LMFDB - Embedded newform 2672.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2672,2,Mod(1,2672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2672 = 2^{4} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2672.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:	$21.3360274201$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	2672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{3} -3.00000 q^{5} -3.30278 q^{7} +2.30278 q^{9} +O(q^{10})$	$q-2.30278 q^{3} -3.00000 q^{5} -3.30278 q^{7} +2.30278 q^{9} +6.30278 q^{13} +6.90833 q^{15} -1.30278 q^{17} -2.00000 q^{19} +7.60555 q^{21} -1.30278 q^{23} +4.00000 q^{25} +1.60555 q^{27} +0.394449 q^{29} +0.605551 q^{31} +9.90833 q^{35} +7.60555 q^{37} -14.5139 q^{39} +8.21110 q^{41} -2.39445 q^{43} -6.90833 q^{45} +5.60555 q^{47} +3.90833 q^{49} +3.00000 q^{51} +2.60555 q^{53} +4.60555 q^{57} +13.8167 q^{59} -14.8167 q^{61} -7.60555 q^{63} -18.9083 q^{65} +6.21110 q^{67} +3.00000 q^{69} +6.90833 q^{71} -13.9083 q^{73} -9.21110 q^{75} -13.6056 q^{79} -10.6056 q^{81} -5.60555 q^{83} +3.90833 q^{85} -0.908327 q^{87} +7.81665 q^{89} -20.8167 q^{91} -1.39445 q^{93} +6.00000 q^{95} -13.5139 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 6 q^{5} - 3 q^{7} + q^{9}+O(q^{10}) $ 2 * q - q^3 - 6 * q^5 - 3 * q^7 + q^9	$ 2 q - q^{3} - 6 q^{5} - 3 q^{7} + q^{9} + 9 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 8 q^{21} + q^{23} + 8 q^{25} - 4 q^{27} + 8 q^{29} - 6 q^{31} + 9 q^{35} + 8 q^{37} - 11 q^{39} + 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} - 3 q^{49} + 6 q^{51} - 2 q^{53} + 2 q^{57} + 6 q^{59} - 8 q^{61} - 8 q^{63} - 27 q^{65} - 2 q^{67} + 6 q^{69} + 3 q^{71} - 17 q^{73} - 4 q^{75} - 20 q^{79} - 14 q^{81} - 4 q^{83} - 3 q^{85} + 9 q^{87} - 6 q^{89} - 20 q^{91} - 10 q^{93} + 12 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q - q^3 - 6 * q^5 - 3 * q^7 + q^9 + 9 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 8 * q^21 + q^23 + 8 * q^25 - 4 * q^27 + 8 * q^29 - 6 * q^31 + 9 * q^35 + 8 * q^37 - 11 * q^39 + 2 * q^41 - 12 * q^43 - 3 * q^45 + 4 * q^47 - 3 * q^49 + 6 * q^51 - 2 * q^53 + 2 * q^57 + 6 * q^59 - 8 * q^61 - 8 * q^63 - 27 * q^65 - 2 * q^67 + 6 * q^69 + 3 * q^71 - 17 * q^73 - 4 * q^75 - 20 * q^79 - 14 * q^81 - 4 * q^83 - 3 * q^85 + 9 * q^87 - 6 * q^89 - 20 * q^91 - 10 * q^93 + 12 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−2.30278	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$3$	−2.30278	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−3.30278	−1.24833	−0.624166	−	0.781292i	$-0.714561\pi$
$7$	−3.30278	−1.24833	−0.624166	+	0.781292i	$0.714561\pi$
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	6.30278	1.74808	0.874038	−	0.485858i	$-0.161493\pi$
$13$	6.30278	1.74808	0.874038	+	0.485858i	$0.161493\pi$
$14$	0	0
$15$	6.90833	1.78372
$16$	0	0
$17$	−1.30278	−0.315970	−0.157985	−	0.987442i	$-0.550500\pi$
$17$	−1.30278	−0.315970	−0.157985	+	0.987442i	$0.550500\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	7.60555	1.65967
$22$	0	0
$23$	−1.30278	−0.271647	−0.135824	−	0.990733i	$-0.543368\pi$
$23$	−1.30278	−0.271647	−0.135824	+	0.990733i	$0.543368\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.60555	0.308988
$28$	0	0
$29$	0.394449	0.0732473	0.0366236	−	0.999329i	$-0.488340\pi$
$29$	0.394449	0.0732473	0.0366236	+	0.999329i	$0.488340\pi$
$30$	0	0
$31$	0.605551	0.108760	0.0543801	−	0.998520i	$-0.482682\pi$
$31$	0.605551	0.108760	0.0543801	+	0.998520i	$0.482682\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.90833	1.67481
$36$	0	0
$37$	7.60555	1.25034	0.625172	−	0.780487i	$-0.285029\pi$
$37$	7.60555	1.25034	0.625172	+	0.780487i	$0.285029\pi$
$38$	0	0
$39$	−14.5139	−2.32408
$40$	0	0
$41$	8.21110	1.28236	0.641179	−	0.767391i	$-0.278445\pi$
$41$	8.21110	1.28236	0.641179	+	0.767391i	$0.278445\pi$
$42$	0	0
$43$	−2.39445	−0.365150	−0.182575	−	0.983192i	$-0.558443\pi$
$43$	−2.39445	−0.365150	−0.182575	+	0.983192i	$0.558443\pi$
$44$	0	0
$45$	−6.90833	−1.02983
$46$	0	0
$47$	5.60555	0.817654	0.408827	−	0.912612i	$-0.365938\pi$
$47$	5.60555	0.817654	0.408827	+	0.912612i	$0.365938\pi$
$48$	0	0
$49$	3.90833	0.558332
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	2.60555	0.357900	0.178950	−	0.983858i	$-0.442730\pi$
