LMFDB - Embedded newform 4320.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.a (trivial)

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:	yes
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	4320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -5.16228 q^{7} -1.16228 q^{11} -3.16228 q^{13} -4.16228 q^{17} +4.16228 q^{19} +1.00000 q^{23} +1.00000 q^{25} -5.16228 q^{29} +2.16228 q^{31} +5.16228 q^{35} -8.32456 q^{37} -9.48683 q^{41} -7.16228 q^{43} +6.00000 q^{47} +19.6491 q^{49} -3.83772 q^{53} +1.16228 q^{55} +11.1623 q^{59} -7.00000 q^{61} +3.16228 q^{65} +6.00000 q^{67} +5.16228 q^{71} -9.16228 q^{73} +6.00000 q^{77} -14.4868 q^{79} +0.675445 q^{83} +4.16228 q^{85} +7.16228 q^{89} +16.3246 q^{91} -4.16228 q^{95} +2.32456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 2 q^{17} + 2 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{29} - 2 q^{31} + 4 q^{35} - 4 q^{37} - 8 q^{43} + 12 q^{47} + 14 q^{49} - 14 q^{53} - 4 q^{55} + 16 q^{59} - 14 q^{61}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 2 * q^17 + 2 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^29 - 2 * q^31 + 4 * q^35 - 4 * q^37 - 8 * q^43 + 12 * q^47 + 14 * q^49 - 14 * q^53 - 4 * q^55 + 16 * q^59 - 14 * q^61 + 12 * q^67 + 4 * q^71 - 12 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 2 * q^85 + 8 * q^89 + 20 * q^91 - 2 * q^95 - 8 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−5.16228	−1.95116	−0.975579	−	0.219650i	$-0.929509\pi$
$7$	−5.16228	−1.95116	−0.975579	+	0.219650i	$0.929509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.16228	−0.350440	−0.175220	−	0.984529i	$-0.556064\pi$
$11$	−1.16228	−0.350440	−0.175220	+	0.984529i	$0.556064\pi$
$12$	0	0
$13$	−3.16228	−0.877058	−0.438529	−	0.898717i	$-0.644500\pi$
$13$	−3.16228	−0.877058	−0.438529	+	0.898717i	$0.644500\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.16228	−1.00950	−0.504750	−	0.863265i	$-0.668415\pi$
$17$	−4.16228	−1.00950	−0.504750	+	0.863265i	$0.668415\pi$
$18$	0	0
$19$	4.16228	0.954892	0.477446	−	0.878661i	$-0.341563\pi$
$19$	4.16228	0.954892	0.477446	+	0.878661i	$0.341563\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.16228	−0.958611	−0.479305	−	0.877648i	$-0.659111\pi$
$29$	−5.16228	−0.958611	−0.479305	+	0.877648i	$0.659111\pi$
$30$	0	0
$31$	2.16228	0.388357	0.194178	−	0.980966i	$-0.437796\pi$
$31$	2.16228	0.388357	0.194178	+	0.980966i	$0.437796\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.16228	0.872584
$36$	0	0
$37$	−8.32456	−1.36855	−0.684274	−	0.729225i	$-0.739881\pi$
$37$	−8.32456	−1.36855	−0.684274	+	0.729225i	$0.739881\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.48683	−1.48159	−0.740797	−	0.671729i	$-0.765552\pi$
$41$	−9.48683	−1.48159	−0.740797	+	0.671729i	$0.765552\pi$
$42$	0	0
$43$	−7.16228	−1.09224	−0.546119	−	0.837708i	$-0.683895\pi$
$43$	−7.16228	−1.09224	−0.546119	+	0.837708i	$0.683895\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	19.6491	2.80702
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.83772	−0.527152	−0.263576	−	0.964639i	$-0.584902\pi$
$53$	−3.83772	−0.527152	−0.263576	+	0.964639i	$0.584902\pi$
$54$	0	0
$55$	1.16228	0.156721
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.1623	1.45320	0.726602	−	0.687058i	$-0.241098\pi$
$59$	11.1623	1.45320	0.726602	+	0.687058i	$0.241098\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.16228	0.392232
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.16228	0.612650	0.306325	−	0.951927i	$-0.400901\pi$
$71$	5.16228	0.612650	0.306325	+	0.951927i	$0.400901\pi$
$72$	0	0
$73$	−9.16228	−1.07236	−0.536182	−	0.844103i	$-0.680134\pi$
$73$	−9.16228	−1.07236	−0.536182	+	0.844103i	$0.680134\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−14.4868	−1.62990	−0.814948	−	0.579534i	$-0.803235\pi$
$79$	−14.4868	−1.62990	−0.814948	+	0.579534i	$0.803235\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.675445	0.0741397	0.0370698	−	0.999313i	$-0.488198\pi$
$83$	0.675445	0.0741397	0.0370698	+	0.999313i	$0.488198\pi$
$84$	0	0
$85$	4.16228	0.451462
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.16228	0.759200	0.379600	−	0.925151i	$-0.376062\pi$
$89$	7.16228	0.759200	0.379600	+	0.925151i	$0.376062\pi$
$90$	0	0
$91$	16.3246	1.71128
