
2208.2.j.d.47.38		42	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.59465 - 0.676097i) q^{3} +1.67234 q^{5} -2.28745i q^{7} +(2.08579 - 2.15627i) q^{9} +O(q^{10})\)	\(q+(1.59465 - 0.676097i) q^{3} +1.67234 q^{5} -2.28745i q^{7} +(2.08579 - 2.15627i) q^{9} +2.91064i q^{11} -4.63743i q^{13} +(2.66679 - 1.13066i) q^{15} -3.10443i q^{17} -7.70658 q^{19} +(-1.54654 - 3.64767i) q^{21} +1.00000 q^{23} -2.20328 q^{25} +(1.86824 - 4.84868i) q^{27} +8.53054 q^{29} -7.88372i q^{31} +(1.96787 + 4.64143i) q^{33} -3.82539i q^{35} +3.97637i q^{37} +(-3.13535 - 7.39505i) q^{39} -5.01274i q^{41} -7.62705 q^{43} +(3.48814 - 3.60601i) q^{45} +4.61765 q^{47} +1.76758 q^{49} +(-2.09889 - 4.95046i) q^{51} +0.427900 q^{53} +4.86757i q^{55} +(-12.2893 + 5.21039i) q^{57} +9.10118i q^{59} +9.06577i q^{61} +(-4.93235 - 4.77113i) q^{63} -7.75536i q^{65} +8.29515 q^{67} +(1.59465 - 0.676097i) q^{69} -4.33432 q^{71} +6.91010 q^{73} +(-3.51345 + 1.48963i) q^{75} +6.65793 q^{77} -6.46025i q^{79} +(-0.298990 - 8.99503i) q^{81} +12.7870i q^{83} -5.19165i q^{85} +(13.6032 - 5.76747i) q^{87} +4.52095i q^{89} -10.6079 q^{91} +(-5.33016 - 12.5717i) q^{93} -12.8880 q^{95} -8.58440 q^{97} +(6.27612 + 6.07097i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{3} - 8 q^{5} + 2 q^{9}+O(q^{10}) \) 42 * q - 2 * q^3 - 8 * q^5 + 2 * q^9	\( 42 q - 2 q^{3} - 8 q^{5} + 2 q^{9} - 8 q^{15} - 4 q^{19} - 8 q^{21} + 42 q^{23} + 22 q^{25} + 16 q^{27} + 12 q^{33} + 8 q^{39} - 28 q^{43} + 8 q^{45} - 50 q^{49} - 28 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63} + 4 q^{67} - 2 q^{69} + 4 q^{73} + 6 q^{75} - 32 q^{77} + 18 q^{81} - 48 q^{87} + 8 q^{91} + 22 q^{93} - 16 q^{95} + 20 q^{97}+O(q^{100}) \) 42 * q - 2 * q^3 - 8 * q^5 + 2 * q^9 - 8 * q^15 - 4 * q^19 - 8 * q^21 + 42 * q^23 + 22 * q^25 + 16 * q^27 + 12 * q^33 + 8 * q^39 - 28 * q^43 + 8 * q^45 - 50 * q^49 - 28 * q^51 - 24 * q^53 - 8 * q^57 - 16 * q^63 + 4 * q^67 - 2 * q^69 + 4 * q^73 + 6 * q^75 - 32 * q^77 + 18 * q^81 - 48 * q^87 + 8 * q^91 + 22 * q^93 - 16 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more