$53$	2.60555	0.357900	0.178950	+	0.983858i	$0.442730\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.60555	0.610020
$58$	0	0
$59$	13.8167	1.79878	0.899388	−	0.437152i	$-0.144013\pi$
$59$	13.8167	1.79878	0.899388	+	0.437152i	$0.144013\pi$
$60$	0	0
$61$	−14.8167	−1.89708	−0.948539	−	0.316660i	$-0.897439\pi$
$61$	−14.8167	−1.89708	−0.948539	+	0.316660i	$0.897439\pi$
$62$	0	0
$63$	−7.60555	−0.958209
$64$	0	0
$65$	−18.9083	−2.34529
$66$	0	0
$67$	6.21110	0.758807	0.379403	−	0.925231i	$-0.376129\pi$
$67$	6.21110	0.758807	0.379403	+	0.925231i	$0.376129\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	6.90833	0.819868	0.409934	−	0.912115i	$-0.365552\pi$
$71$	6.90833	0.819868	0.409934	+	0.912115i	$0.365552\pi$
$72$	0	0
$73$	−13.9083	−1.62785	−0.813923	−	0.580972i	$-0.802672\pi$
$73$	−13.9083	−1.62785	−0.813923	+	0.580972i	$0.802672\pi$
$74$	0	0
$75$	−9.21110	−1.06361
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.6056	−1.53074	−0.765372	−	0.643588i	$-0.777445\pi$
$79$	−13.6056	−1.53074	−0.765372	+	0.643588i	$0.777445\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−5.60555	−0.615289	−0.307645	−	0.951501i	$-0.599541\pi$
$83$	−5.60555	−0.615289	−0.307645	+	0.951501i	$0.599541\pi$
$84$	0	0
$85$	3.90833	0.423918
$86$	0	0
$87$	−0.908327	−0.0973829
$88$	0	0
$89$	7.81665	0.828564	0.414282	−	0.910149i	$-0.364033\pi$
$89$	7.81665	0.828564	0.414282	+	0.910149i	$0.364033\pi$
$90$	0	0
$91$	−20.8167	−2.18218
$92$	0	0
$93$	−1.39445	−0.144598
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−13.5139	−1.37213	−0.686063	−	0.727542i	$-0.740663\pi$
$97$	−13.5139	−1.37213	−0.686063	+	0.727542i	$0.740663\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.8167	1.07630	0.538149	−	0.842850i	$-0.319124\pi$
$101$	10.8167	1.07630	0.538149	+	0.842850i	$0.319124\pi$
$102$	0	0
$103$	−0.302776	−0.0298334	−0.0149167	−	0.999889i	$-0.504748\pi$
$103$	−0.302776	−0.0298334	−0.0149167	+	0.999889i	$0.504748\pi$
$104$	0	0
$105$	−22.8167	−2.22668
$106$	0	0
$107$	−0.394449	−0.0381328	−0.0190664	−	0.999818i	$-0.506069\pi$
$107$	−0.394449	−0.0381328	−0.0190664	+	0.999818i	$0.506069\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−17.5139	−1.66234
$112$	0	0
$113$	−19.4222	−1.82709	−0.913544	−	0.406741i	$-0.866665\pi$
$113$	−19.4222	−1.82709	−0.913544	+	0.406741i	$0.866665\pi$
$114$	0	0
$115$	3.90833	0.364453
$116$	0	0
$117$	14.5139	1.34181
$118$	0	0
$119$	4.30278	0.394435
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−18.9083	−1.70491
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	0.211103	0.0187323	0.00936616	−	0.999956i	$-0.497019\pi$
$127$	0.211103	0.0187323	0.00936616	+	0.999956i	$0.497019\pi$
$128$	0	0
$129$	5.51388	0.485470
$130$	0	0
$131$	−16.0278	−1.40035	−0.700176	−	0.713970i	$-0.746895\pi$
$131$	−16.0278	−1.40035	−0.700176	+	0.713970i	$0.746895\pi$
$132$	0	0
$133$	6.60555	0.572774
$134$	0	0
$135$	−4.81665	−0.414552
$136$	0	0
$137$	−17.2111	−1.47044	−0.735222	−	0.677827i	$-0.762922\pi$
$137$	−17.2111	−1.47044	−0.735222	+	0.677827i	$0.762922\pi$
$138$	0	0
$139$	4.90833	0.416319	0.208159	−	0.978095i	$-0.433253\pi$
$139$	4.90833	0.416319	0.208159	+	0.978095i	$0.433253\pi$
$140$	0	0
$141$	−12.9083	−1.08708
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.18335	−0.0982716
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	−0.513878	−0.0420985	−0.0210493	−	0.999778i	$-0.506701\pi$
$149$	−0.513878	−0.0420985	−0.0210493	+	0.999778i	$0.506701\pi$
$150$	0	0
$151$	−1.48612	−0.120939	−0.0604694	−	0.998170i	$-0.519260\pi$
$151$	−1.48612	−0.120939	−0.0604694	+	0.998170i	$0.519260\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−1.81665	−0.145917
$156$	0	0
$157$	−20.8167	−1.66135	−0.830675	−	0.556758i	$-0.812045\pi$
$157$	−20.8167	−1.66135	−0.830675	+	0.556758i	$0.812045\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	4.30278	0.339106
$162$	0	0
$163$	12.6056	0.987343	0.493671	−	0.869648i	$-0.335655\pi$
$163$	12.6056	0.987343	0.493671	+	0.869648i	$0.335655\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	26.7250	2.05577
$170$	0	0
$171$	−4.60555	−0.352195
$172$	0	0
$173$	20.2111	1.53662	0.768311	−	0.640077i	$-0.221098\pi$
$173$	20.2111	1.53662	0.768311	+	0.640077i	$0.221098\pi$
$174$	0	0
$175$	−13.2111	−0.998665
$176$	0	0
$177$	−31.8167	−2.39149
$178$	0	0
$179$	9.90833	0.740583	0.370292	−	0.928916i	$-0.379258\pi$