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.16228	−0.427041
$96$	0	0
$97$	2.32456	0.236023	0.118011	−	0.993012i	$-0.462348\pi$
$97$	2.32456	0.236023	0.118011	+	0.993012i	$0.462348\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.3246	1.42535	0.712673	−	0.701496i	$-0.247484\pi$
$101$	14.3246	1.42535	0.712673	+	0.701496i	$0.247484\pi$
$102$	0	0
$103$	−5.67544	−0.559218	−0.279609	−	0.960114i	$-0.590205\pi$
$103$	−5.67544	−0.559218	−0.279609	+	0.960114i	$0.590205\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.3246	1.57815	0.789077	−	0.614294i	$-0.210559\pi$
$107$	16.3246	1.57815	0.789077	+	0.614294i	$0.210559\pi$
$108$	0	0
$109$	19.6491	1.88204	0.941022	−	0.338346i	$-0.109867\pi$
$109$	19.6491	1.88204	0.941022	+	0.338346i	$0.109867\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.9737	1.78489	0.892446	−	0.451154i	$-0.148987\pi$
$113$	18.9737	1.78489	0.892446	+	0.451154i	$0.148987\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.4868	1.96969
$120$	0	0
$121$	−9.64911	−0.877192
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.3246	1.77576	0.887882	−	0.460071i	$-0.152176\pi$
$131$	20.3246	1.77576	0.887882	+	0.460071i	$0.152176\pi$
$132$	0	0
$133$	−21.4868	−1.86314
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.16228	0.526479	0.263239	−	0.964731i	$-0.415209\pi$
$137$	6.16228	0.526479	0.263239	+	0.964731i	$0.415209\pi$
$138$	0	0
$139$	−4.64911	−0.394332	−0.197166	−	0.980370i	$-0.563174\pi$
$139$	−4.64911	−0.394332	−0.197166	+	0.980370i	$0.563174\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.67544	0.307356
$144$	0	0
$145$	5.16228	0.428704
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.1623	0.914449	0.457225	−	0.889351i	$-0.348844\pi$
$149$	11.1623	0.914449	0.457225	+	0.889351i	$0.348844\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.16228	−0.173678
$156$	0	0
$157$	10.8377	0.864944	0.432472	−	0.901647i	$-0.357641\pi$
$157$	10.8377	0.864944	0.432472	+	0.901647i	$0.357641\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.16228	−0.406844
$162$	0	0
$163$	−0.324555	−0.0254211	−0.0127106	−	0.999919i	$-0.504046\pi$
$163$	−0.324555	−0.0254211	−0.0127106	+	0.999919i	$0.504046\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.9737	−1.23608	−0.618040	−	0.786147i	$-0.712073\pi$
$167$	−15.9737	−1.23608	−0.618040	+	0.786147i	$0.712073\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.48683	−0.645242	−0.322621	−	0.946528i	$-0.604564\pi$
$173$	−8.48683	−0.645242	−0.322621	+	0.946528i	$0.604564\pi$
$174$	0	0
$175$	−5.16228	−0.390232
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.9737	−1.41816	−0.709079	−	0.705129i	$-0.750889\pi$
$179$	−18.9737	−1.41816	−0.709079	+	0.705129i	$0.750889\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.32456	0.612033
$186$	0	0
$187$	4.83772	0.353769
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.324555	−0.0234840	−0.0117420	−	0.999931i	$-0.503738\pi$
$191$	−0.324555	−0.0234840	−0.0117420	+	0.999931i	$0.503738\pi$
$192$	0	0
$193$	−1.48683	−0.107025	−0.0535123	−	0.998567i	$-0.517042\pi$
$193$	−1.48683	−0.107025	−0.0535123	+	0.998567i	$0.517042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.83772	−0.130932	−0.0654661	−	0.997855i	$-0.520853\pi$
$197$	−1.83772	−0.130932	−0.0654661	+	0.997855i	$0.520853\pi$
$198$	0	0
$199$	10.9737	0.777903	0.388951	−	0.921258i	$-0.372837\pi$
$199$	10.9737	0.777903	0.388951	+	0.921258i	$0.372837\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	26.6491	1.87040
$204$	0	0
$205$	9.48683	0.662589
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.83772	−0.334632
$210$	0	0
$211$	24.1623	1.66340	0.831700	−	0.555225i	$-0.187368\pi$
$211$	24.1623	1.66340	0.831700	+	0.555225i	$0.187368\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.16228	0.488463
$216$	0	0
$217$	−11.1623	−0.757745
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.1623	0.885391
$222$	0	0
$223$	−7.67544	−0.513986	−0.256993	−	0.966413i	$-0.582732\pi$
$223$	−7.67544	−0.513986	−0.256993	+	0.966413i	$0.582732\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.32456	0.220658	0.110329	−	0.993895i	$-0.464809\pi$
$227$	3.32456	0.220658	0.110329	+	0.993895i	$0.464809\pi$
$228$	0	0
$229$	10.6754	0.705453	0.352727	−	0.935726i	$-0.385255\pi$