$179$	9.90833	0.740583	0.370292	+	0.928916i	$0.379258\pi$
$180$	0	0
$181$	7.21110	0.535997	0.267999	−	0.963419i	$-0.413638\pi$
$181$	7.21110	0.535997	0.267999	+	0.963419i	$0.413638\pi$
$182$	0	0
$183$	34.1194	2.52218
$184$	0	0
$185$	−22.8167	−1.67751
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.30278	−0.385720
$190$	0	0
$191$	7.69722	0.556952	0.278476	−	0.960443i	$-0.410171\pi$
$191$	7.69722	0.556952	0.278476	+	0.960443i	$0.410171\pi$
$192$	0	0
$193$	−4.90833	−0.353309	−0.176655	−	0.984273i	$-0.556528\pi$
$193$	−4.90833	−0.353309	−0.176655	+	0.984273i	$0.556528\pi$
$194$	0	0
$195$	43.5416	3.11808
$196$	0	0
$197$	7.30278	0.520301	0.260151	−	0.965568i	$-0.416228\pi$
$197$	7.30278	0.520301	0.260151	+	0.965568i	$0.416228\pi$
$198$	0	0
$199$	14.3028	1.01390	0.506948	−	0.861976i	$-0.330773\pi$
$199$	14.3028	1.01390	0.506948	+	0.861976i	$0.330773\pi$
$200$	0	0
$201$	−14.3028	−1.00884
$202$	0	0
$203$	−1.30278	−0.0914369
$204$	0	0
$205$	−24.6333	−1.72046
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.51388	0.310748	0.155374	−	0.987856i	$-0.450342\pi$
$211$	4.51388	0.310748	0.155374	+	0.987856i	$0.450342\pi$
$212$	0	0
$213$	−15.9083	−1.09002
$214$	0	0
$215$	7.18335	0.489900
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	32.0278	2.16424
$220$	0	0
$221$	−8.21110	−0.552339
$222$	0	0
$223$	−17.5139	−1.17282	−0.586408	−	0.810016i	$-0.699458\pi$
$223$	−17.5139	−1.17282	−0.586408	+	0.810016i	$0.699458\pi$
$224$	0	0
$225$	9.21110	0.614074
$226$	0	0
$227$	−22.8167	−1.51439	−0.757197	−	0.653186i	$-0.773432\pi$
$227$	−22.8167	−1.51439	−0.757197	+	0.653186i	$0.773432\pi$
$228$	0	0
$229$	12.3028	0.812990	0.406495	−	0.913653i	$-0.366751\pi$
$229$	12.3028	0.812990	0.406495	+	0.913653i	$0.366751\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.90833	−0.256043	−0.128022	−	0.991771i	$-0.540863\pi$
$233$	−3.90833	−0.256043	−0.128022	+	0.991771i	$0.540863\pi$
$234$	0	0
$235$	−16.8167	−1.09700
$236$	0	0
$237$	31.3305	2.03514
$238$	0	0
$239$	−21.9083	−1.41713	−0.708566	−	0.705645i	$-0.750658\pi$
$239$	−21.9083	−1.41713	−0.708566	+	0.705645i	$0.750658\pi$
$240$	0	0
$241$	6.30278	0.405997	0.202999	−	0.979179i	$-0.434931\pi$
$241$	6.30278	0.405997	0.202999	+	0.979179i	$0.434931\pi$
$242$	0	0
$243$	19.6056	1.25770
$244$	0	0
$245$	−11.7250	−0.749082
$246$	0	0
$247$	−12.6056	−0.802072
$248$	0	0
$249$	12.9083	0.818032
$250$	0	0
$251$	17.6056	1.11125	0.555626	−	0.831432i	$-0.312479\pi$
$251$	17.6056	1.11125	0.555626	+	0.831432i	$0.312479\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.00000	−0.563602
$256$	0	0
$257$	−6.11943	−0.381720	−0.190860	−	0.981617i	$-0.561128\pi$
$257$	−6.11943	−0.381720	−0.190860	+	0.981617i	$0.561128\pi$
$258$	0	0
$259$	−25.1194	−1.56085
$260$	0	0
$261$	0.908327	0.0562240
$262$	0	0
$263$	2.21110	0.136342	0.0681712	−	0.997674i	$-0.478284\pi$
$263$	2.21110	0.136342	0.0681712	+	0.997674i	$0.478284\pi$
$264$	0	0
$265$	−7.81665	−0.480173
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	1.69722	0.103482	0.0517408	−	0.998661i	$-0.483523\pi$
$269$	1.69722	0.103482	0.0517408	+	0.998661i	$0.483523\pi$
$270$	0	0
$271$	14.4222	0.876087	0.438043	−	0.898954i	$-0.355672\pi$
$271$	14.4222	0.876087	0.438043	+	0.898954i	$0.355672\pi$
$272$	0	0
$273$	47.9361	2.90122
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.48612	0.449797	0.224899	−	0.974382i	$-0.427795\pi$
$277$	7.48612	0.449797	0.224899	+	0.974382i	$0.427795\pi$
$278$	0	0
$279$	1.39445	0.0834835
$280$	0	0
$281$	13.4222	0.800702	0.400351	−	0.916362i	$-0.368888\pi$
$281$	13.4222	0.800702	0.400351	+	0.916362i	$0.368888\pi$
$282$	0	0
$283$	−18.4222	−1.09509	−0.547543	−	0.836777i	$-0.684437\pi$
$283$	−18.4222	−1.09509	−0.547543	+	0.836777i	$0.684437\pi$
$284$	0	0
$285$	−13.8167	−0.818428
$286$	0	0
$287$	−27.1194	−1.60081
$288$	0	0
$289$	−15.3028	−0.900163
$290$	0	0
$291$	31.1194	1.82425
$292$	0	0
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	−41.4500	−2.41331
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.21110	−0.474860
$300$	0	0
$301$	7.90833	0.455828
$302$	0	0
$303$	−24.9083	−1.43095
$304$	0	0
$305$	44.4500	2.54520
$306$	0	0
$307$	20.9361	1.19489	0.597443	−	0.801912i	$-0.296184\pi$
$307$	20.9361	1.19489	0.597443	+	0.801912i	$0.296184\pi$
$308$	0	0
$309$	0.697224	0.0396637
$310$	0	0