$229$	10.6754	0.705453	0.352727	+	0.935726i	$0.385255\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.3246	1.31469	0.657343	−	0.753591i	$-0.271680\pi$
$239$	20.3246	1.31469	0.657343	+	0.753591i	$0.271680\pi$
$240$	0	0
$241$	−15.6491	−1.00805	−0.504024	−	0.863690i	$-0.668148\pi$
$241$	−15.6491	−1.00805	−0.504024	+	0.863690i	$0.668148\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−19.6491	−1.25534
$246$	0	0
$247$	−13.1623	−0.837496
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.6754	0.863186	0.431593	−	0.902068i	$-0.357952\pi$
$251$	13.6754	0.863186	0.431593	+	0.902068i	$0.357952\pi$
$252$	0	0
$253$	−1.16228	−0.0730718
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.8114	−1.79721	−0.898603	−	0.438764i	$-0.855417\pi$
$257$	−28.8114	−1.79721	−0.898603	+	0.438764i	$0.855417\pi$
$258$	0	0
$259$	42.9737	2.67025
$260$	0	0
$261$	0	0
$262$	0	0
$263$	31.2982	1.92993	0.964966	−	0.262377i	$-0.0845063\pi$
$263$	31.2982	1.92993	0.964966	+	0.262377i	$0.0845063\pi$
$264$	0	0
$265$	3.83772	0.235749
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.3246	−0.751441	−0.375721	−	0.926733i	$-0.622605\pi$
$269$	−12.3246	−0.751441	−0.375721	+	0.926733i	$0.622605\pi$
$270$	0	0
$271$	−19.1359	−1.16243	−0.581213	−	0.813752i	$-0.697422\pi$
$271$	−19.1359	−1.16243	−0.581213	+	0.813752i	$0.697422\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.16228	−0.0700880
$276$	0	0
$277$	−6.83772	−0.410839	−0.205419	−	0.978674i	$-0.565856\pi$
$277$	−6.83772	−0.410839	−0.205419	+	0.978674i	$0.565856\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.4868	−1.16249	−0.581243	−	0.813730i	$-0.697434\pi$
$281$	−19.4868	−1.16249	−0.581243	+	0.813730i	$0.697434\pi$
$282$	0	0
$283$	−6.51317	−0.387168	−0.193584	−	0.981084i	$-0.562011\pi$
$283$	−6.51317	−0.387168	−0.193584	+	0.981084i	$0.562011\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	48.9737	2.89082
$288$	0	0
$289$	0.324555	0.0190915
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.8114	−1.21581	−0.607907	−	0.794008i	$-0.707991\pi$
$293$	−20.8114	−1.21581	−0.607907	+	0.794008i	$0.707991\pi$
$294$	0	0
$295$	−11.1623	−0.649893
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.16228	−0.182879
$300$	0	0
$301$	36.9737	2.13113
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.00000	0.400819
$306$	0	0
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.1623	−0.632955	−0.316477	−	0.948600i	$-0.602500\pi$
$311$	−11.1623	−0.632955	−0.316477	+	0.948600i	$0.602500\pi$
$312$	0	0
$313$	−30.3246	−1.71405	−0.857023	−	0.515279i	$-0.827688\pi$
$313$	−30.3246	−1.71405	−0.857023	+	0.515279i	$0.827688\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.4868	0.701330	0.350665	−	0.936501i	$-0.385956\pi$
$317$	12.4868	0.701330	0.350665	+	0.936501i	$0.385956\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.3246	−0.963964
$324$	0	0
$325$	−3.16228	−0.175412
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−30.9737	−1.70763
$330$	0	0
$331$	13.6754	0.751670	0.375835	−	0.926687i	$-0.377356\pi$
$331$	13.6754	0.751670	0.375835	+	0.926687i	$0.377356\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	17.4868	0.952568	0.476284	−	0.879291i	$-0.341983\pi$
$337$	17.4868	0.952568	0.476284	+	0.879291i	$0.341983\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.51317	−0.136096
$342$	0	0
$343$	−65.2982	−3.52577
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.6491	−1.32323	−0.661617	−	0.749842i	$-0.730129\pi$
$347$	−24.6491	−1.32323	−0.661617	+	0.749842i	$0.730129\pi$
$348$	0	0
$349$	−12.6754	−0.678501	−0.339250	−	0.940696i	$-0.610173\pi$
$349$	−12.6754	−0.678501	−0.339250	+	0.940696i	$0.610173\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.6754	0.834320	0.417160	−	0.908833i	$-0.363026\pi$
$353$	15.6754	0.834320	0.417160	+	0.908833i	$0.363026\pi$
$354$	0	0
$355$	−5.16228	−0.273985
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.2982	−1.44075	−0.720373	−	0.693587i	$-0.756029\pi$
$359$	−27.2982	−1.44075	−0.720373	+	0.693587i	$0.756029\pi$
$360$	0	0
$361$	−1.67544	−0.0881813
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.16228	0.479576
$366$	0	0
$367$	34.4605	1.79882	0.899412	−	0.437102i	$-0.143995\pi$
$367$	34.4605	1.79882	0.899412	+	0.437102i	$0.143995\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	19.8114	1.02856
$372$	0	0
$373$	28.6491	1.48339	0.741697	−	0.670735i	$-0.234021\pi$
$373$	28.6491	1.48339	0.741697	+	0.670735i	$0.234021\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.3246	0.840757
$378$	0	0
$379$	−28.4868	−1.46327	−0.731635	−	0.681696i	$-0.761243\pi$
$379$	−28.4868	−1.46327	−0.731635	+	0.681696i	$0.761243\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.0000	0.970855	0.485427	−	0.874277i	$-0.338664\pi$
$383$	19.0000	0.970855	0.485427	+	0.874277i	$0.338664\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.81139	0.396053	0.198027	−	0.980197i	$-0.436547\pi$
$389$	7.81139	0.396053	0.198027	+	0.980197i	$0.436547\pi$
$390$	0	0
$391$	−4.16228	−0.210495
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.4868	0.728912
$396$	0	0
$397$	19.8114	0.994305	0.497153	−	0.867663i	$-0.334379\pi$
$397$	19.8114	0.994305	0.497153	+	0.867663i	$0.334379\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.6754	−0.782794	−0.391397	−	0.920222i	$-0.628008\pi$
$401$	−15.6754	−0.782794	−0.391397	+	0.920222i	$0.628008\pi$
$402$	0	0
$403$	−6.83772	−0.340611
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.67544	0.479594
$408$	0	0
$409$	−7.97367	−0.394272	−0.197136	−	0.980376i	$-0.563164\pi$
$409$	−7.97367	−0.394272	−0.197136	+	0.980376i	$0.563164\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−57.6228	−2.83543
$414$	0	0
$415$	−0.675445	−0.0331563
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.1359	1.17912	0.589559	−	0.807725i	$-0.299302\pi$
$419$	24.1359	1.17912	0.589559	+	0.807725i	$0.299302\pi$
$420$	0	0
$421$	−18.6754	−0.910185	−0.455093	−	0.890444i	$-0.650394\pi$
$421$	−18.6754	−0.910185	−0.455093	+	0.890444i	$0.650394\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.16228	−0.201900
$426$	0	0
$427$	36.1359	1.74874
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.188612	−0.00908511	−0.00454255	−	0.999990i	$-0.501446\pi$
$431$	−0.188612	−0.00908511	−0.00454255	+	0.999990i	$0.501446\pi$
$432$	0	0
$433$	7.16228	0.344197	0.172099	−	0.985080i	$-0.444945\pi$
$433$	7.16228	0.344197	0.172099	+	0.985080i	$0.444945\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.16228	0.199109
$438$	0	0
$439$	−36.8114	−1.75691	−0.878456	−	0.477824i	$-0.841426\pi$
$439$	−36.8114	−1.75691	−0.878456	+	0.477824i	$0.841426\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.64911	−0.458443	−0.229222	−	0.973374i	$-0.573618\pi$
$443$	−9.64911	−0.458443	−0.229222	+	0.973374i	$0.573618\pi$
$444$	0	0
$445$	−7.16228	−0.339525
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.8377	1.64409	0.822047	−	0.569420i	$-0.192832\pi$
$449$	34.8377	1.64409	0.822047	+	0.569420i	$0.192832\pi$
$450$	0	0
$451$	11.0263	0.519210
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16.3246	−0.765307
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.2982	1.27140	0.635702	−	0.771934i	$-0.280711\pi$
$461$	27.2982	1.27140	0.635702	+	0.771934i	$0.280711\pi$
$462$	0	0
$463$	−36.3246	−1.68815	−0.844073	−	0.536229i	$-0.819848\pi$
$463$	−36.3246	−1.68815	−0.844073	+	0.536229i	$0.819848\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.9737	0.739173	0.369587	−	0.929196i	$-0.379499\pi$
$467$	15.9737	0.739173	0.369587	+	0.929196i	$0.379499\pi$
$468$	0	0
$469$	−30.9737	−1.43023
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.32456	0.382763
$474$	0	0
$475$	4.16228	0.190978
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.8114	−0.813823	−0.406912	−	0.913468i	$-0.633394\pi$
$479$	−17.8114	−0.813823	−0.406912	+	0.913468i	$0.633394\pi$
$480$	0	0
$481$	26.3246	1.20030
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.32456	−0.105553
$486$	0	0
$487$	−25.4868	−1.15492	−0.577459	−	0.816419i	$-0.695956\pi$
$487$	−25.4868	−1.15492	−0.577459	+	0.816419i	$0.695956\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.64911	−0.119553	−0.0597763	−	0.998212i	$-0.519039\pi$
$491$	−2.64911	−0.119553	−0.0597763	+	0.998212i	$0.519039\pi$
$492$	0	0
$493$	21.4868	0.967718
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−26.6491	−1.19538
$498$	0	0
$499$	1.51317	0.0677387	0.0338693	−	0.999426i	$-0.489217\pi$