$311$	8.21110	0.465609	0.232804	−	0.972524i	$-0.425210\pi$
$311$	8.21110	0.465609	0.232804	+	0.972524i	$0.425210\pi$
$312$	0	0
$313$	−7.51388	−0.424710	−0.212355	−	0.977193i	$-0.568113\pi$
$313$	−7.51388	−0.424710	−0.212355	+	0.977193i	$0.568113\pi$
$314$	0	0
$315$	22.8167	1.28557
$316$	0	0
$317$	−25.5416	−1.43456	−0.717281	−	0.696784i	$-0.754613\pi$
$317$	−25.5416	−1.43456	−0.717281	+	0.696784i	$0.754613\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0.908327	0.0506979
$322$	0	0
$323$	2.60555	0.144977
$324$	0	0
$325$	25.2111	1.39846
$326$	0	0
$327$	−4.60555	−0.254688
$328$	0	0
$329$	−18.5139	−1.02070
$330$	0	0
$331$	−30.8167	−1.69384	−0.846918	−	0.531723i	$-0.821545\pi$
$331$	−30.8167	−1.69384	−0.846918	+	0.531723i	$0.821545\pi$
$332$	0	0
$333$	17.5139	0.959755
$334$	0	0
$335$	−18.6333	−1.01805
$336$	0	0
$337$	20.3944	1.11096	0.555478	−	0.831531i	$-0.312535\pi$
$337$	20.3944	1.11096	0.555478	+	0.831531i	$0.312535\pi$
$338$	0	0
$339$	44.7250	2.42913
$340$	0	0
$341$	0	0
$342$	0	0
$343$	10.2111	0.551348
$344$	0	0
$345$	−9.00000	−0.484544
$346$	0	0
$347$	−11.0917	−0.595432	−0.297716	−	0.954654i	$-0.596225\pi$
$347$	−11.0917	−0.595432	−0.297716	+	0.954654i	$0.596225\pi$
$348$	0	0
$349$	28.4500	1.52289	0.761446	−	0.648229i	$-0.224490\pi$
$349$	28.4500	1.52289	0.761446	+	0.648229i	$0.224490\pi$
$350$	0	0
$351$	10.1194	0.540135
$352$	0	0
$353$	−25.8167	−1.37408	−0.687041	−	0.726619i	$-0.741091\pi$
$353$	−25.8167	−1.37408	−0.687041	+	0.726619i	$0.741091\pi$
$354$	0	0
$355$	−20.7250	−1.09997
$356$	0	0
$357$	−9.90833	−0.524404
$358$	0	0
$359$	−22.8167	−1.20422	−0.602108	−	0.798414i	$-0.705673\pi$
$359$	−22.8167	−1.20422	−0.602108	+	0.798414i	$0.705673\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	25.3305	1.32951
$364$	0	0
$365$	41.7250	2.18399
$366$	0	0
$367$	2.81665	0.147028	0.0735141	−	0.997294i	$-0.476579\pi$
$367$	2.81665	0.147028	0.0735141	+	0.997294i	$0.476579\pi$
$368$	0	0
$369$	18.9083	0.984328
$370$	0	0
$371$	−8.60555	−0.446778
$372$	0	0
$373$	−34.5139	−1.78706	−0.893530	−	0.449003i	$-0.851779\pi$
$373$	−34.5139	−1.78706	−0.893530	+	0.449003i	$0.851779\pi$
$374$	0	0
$375$	−6.90833	−0.356744
$376$	0	0
$377$	2.48612	0.128042
$378$	0	0
$379$	−36.0278	−1.85062	−0.925311	−	0.379210i	$-0.876196\pi$
$379$	−36.0278	−1.85062	−0.925311	+	0.379210i	$0.876196\pi$
$380$	0	0
$381$	−0.486122	−0.0249048
$382$	0	0
$383$	−29.8444	−1.52498	−0.762489	−	0.647001i	$-0.776023\pi$
$383$	−29.8444	−1.52498	−0.762489	+	0.647001i	$0.776023\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.51388	−0.280286
$388$	0	0
$389$	16.8167	0.852638	0.426319	−	0.904573i	$-0.359810\pi$
$389$	16.8167	0.852638	0.426319	+	0.904573i	$0.359810\pi$
$390$	0	0
$391$	1.69722	0.0858323
$392$	0	0
$393$	36.9083	1.86178
$394$	0	0
$395$	40.8167	2.05371
$396$	0	0
$397$	−22.1194	−1.11014	−0.555071	−	0.831803i	$-0.687309\pi$
$397$	−22.1194	−1.11014	−0.555071	+	0.831803i	$0.687309\pi$
$398$	0	0
$399$	−15.2111	−0.761508
$400$	0	0
$401$	−25.3028	−1.26356	−0.631780	−	0.775148i	$-0.717675\pi$
$401$	−25.3028	−1.26356	−0.631780	+	0.775148i	$0.717675\pi$
$402$	0	0
$403$	3.81665	0.190121
$404$	0	0
$405$	31.8167	1.58098
$406$	0	0
$407$	0	0
$408$	0	0
$409$	29.9083	1.47887	0.739436	−	0.673227i	$-0.235092\pi$
$409$	29.9083	1.47887	0.739436	+	0.673227i	$0.235092\pi$
$410$	0	0
$411$	39.6333	1.95497
$412$	0	0
$413$	−45.6333	−2.24547
$414$	0	0
$415$	16.8167	0.825497
$416$	0	0
$417$	−11.3028	−0.553499
$418$	0	0
$419$	−2.21110	−0.108019	−0.0540097	−	0.998540i	$-0.517200\pi$
$419$	−2.21110	−0.108019	−0.0540097	+	0.998540i	$0.517200\pi$
$420$	0	0
$421$	5.39445	0.262909	0.131455	−	0.991322i	$-0.458035\pi$
$421$	5.39445	0.262909	0.131455	+	0.991322i	$0.458035\pi$
$422$	0	0
$423$	12.9083	0.627624
$424$	0	0
$425$	−5.21110	−0.252776
$426$	0	0
$427$	48.9361	2.36818
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.0278	0.916535	0.458267	−	0.888814i	$-0.348470\pi$
$431$	19.0278	0.916535	0.458267	+	0.888814i	$0.348470\pi$
$432$	0	0
$433$	10.3305	0.496454	0.248227	−	0.968702i	$-0.420152\pi$
$433$	10.3305	0.496454	0.248227	+	0.968702i	$0.420152\pi$
$434$	0	0
$435$	2.72498	0.130653
$436$	0	0
$437$	2.60555	0.124640
$438$	0	0
$439$	7.78890	0.371744	0.185872	−	0.982574i	$-0.440489\pi$
$439$	7.78890	0.371744	0.185872	+	0.982574i	$0.440489\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−32.7250	−1.55481	−0.777405	−	0.629000i	$-0.783465\pi$
$443$	−32.7250	−1.55481	−0.777405	+	0.629000i	$0.783465\pi$
$444$	0	0
$445$	−23.4500	−1.11163
$446$	0	0
$447$	1.18335	0.0559704
$448$	0	0
$449$	40.0278	1.88903	0.944513	−	0.328473i	$-0.106534\pi$
$449$	40.0278	1.88903	0.944513	+	0.328473i	$0.106534\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.42221	0.160789
$454$	0	0
$455$	62.4500	2.92770
$456$	0	0
$457$	32.1194	1.50248	0.751242	−	0.660027i	$-0.229455\pi$
$457$	32.1194	1.50248	0.751242	+	0.660027i	$0.229455\pi$
$458$	0	0
$459$	−2.09167	−0.0976309
$460$	0	0
$461$	−25.3028	−1.17847	−0.589234	−	0.807963i	$-0.700570\pi$
$461$	−25.3028	−1.17847	−0.589234	+	0.807963i	$0.700570\pi$
$462$	0	0
$463$	−16.2111	−0.753394	−0.376697	−	0.926337i	$-0.622940\pi$
$463$	−16.2111	−0.753394	−0.376697	+	0.926337i	$0.622940\pi$
$464$	0	0
$465$	4.18335	0.193998
$466$	0	0
$467$	−12.1194	−0.560820	−0.280410	−	0.959880i	$-0.590470\pi$
$467$	−12.1194	−0.560820	−0.280410	+	0.959880i	$0.590470\pi$
$468$	0	0
$469$	−20.5139	−0.947243
$470$	0	0
$471$	47.9361	2.20878
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	39.6333	1.81089	0.905446	−	0.424461i	$-0.139537\pi$
$479$	39.6333	1.81089	0.905446	+	0.424461i	$0.139537\pi$
$480$	0	0
$481$	47.9361	2.18570
$482$	0	0
$483$	−9.90833	−0.450844
$484$	0	0
$485$	40.5416	1.84090
$486$	0	0
$487$	−9.18335	−0.416137	−0.208069	−	0.978114i	$-0.566718\pi$
$487$	−9.18335	−0.416137	−0.208069	+	0.978114i	$0.566718\pi$
$488$	0	0
$489$	−29.0278	−1.31268
$490$	0	0
$491$	42.6333	1.92401	0.962007	−	0.273024i	$-0.0880240\pi$
$491$	42.6333	1.92401	0.962007	+	0.273024i	$0.0880240\pi$
$492$	0	0
$493$	−0.513878	−0.0231439
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.8167	−1.02347
$498$	0	0
$499$	−40.3305	−1.80544	−0.902721	−	0.430226i	$-0.858434\pi$
$499$	−40.3305	−1.80544	−0.902721	+	0.430226i	$0.858434\pi$
$500$	0	0
$501$	−2.30278	−0.102880
$502$	0	0
$503$	18.5139	0.825493	0.412747	−	0.910846i	$-0.364570\pi$
$503$	18.5139	0.825493	0.412747	+	0.910846i	$0.364570\pi$
$504$	0	0
$505$	−32.4500	−1.44400
$506$	0	0
$507$	−61.5416	−2.73316
$508$	0	0
$509$	14.0917	0.624602	0.312301	−	0.949983i	$-0.398900\pi$
$509$	14.0917	0.624602	0.312301	+	0.949983i	$0.398900\pi$
$510$	0	0
$511$	45.9361	2.03209
$512$	0	0
$513$	−3.21110	−0.141774
$514$	0	0
$515$	0.908327	0.0400257
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−46.5416	−2.04295
$520$	0	0
$521$	−22.1833	−0.971870	−0.485935	−	0.873995i	$-0.661521\pi$
$521$	−22.1833	−0.971870	−0.485935	+	0.873995i	$0.661521\pi$
$522$	0	0
$523$	−12.8167	−0.560433	−0.280217	−	0.959937i	$-0.590406\pi$
$523$	−12.8167	−0.560433	−0.280217	+	0.959937i	$0.590406\pi$
$524$	0	0
$525$	30.4222	1.32773
$526$	0	0
$527$	−0.788897	−0.0343649
$528$	0	0
$529$	−21.3028	−0.926208
$530$	0	0
$531$	31.8167	1.38073
$532$	0	0
$533$	51.7527	2.24166
$534$	0	0
$535$	1.18335	0.0511605
$536$	0	0
$537$	−22.8167	−0.984611
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−28.3944	−1.22077	−0.610386	−	0.792104i	$-0.708986\pi$
$541$	−28.3944	−1.22077	−0.610386	+	0.792104i	$0.708986\pi$
$542$	0	0
$543$	−16.6056	−0.712612
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−41.3583	−1.76835	−0.884176	−	0.467153i	$-0.845280\pi$
$547$	−41.3583	−1.76835	−0.884176	+	0.467153i	$0.845280\pi$
$548$	0	0
$549$	−34.1194	−1.45618
$550$	0	0
$551$	−0.788897	−0.0336082
$552$	0	0
$553$	44.9361	1.91088
$554$	0	0
$555$	52.5416	2.23027
$556$	0	0
$557$	−10.8167	−0.458316	−0.229158	−	0.973389i	$-0.573597\pi$
$557$	−10.8167	−0.458316	−0.229158	+	0.973389i	$0.573597\pi$
$558$	0	0
$559$	−15.0917	−0.638310
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.48612	−0.231212	−0.115606	−	0.993295i	$-0.536881\pi$
$563$	−5.48612	−0.231212	−0.115606	+	0.993295i	$0.536881\pi$
$564$	0	0
$565$	58.2666	2.45129
$566$	0	0
$567$	35.0278	1.47103
$568$	0	0
$569$	21.7889	0.913438	0.456719	−	0.889611i	$-0.349024\pi$
$569$	21.7889	0.913438	0.456719	+	0.889611i	$0.349024\pi$
$570$	0	0
$571$	−39.5416	−1.65477	−0.827383	−	0.561638i	$-0.810171\pi$
$571$	−39.5416	−1.65477	−0.827383	+	0.561638i	$0.810171\pi$
$572$	0	0
$573$	−17.7250	−0.740472
$574$	0	0
$575$	−5.21110	−0.217318
$576$	0	0
$577$	−6.48612	−0.270021	−0.135010	−	0.990844i	$-0.543107\pi$
$577$	−6.48612	−0.270021	−0.135010	+	0.990844i	$0.543107\pi$