$499$	1.51317	0.0677387	0.0338693	+	0.999426i	$0.489217\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−36.2982	−1.61846	−0.809229	−	0.587494i	$-0.800115\pi$
$503$	−36.2982	−1.61846	−0.809229	+	0.587494i	$0.800115\pi$
$504$	0	0
$505$	−14.3246	−0.637434
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.6491	1.09255	0.546276	−	0.837605i	$-0.316045\pi$
$509$	24.6491	1.09255	0.546276	+	0.837605i	$0.316045\pi$
$510$	0	0
$511$	47.2982	2.09235
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.67544	0.250090
$516$	0	0
$517$	−6.97367	−0.306701
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11.1623	0.489028	0.244514	−	0.969646i	$-0.421372\pi$
$521$	11.1623	0.489028	0.244514	+	0.969646i	$0.421372\pi$
$522$	0	0
$523$	9.81139	0.429022	0.214511	−	0.976722i	$-0.431184\pi$
$523$	9.81139	0.429022	0.214511	+	0.976722i	$0.431184\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.00000	−0.392046
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	30.0000	1.29944
$534$	0	0
$535$	−16.3246	−0.705772
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−22.8377	−0.983690
$540$	0	0
$541$	−27.6228	−1.18760	−0.593798	−	0.804614i	$-0.702372\pi$
$541$	−27.6228	−1.18760	−0.593798	+	0.804614i	$0.702372\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.6491	−0.841675
$546$	0	0
$547$	6.51317	0.278483	0.139241	−	0.990258i	$-0.455534\pi$
$547$	6.51317	0.278483	0.139241	+	0.990258i	$0.455534\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21.4868	−0.915370
$552$	0	0
$553$	74.7851	3.18018
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.9737	0.549712	0.274856	−	0.961485i	$-0.411370\pi$
$557$	12.9737	0.549712	0.274856	+	0.961485i	$0.411370\pi$
$558$	0	0
$559$	22.6491	0.957955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.3246	−1.19374	−0.596869	−	0.802339i	$-0.703589\pi$
$563$	−28.3246	−1.19374	−0.596869	+	0.802339i	$0.703589\pi$
$564$	0	0
$565$	−18.9737	−0.798228
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.8377	−1.12510	−0.562548	−	0.826765i	$-0.690179\pi$
$569$	−26.8377	−1.12510	−0.562548	+	0.826765i	$0.690179\pi$
$570$	0	0
$571$	−10.4868	−0.438860	−0.219430	−	0.975628i	$-0.570420\pi$
$571$	−10.4868	−0.438860	−0.219430	+	0.975628i	$0.570420\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−16.1359	−0.671748	−0.335874	−	0.941907i	$-0.609032\pi$
$577$	−16.1359	−0.671748	−0.335874	+	0.941907i	$0.609032\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.48683	−0.144658
$582$	0	0
$583$	4.46050	0.184735
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.9737	−1.89754	−0.948768	−	0.315973i	$-0.897669\pi$
$587$	−45.9737	−1.89754	−0.948768	+	0.315973i	$0.897669\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	0	0
$591$	0	0
$592$	0	0
$593$	33.1359	1.36073	0.680365	−	0.732873i	$-0.261821\pi$
$593$	33.1359	1.36073	0.680365	+	0.732873i	$0.261821\pi$
$594$	0	0
$595$	−21.4868	−0.880874
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.4605	1.81661	0.908303	−	0.418312i	$-0.137378\pi$
$599$	44.4605	1.81661	0.908303	+	0.418312i	$0.137378\pi$
$600$	0	0
$601$	24.2982	0.991145	0.495573	−	0.868567i	$-0.334958\pi$
$601$	24.2982	0.991145	0.495573	+	0.868567i	$0.334958\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.64911	0.392292
$606$	0	0
$607$	0.188612	0.00765551	0.00382776	−	0.999993i	$-0.498782\pi$
$607$	0.188612	0.00765551	0.00382776	+	0.999993i	$0.498782\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.9737	−0.767592
$612$	0	0
$613$	0.649111	0.0262173	0.0131087	−	0.999914i	$-0.495827\pi$
$613$	0.649111	0.0262173	0.0131087	+	0.999914i	$0.495827\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.7851	1.44065	0.720326	−	0.693636i	$-0.243992\pi$
$617$	35.7851	1.44065	0.720326	+	0.693636i	$0.243992\pi$
$618$	0	0
$619$	38.9737	1.56648	0.783242	−	0.621717i	$-0.213565\pi$
$619$	38.9737	1.56648	0.783242	+	0.621717i	$0.213565\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−36.9737	−1.48132
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.6491	1.38155
$630$	0	0
$631$	35.4605	1.41166	0.705830	−	0.708381i	$-0.250574\pi$
$631$	35.4605	1.41166	0.705830	+	0.708381i	$0.250574\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	−62.1359	−2.46192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.6491	1.60554	0.802772	−	0.596287i	$-0.203358\pi$