$578$	0	0
$579$	11.3028	0.469727
$580$	0	0
$581$	18.5139	0.768085
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−43.5416	−1.80023
$586$	0	0
$587$	32.0917	1.32457	0.662283	−	0.749254i	$-0.269588\pi$
$587$	32.0917	1.32457	0.662283	+	0.749254i	$0.269588\pi$
$588$	0	0
$589$	−1.21110	−0.0499026
$590$	0	0
$591$	−16.8167	−0.691745
$592$	0	0
$593$	−35.6056	−1.46214	−0.731072	−	0.682300i	$-0.760980\pi$
$593$	−35.6056	−1.46214	−0.731072	+	0.682300i	$0.760980\pi$
$594$	0	0
$595$	−12.9083	−0.529190
$596$	0	0
$597$	−32.9361	−1.34798
$598$	0	0
$599$	36.9083	1.50803	0.754017	−	0.656855i	$-0.228114\pi$
$599$	36.9083	1.50803	0.754017	+	0.656855i	$0.228114\pi$
$600$	0	0
$601$	−29.5416	−1.20503	−0.602514	−	0.798108i	$-0.705834\pi$
$601$	−29.5416	−1.20503	−0.602514	+	0.798108i	$0.705834\pi$
$602$	0	0
$603$	14.3028	0.582454
$604$	0	0
$605$	33.0000	1.34164
$606$	0	0
$607$	18.7250	0.760024	0.380012	−	0.924982i	$-0.375920\pi$
$607$	18.7250	0.760024	0.380012	+	0.924982i	$0.375920\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	35.3305	1.42932
$612$	0	0
$613$	1.33053	0.0537397	0.0268698	−	0.999639i	$-0.491446\pi$
$613$	1.33053	0.0537397	0.0268698	+	0.999639i	$0.491446\pi$
$614$	0	0
$615$	56.7250	2.28737
$616$	0	0
$617$	26.4500	1.06484	0.532418	−	0.846482i	$-0.321284\pi$
$617$	26.4500	1.06484	0.532418	+	0.846482i	$0.321284\pi$
$618$	0	0
$619$	11.0278	0.443243	0.221621	−	0.975133i	$-0.428865\pi$
$619$	11.0278	0.443243	0.221621	+	0.975133i	$0.428865\pi$
$620$	0	0
$621$	−2.09167	−0.0839359
$622$	0	0
$623$	−25.8167	−1.03432
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.90833	−0.395071
$630$	0	0
$631$	−12.9361	−0.514977	−0.257489	−	0.966281i	$-0.582895\pi$
$631$	−12.9361	−0.514977	−0.257489	+	0.966281i	$0.582895\pi$
$632$	0	0
$633$	−10.3944	−0.413142
$634$	0	0
$635$	−0.633308	−0.0251320
$636$	0	0
$637$	24.6333	0.976007
$638$	0	0
$639$	15.9083	0.629324
$640$	0	0
$641$	−7.69722	−0.304022	−0.152011	−	0.988379i	$-0.548575\pi$
$641$	−7.69722	−0.304022	−0.152011	+	0.988379i	$0.548575\pi$
$642$	0	0
$643$	−5.51388	−0.217446	−0.108723	−	0.994072i	$-0.534676\pi$
$643$	−5.51388	−0.217446	−0.108723	+	0.994072i	$0.534676\pi$
$644$	0	0
$645$	−16.5416	−0.651326
$646$	0	0
$647$	−10.9361	−0.429942	−0.214971	−	0.976620i	$-0.568966\pi$
$647$	−10.9361	−0.429942	−0.214971	+	0.976620i	$0.568966\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.60555	0.180506
$652$	0	0
$653$	12.1194	0.474270	0.237135	−	0.971477i	$-0.423792\pi$
$653$	12.1194	0.474270	0.237135	+	0.971477i	$0.423792\pi$
$654$	0	0
$655$	48.0833	1.87877
$656$	0	0
$657$	−32.0278	−1.24952
$658$	0	0
$659$	−1.81665	−0.0707668	−0.0353834	−	0.999374i	$-0.511265\pi$
$659$	−1.81665	−0.0707668	−0.0353834	+	0.999374i	$0.511265\pi$
$660$	0	0
$661$	−12.8806	−0.500996	−0.250498	−	0.968117i	$-0.580594\pi$
$661$	−12.8806	−0.500996	−0.250498	+	0.968117i	$0.580594\pi$
$662$	0	0
$663$	18.9083	0.734339
$664$	0	0
$665$	−19.8167	−0.768457
$666$	0	0
$667$	−0.513878	−0.0198974
$668$	0	0
$669$	40.3305	1.55927
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.63331	−0.0629594	−0.0314797	−	0.999504i	$-0.510022\pi$
$673$	−1.63331	−0.0629594	−0.0314797	+	0.999504i	$0.510022\pi$
$674$	0	0
$675$	6.42221	0.247191
$676$	0	0
$677$	7.81665	0.300418	0.150209	−	0.988654i	$-0.452005\pi$
$677$	7.81665	0.300418	0.150209	+	0.988654i	$0.452005\pi$
$678$	0	0
$679$	44.6333	1.71287
$680$	0	0
$681$	52.5416	2.01340
$682$	0	0
$683$	−4.54163	−0.173781	−0.0868904	−	0.996218i	$-0.527693\pi$
$683$	−4.54163	−0.173781	−0.0868904	+	0.996218i	$0.527693\pi$
$684$	0	0
$685$	51.6333	1.97281
$686$	0	0
$687$	−28.3305	−1.08088
$688$	0	0
$689$	16.4222	0.625636
$690$	0	0
$691$	−22.4500	−0.854037	−0.427018	−	0.904243i	$-0.640436\pi$
$691$	−22.4500	−0.854037	−0.427018	+	0.904243i	$0.640436\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.7250	−0.558550
$696$	0	0
$697$	−10.6972	−0.405186
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	−28.0278	−1.05859	−0.529297	−	0.848437i	$-0.677544\pi$
$701$	−28.0278	−1.05859	−0.529297	+	0.848437i	$0.677544\pi$
$702$	0	0
$703$	−15.2111	−0.573698
$704$	0	0
$705$	38.7250	1.45847
$706$	0	0
$707$	−35.7250	−1.34358
$708$	0	0
$709$	−11.1472	−0.418641	−0.209321	−	0.977847i	$-0.567125\pi$
$709$	−11.1472	−0.418641	−0.209321	+	0.977847i	$0.567125\pi$
$710$	0	0
$711$	−31.3305	−1.17499
$712$	0	0
$713$	−0.788897	−0.0295444
$714$	0	0
$715$	0	0
$716$	0	0
$717$	50.4500	1.88409
$718$	0	0
$719$	48.1194	1.79455	0.897276	−	0.441470i	$-0.145543\pi$
$719$	48.1194	1.79455	0.897276	+	0.441470i	$0.145543\pi$
$720$	0	0
$721$	1.00000	0.0372419
$722$	0	0
$723$	−14.5139	−0.539777
$724$	0	0
$725$	1.57779	0.0585978
$726$	0	0
$727$	3.72498	0.138152	0.0690759	−	0.997611i	$-0.477995\pi$
$727$	3.72498	0.138152	0.0690759	+	0.997611i	$0.477995\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	3.11943	0.115376
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	27.0000	0.995910
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−42.9361	−1.57943	−0.789715	−	0.613474i	$-0.789771\pi$
$739$	−42.9361	−1.57943	−0.789715	+	0.613474i	$0.789771\pi$
$740$	0	0
$741$	29.0278	1.06636
$742$	0	0
$743$	−26.8806	−0.986152	−0.493076	−	0.869986i	$-0.664128\pi$
$743$	−26.8806	−0.986152	−0.493076	+	0.869986i	$0.664128\pi$
$744$	0	0
$745$	1.54163	0.0564811
$746$	0	0
$747$	−12.9083	−0.472291
$748$	0	0
$749$	1.30278	0.0476024
$750$	0	0
$751$	16.3944	0.598242	0.299121	−	0.954215i	$-0.403307\pi$
$751$	16.3944	0.598242	0.299121	+	0.954215i	$0.403307\pi$
$752$	0	0
$753$	−40.5416	−1.47742
$754$	0	0
$755$	4.45837	0.162257
$756$	0	0
$757$	−9.72498	−0.353460	−0.176730	−	0.984259i	$-0.556552\pi$
$757$	−9.72498	−0.353460	−0.176730	+	0.984259i	$0.556552\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.30278	−0.264725	−0.132363	−	0.991201i	$-0.542256\pi$
$761$	−7.30278	−0.264725	−0.132363	+	0.991201i	$0.542256\pi$
$762$	0	0
$763$	−6.60555	−0.239137
$764$	0	0
$765$	9.00000	0.325396
$766$	0	0
$767$	87.0833	3.14439
$768$	0	0
$769$	6.81665	0.245815	0.122907	−	0.992418i	$-0.460778\pi$
$769$	6.81665	0.245815	0.122907	+	0.992418i	$0.460778\pi$
$770$	0	0
$771$	14.0917	0.507499
$772$	0	0
$773$	3.39445	0.122090	0.0610449	−	0.998135i	$-0.480557\pi$
$773$	3.39445	0.122090	0.0610449	+	0.998135i	$0.480557\pi$
$774$	0	0
$775$	2.42221	0.0870082
$776$	0	0
$777$	57.8444	2.07516
$778$	0	0
$779$	−16.4222	−0.588387
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.633308	0.0226326
$784$	0	0
$785$	62.4500	2.22893
$786$	0	0
$787$	21.4500	0.764609	0.382304	−	0.924036i	$-0.375131\pi$
$787$	21.4500	0.764609	0.382304	+	0.924036i	$0.375131\pi$
$788$	0	0
$789$	−5.09167	−0.181268
$790$	0	0
$791$	64.1472	2.28081
$792$	0	0
$793$	−93.3860	−3.31624
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	−6.63331	−0.234964	−0.117482	−	0.993075i	$-0.537482\pi$
$797$	−6.63331	−0.234964	−0.117482	+	0.993075i	$0.537482\pi$
$798$	0	0
$799$	−7.30278	−0.258354
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−12.9083	−0.454959
$806$	0	0
$807$	−3.90833	−0.137580
$808$	0	0
$809$	−8.72498	−0.306754	−0.153377	−	0.988168i	$-0.549015\pi$
$809$	−8.72498	−0.306754	−0.153377	+	0.988168i	$0.549015\pi$
$810$	0	0
$811$	−19.8444	−0.696831	−0.348416	−	0.937340i	$-0.613280\pi$
$811$	−19.8444	−0.696831	−0.348416	+	0.937340i	$0.613280\pi$
$812$	0	0
$813$	−33.2111	−1.16476
$814$	0	0
$815$	−37.8167	−1.32466
$816$	0	0
$817$	4.78890	0.167542
$818$	0	0
$819$	−47.9361	−1.67502
$820$	0	0
$821$	−17.7250	−0.618606	−0.309303	−	0.950964i	$-0.600096\pi$
$821$	−17.7250	−0.618606	−0.309303	+	0.950964i	$0.600096\pi$
$822$	0	0
$823$	17.6972	0.616886	0.308443	−	0.951243i	$-0.400192\pi$
$823$	17.6972	0.616886	0.308443	+	0.951243i	$0.400192\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−55.5416	−1.93137	−0.965686	−	0.259713i	$-0.916372\pi$
$827$	−55.5416	−1.93137	−0.965686	+	0.259713i	$0.916372\pi$
$828$	0	0
$829$	−55.8722	−1.94052	−0.970260	−	0.242064i	$-0.922176\pi$
$829$	−55.8722	−1.94052	−0.970260	+	0.242064i	$0.922176\pi$
$830$	0	0
$831$	−17.2389	−0.598009
$832$	0	0
$833$	−5.09167	−0.176416
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	0.972244	0.0336057
$838$	0	0
$839$	−47.6056	−1.64353	−0.821763	−	0.569829i	$-0.807009\pi$
$839$	−47.6056	−1.64353	−0.821763	+	0.569829i	$0.807009\pi$
$840$	0	0
$841$	−28.8444	−0.994635
$842$	0	0
$843$	−30.9083	−1.06454
$844$	0	0
$845$	−80.1749	−2.75810
$846$	0	0
$847$	36.3305	1.24833
$848$	0	0
$849$	42.4222	1.45593
$850$	0	0
$851$	−9.90833	−0.339653
$852$	0	0
$853$	8.27502	0.283331	0.141666	−	0.989915i	$-0.454754\pi$
$853$	8.27502	0.283331	0.141666	+	0.989915i	$0.454754\pi$
$854$	0	0
$855$	13.8167	0.472520
$856$	0	0