$641$	40.6491	1.60554	0.802772	+	0.596287i	$0.203358\pi$
$642$	0	0
$643$	3.48683	0.137507	0.0687536	−	0.997634i	$-0.478098\pi$
$643$	3.48683	0.137507	0.0687536	+	0.997634i	$0.478098\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.9473	1.45255	0.726275	−	0.687405i	$-0.241250\pi$
$647$	36.9473	1.45255	0.726275	+	0.687405i	$0.241250\pi$
$648$	0	0
$649$	−12.9737	−0.509261
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.1623	0.710745	0.355372	−	0.934725i	$-0.384354\pi$
$653$	18.1623	0.710745	0.355372	+	0.934725i	$0.384354\pi$
$654$	0	0
$655$	−20.3246	−0.794146
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.1359	1.01811	0.509056	−	0.860733i	$-0.329995\pi$
$659$	26.1359	1.01811	0.509056	+	0.860733i	$0.329995\pi$
$660$	0	0
$661$	29.6754	1.15424	0.577121	−	0.816659i	$-0.304176\pi$
$661$	29.6754	1.15424	0.577121	+	0.816659i	$0.304176\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	21.4868	0.833224
$666$	0	0
$667$	−5.16228	−0.199884
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.13594	0.314085
$672$	0	0
$673$	18.1359	0.699089	0.349545	−	0.936920i	$-0.386336\pi$
$673$	18.1359	0.699089	0.349545	+	0.936920i	$0.386336\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.97367	−0.268020	−0.134010	−	0.990980i	$-0.542785\pi$
$677$	−6.97367	−0.268020	−0.134010	+	0.990980i	$0.542785\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.9473	1.33722	0.668611	−	0.743612i	$-0.266889\pi$
$683$	34.9473	1.33722	0.668611	+	0.743612i	$0.266889\pi$
$684$	0	0
$685$	−6.16228	−0.235449
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.1359	0.462343
$690$	0	0
$691$	27.5132	1.04665	0.523325	−	0.852133i	$-0.324691\pi$
$691$	27.5132	1.04665	0.523325	+	0.852133i	$0.324691\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.64911	0.176351
$696$	0	0
$697$	39.4868	1.49567
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15.6754	−0.592053	−0.296027	−	0.955180i	$-0.595662\pi$
$701$	−15.6754	−0.592053	−0.296027	+	0.955180i	$0.595662\pi$
$702$	0	0
$703$	−34.6491	−1.30682
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−73.9473	−2.78108
$708$	0	0
$709$	−47.6228	−1.78851	−0.894255	−	0.447557i	$-0.852294\pi$
$709$	−47.6228	−1.78851	−0.894255	+	0.447557i	$0.852294\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.16228	0.0809779
$714$	0	0
$715$	−3.67544	−0.137454
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.35089	−0.0503797	−0.0251898	−	0.999683i	$-0.508019\pi$
$719$	−1.35089	−0.0503797	−0.0251898	+	0.999683i	$0.508019\pi$
$720$	0	0
$721$	29.2982	1.09112
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.16228	−0.191722
$726$	0	0
$727$	−14.9737	−0.555343	−0.277671	−	0.960676i	$-0.589563\pi$
$727$	−14.9737	−0.555343	−0.277671	+	0.960676i	$0.589563\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29.8114	1.10261
$732$	0	0
$733$	−26.9737	−0.996295	−0.498148	−	0.867092i	$-0.665986\pi$
$733$	−26.9737	−0.996295	−0.498148	+	0.867092i	$0.665986\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.97367	−0.256878
$738$	0	0
$739$	17.1886	0.632293	0.316147	−	0.948710i	$-0.397611\pi$
$739$	17.1886	0.632293	0.316147	+	0.948710i	$0.397611\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.35089	−0.122932	−0.0614661	−	0.998109i	$-0.519578\pi$
$743$	−3.35089	−0.122932	−0.0614661	+	0.998109i	$0.519578\pi$
$744$	0	0
$745$	−11.1623	−0.408954
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−84.2719	−3.07923
$750$	0	0
$751$	46.1623	1.68449	0.842243	−	0.539099i	$-0.181235\pi$
$751$	46.1623	1.68449	0.842243	+	0.539099i	$0.181235\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	35.9473	1.30653	0.653264	−	0.757130i	$-0.273399\pi$
$757$	35.9473	1.30653	0.653264	+	0.757130i	$0.273399\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.8377	−0.682867	−0.341433	−	0.939906i	$-0.610912\pi$
$761$	−18.8377	−0.682867	−0.341433	+	0.939906i	$0.610912\pi$
$762$	0	0
$763$	−101.434	−3.67216
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.2982	−1.27454
$768$	0	0
$769$	−15.3246	−0.552618	−0.276309	−	0.961069i	$-0.589111\pi$
$769$	−15.3246	−0.552618	−0.276309	+	0.961069i	$0.589111\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.18861	−0.330491	−0.165246	−	0.986252i	$-0.552842\pi$
$773$	−9.18861	−0.330491	−0.165246	+	0.986252i	$0.552842\pi$
$774$	0	0
$775$	2.16228	0.0776713
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−39.4868	−1.41476
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.8377	−0.386815
$786$	0	0
$787$	−8.46050	−0.301584	−0.150792	−	0.988565i	$-0.548182\pi$
$787$	−8.46050	−0.301584	−0.150792	+	0.988565i	$0.548182\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−97.9473	−3.48261
$792$	0	0
$793$	22.1359	0.786070
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.8377	0.702688	0.351344	−	0.936246i	$-0.385725\pi$
$797$	19.8377	0.702688	0.351344	+	0.936246i	$0.385725\pi$
$798$	0	0
$799$	−24.9737	−0.883505
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.6491	0.375799
$804$	0	0
$805$	5.16228	0.181946
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.5132	0.369623	0.184812	−	0.982774i	$-0.440832\pi$
$809$	10.5132	0.369623	0.184812	+	0.982774i	$0.440832\pi$
$810$	0	0
$811$	47.2982	1.66087	0.830433	−	0.557119i	$-0.188093\pi$
$811$	47.2982	1.66087	0.830433	+	0.557119i	$0.188093\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.324555	0.0113687
$816$	0	0
$817$	−29.8114	−1.04297
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.9473	−0.696167	−0.348083	−	0.937464i	$-0.613167\pi$
$821$	−19.9473	−0.696167	−0.348083	+	0.937464i	$0.613167\pi$
$822$	0	0
$823$	−41.4868	−1.44614	−0.723070	−	0.690775i	$-0.757270\pi$
$823$	−41.4868	−1.44614	−0.723070	+	0.690775i	$0.757270\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.3246	−0.671981	−0.335990	−	0.941865i	$-0.609071\pi$
$827$	−19.3246	−0.671981	−0.335990	+	0.941865i	$0.609071\pi$
$828$	0	0
$829$	−44.6491	−1.55073	−0.775364	−	0.631515i	$-0.782434\pi$
$829$	−44.6491	−1.55073	−0.775364	+	0.631515i	$0.782434\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−81.7851	−2.83368
$834$	0	0
$835$	15.9737	0.552792
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.3246	−0.356443	−0.178222	−	0.983990i	$-0.557034\pi$
$839$	−10.3246	−0.356443	−0.178222	+	0.983990i	$0.557034\pi$
$840$	0	0
$841$	−2.35089	−0.0810652
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	49.8114	1.71154
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.32456	−0.285362
$852$	0	0
$853$	48.2719	1.65280	0.826399	−	0.563085i	$-0.190386\pi$
$853$	48.2719	1.65280	0.826399	+	0.563085i	$0.190386\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.8114	0.642585	0.321292	−	0.946980i	$-0.395883\pi$
$857$	18.8114	0.642585	0.321292	+	0.946980i	$0.395883\pi$
$858$	0	0
$859$	7.13594	0.243475	0.121738	−	0.992562i	$-0.461153\pi$
$859$	7.13594	0.243475	0.121738	+	0.992562i	$0.461153\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.32456	−0.317412	−0.158706	−	0.987326i	$-0.550732\pi$
$863$	−9.32456	−0.317412	−0.158706	+	0.987326i	$0.550732\pi$
$864$	0	0
$865$	8.48683	0.288561
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.8377	0.571181
$870$	0	0
$871$	−18.9737	−0.642898
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.16228	0.174517
$876$	0	0
$877$	19.4868	0.658024	0.329012	−	0.944326i	$-0.393284\pi$
$877$	19.4868	0.658024	0.329012	+	0.944326i	$0.393284\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10.4605	−0.352423	−0.176212	−	0.984352i	$-0.556384\pi$
$881$	−10.4605	−0.352423	−0.176212	+	0.984352i	$0.556384\pi$
$882$	0	0
$883$	16.7851	0.564862	0.282431	−	0.959288i	$-0.408859\pi$
$883$	16.7851	0.564862	0.282431	+	0.959288i	$0.408859\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.9737	1.47649	0.738246	−	0.674532i	$-0.235655\pi$
$887$	43.9737	1.47649	0.738246	+	0.674532i	$0.235655\pi$
$888$	0	0
$889$	−10.3246	−0.346274
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.9737	0.835712
$894$	0	0
$895$	18.9737	0.634220
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.1623	−0.372283
$900$	0	0
$901$	15.9737	0.532160
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.00000	0.166206
$906$	0	0
$907$	−10.3246	−0.342821	−0.171411	−	0.985200i	$-0.554832\pi$
$907$	−10.3246	−0.342821	−0.171411	+	0.985200i	$0.554832\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.1623	−1.43003	−0.715015	−	0.699109i	$-0.753580\pi$
$911$	−43.1623	−1.43003	−0.715015	+	0.699109i	$0.753580\pi$
$912$	0	0
$913$	−0.785054	−0.0259815
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−104.921	−3.46480
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16.3246	−0.537329
$924$	0	0
$925$	−8.32456	−0.273710
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23.1623	−0.759930	−0.379965	−	0.925001i	$-0.624064\pi$
$929$	−23.1623	−0.759930	−0.379965	+	0.925001i	$0.624064\pi$
$930$	0	0
$931$	81.7851	2.68040
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.83772	−0.158210
$936$	0	0
$937$	8.64911	0.282554	0.141277	−	0.989970i	$-0.454879\pi$
$937$	8.64911	0.282554	0.141277	+	0.989970i	$0.454879\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.35089	0.0440377	0.0220189	−	0.999758i	$-0.492991\pi$
$941$	1.35089	0.0440377	0.0220189	+	0.999758i	$0.492991\pi$
$942$	0	0
$943$	−9.48683	−0.308934
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.02633	−0.195830	−0.0979148	−	0.995195i	$-0.531217\pi$
$947$	−6.02633	−0.195830	−0.0979148	+	0.995195i	$0.531217\pi$
$948$	0	0
$949$	28.9737	0.940525
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.6491	−1.25197	−0.625984	−	0.779836i	$-0.715302\pi$
$953$	−38.6491	−1.25197	−0.625984	+	0.779836i	$0.715302\pi$
$954$	0	0
$955$	0.324555	0.0105024
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−31.8114	−1.02724
$960$	0	0
$961$	−26.3246	−0.849179
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.48683	0.0478628
$966$	0	0
$967$	37.6228	1.20987	0.604933	−	0.796276i	$-0.293200\pi$
$967$	37.6228	1.20987	0.604933	+	0.796276i	$0.293200\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.3246	1.03734	0.518672	−	0.854973i	$-0.326427\pi$
$971$	32.3246	1.03734	0.518672	+	0.854973i	$0.326427\pi$
$972$	0	0
$973$	24.0000	0.769405
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.9737	−1.56681	−0.783403	−	0.621514i	$-0.786518\pi$
$977$	−48.9737	−1.56681	−0.783403	+	0.621514i	$0.786518\pi$
$978$	0	0
$979$	−8.32456	−0.266054
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20.6754	−0.659444	−0.329722	−	0.944078i	$-0.606955\pi$
$983$	−20.6754	−0.659444	−0.329722	+	0.944078i	$0.606955\pi$
$984$	0	0
$985$	1.83772	0.0585547
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.16228	−0.227747
$990$	0	0
$991$	21.5132	0.683388	0.341694	−	0.939811i	$-0.388999\pi$
$991$	21.5132	0.683388	0.341694	+	0.939811i	$0.388999\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.9737	−0.347889
$996$	0	0
$997$	46.4605	1.47142	0.735709	−	0.677297i	$-0.236849\pi$
$997$	46.4605	1.47142	0.735709	+	0.677297i	$0.236849\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.a.m.1.1	✓	2
3.2	odd	2		4320.2.a.w.1.1	yes	2
4.3	odd	2		4320.2.a.v.1.2	yes	2
8.3	odd	2		8640.2.a.dj.1.2		2
8.5	even	2		8640.2.a.cw.1.1		2
12.11	even	2		4320.2.a.bf.1.2	yes	2
24.5	odd	2		8640.2.a.ci.1.1		2
24.11	even	2		8640.2.a.cv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.a.m.1.1	✓	2	1.1	even	1	trivial
4320.2.a.v.1.2	yes	2	4.3	odd	2
4320.2.a.w.1.1	yes	2	3.2	odd	2
4320.2.a.bf.1.2	yes	2	12.11	even	2
8640.2.a.ci.1.1		2	24.5	odd	2
8640.2.a.cv.1.2		2	24.11	even	2
8640.2.a.cw.1.1		2	8.5	even	2
8640.2.a.dj.1.2		2	8.3	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -5.16228 q^{7} -1.16228 q^{11} -3.16228 q^{13} -4.16228 q^{17} +4.16228 q^{19} +1.00000 q^{23} +1.00000 q^{25} -5.16228 q^{29} +2.16228 q^{31} +5.16228 q^{35} -8.32456 q^{37} -9.48683 q^{41} -7.16228 q^{43} +6.00000 q^{47} +19.6491 q^{49} -3.83772 q^{53} +1.16228 q^{55} +11.1623 q^{59} -7.00000 q^{61} +3.16228 q^{65} +6.00000 q^{67} +5.16228 q^{71} -9.16228 q^{73} +6.00000 q^{77} -14.4868 q^{79} +0.675445 q^{83} +4.16228 q^{85} +7.16228 q^{89} +16.3246 q^{91} -4.16228 q^{95} +2.32456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 2 q^{17} + 2 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{29} - 2 q^{31} + 4 q^{35} - 4 q^{37} - 8 q^{43} + 12 q^{47} + 14 q^{49} - 14 q^{53} - 4 q^{55} + 16 q^{59} - 14 q^{61}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 2 * q^17 + 2 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^29 - 2 * q^31 + 4 * q^35 - 4 * q^37 - 8 * q^43 + 12 * q^47 + 14 * q^49 - 14 * q^53 - 4 * q^55 + 16 * q^59 - 14 * q^61 + 12 * q^67 + 4 * q^71 - 12 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 2 * q^85 + 8 * q^89 + 20 * q^91 - 2 * q^95 - 8 * q^97

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