$857$	−28.5416	−0.974964	−0.487482	−	0.873133i	$-0.662084\pi$
$857$	−28.5416	−0.974964	−0.487482	+	0.873133i	$0.662084\pi$
$858$	0	0
$859$	48.6056	1.65840	0.829200	−	0.558952i	$-0.188796\pi$
$859$	48.6056	1.65840	0.829200	+	0.558952i	$0.188796\pi$
$860$	0	0
$861$	62.4500	2.12829
$862$	0	0
$863$	−6.51388	−0.221735	−0.110867	−	0.993835i	$-0.535363\pi$
$863$	−6.51388	−0.221735	−0.110867	+	0.993835i	$0.535363\pi$
$864$	0	0
$865$	−60.6333	−2.06159
$866$	0	0
$867$	35.2389	1.19677
$868$	0	0
$869$	0	0
$870$	0	0
$871$	39.1472	1.32645
$872$	0	0
$873$	−31.1194	−1.05323
$874$	0	0
$875$	−9.90833	−0.334963
$876$	0	0
$877$	1.48612	0.0501828	0.0250914	−	0.999685i	$-0.492012\pi$
$877$	1.48612	0.0501828	0.0250914	+	0.999685i	$0.492012\pi$
$878$	0	0
$879$	−41.4500	−1.39807
$880$	0	0
$881$	−29.2111	−0.984147	−0.492074	−	0.870554i	$-0.663761\pi$
$881$	−29.2111	−0.984147	−0.492074	+	0.870554i	$0.663761\pi$
$882$	0	0
$883$	−18.9361	−0.637250	−0.318625	−	0.947881i	$-0.603221\pi$
$883$	−18.9361	−0.637250	−0.318625	+	0.947881i	$0.603221\pi$
$884$	0	0
$885$	95.4500	3.20852
$886$	0	0
$887$	10.8167	0.363188	0.181594	−	0.983374i	$-0.441874\pi$
$887$	10.8167	0.363188	0.181594	+	0.983374i	$0.441874\pi$
$888$	0	0
$889$	−0.697224	−0.0233842
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.2111	−0.375165
$894$	0	0
$895$	−29.7250	−0.993597
$896$	0	0
$897$	18.9083	0.631331
$898$	0	0
$899$	0.238859	0.00796639
$900$	0	0
$901$	−3.39445	−0.113085
$902$	0	0
$903$	−18.2111	−0.606028
$904$	0	0
$905$	−21.6333	−0.719115
$906$	0	0
$907$	−38.6333	−1.28280	−0.641399	−	0.767208i	$-0.721646\pi$
$907$	−38.6333	−1.28280	−0.641399	+	0.767208i	$0.721646\pi$
$908$	0	0
$909$	24.9083	0.826157
$910$	0	0
$911$	−28.8167	−0.954738	−0.477369	−	0.878703i	$-0.658410\pi$
$911$	−28.8167	−0.954738	−0.477369	+	0.878703i	$0.658410\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−102.358	−3.38386
$916$	0	0
$917$	52.9361	1.74810
$918$	0	0
$919$	10.9083	0.359833	0.179916	−	0.983682i	$-0.442417\pi$
$919$	10.9083	0.359833	0.179916	+	0.983682i	$0.442417\pi$
$920$	0	0
$921$	−48.2111	−1.58861
$922$	0	0
$923$	43.5416	1.43319
$924$	0	0
$925$	30.4222	1.00028
$926$	0	0
$927$	−0.697224	−0.0228999
$928$	0	0
$929$	33.5139	1.09955	0.549777	−	0.835311i	$-0.314713\pi$
$929$	33.5139	1.09955	0.549777	+	0.835311i	$0.314713\pi$
$930$	0	0
$931$	−7.81665	−0.256180
$932$	0	0
$933$	−18.9083	−0.619031
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13.4500	0.439391	0.219696	−	0.975568i	$-0.429494\pi$
$937$	13.4500	0.439391	0.219696	+	0.975568i	$0.429494\pi$
$938$	0	0
$939$	17.3028	0.564655
$940$	0	0
$941$	8.76114	0.285605	0.142803	−	0.989751i	$-0.454389\pi$
$941$	8.76114	0.285605	0.142803	+	0.989751i	$0.454389\pi$
$942$	0	0
$943$	−10.6972	−0.348350
$944$	0	0
$945$	15.9083	0.517498
$946$	0	0
$947$	−24.9083	−0.809412	−0.404706	−	0.914447i	$-0.632626\pi$
$947$	−24.9083	−0.809412	−0.404706	+	0.914447i	$0.632626\pi$
$948$	0	0
$949$	−87.6611	−2.84560
$950$	0	0
$951$	58.8167	1.90726
$952$	0	0
$953$	−33.3583	−1.08058	−0.540290	−	0.841479i	$-0.681686\pi$
$953$	−33.3583	−1.08058	−0.540290	+	0.841479i	$0.681686\pi$
$954$	0	0
$955$	−23.0917	−0.747229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	56.8444	1.83560
$960$	0	0
$961$	−30.6333	−0.988171
$962$	0	0
$963$	−0.908327	−0.0292704
$964$	0	0
$965$	14.7250	0.474014
$966$	0	0
$967$	43.5139	1.39931	0.699656	−	0.714480i	$-0.253337\pi$
$967$	43.5139	1.39931	0.699656	+	0.714480i	$0.253337\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	51.4777	1.65200	0.825999	−	0.563671i	$-0.190611\pi$
$971$	51.4777	1.65200	0.825999	+	0.563671i	$0.190611\pi$
$972$	0	0
$973$	−16.2111	−0.519704
$974$	0	0
$975$	−58.0555	−1.85926
$976$	0	0
$977$	47.5694	1.52188	0.760940	−	0.648822i	$-0.224738\pi$
$977$	47.5694	1.52188	0.760940	+	0.648822i	$0.224738\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.60555	0.147044
$982$	0	0
$983$	−55.2666	−1.76273	−0.881366	−	0.472435i	$-0.843375\pi$
$983$	−55.2666	−1.76273	−0.881366	+	0.472435i	$0.843375\pi$
$984$	0	0
$985$	−21.9083	−0.698057
$986$	0	0
$987$	42.6333	1.35703
$988$	0	0
$989$	3.11943	0.0991921
$990$	0	0
$991$	−23.6333	−0.750737	−0.375368	−	0.926876i	$-0.622484\pi$
$991$	−23.6333	−0.750737	−0.375368	+	0.926876i	$0.622484\pi$
$992$	0	0
$993$	70.9638	2.25197
$994$	0	0
$995$	−42.9083	−1.36029
$996$	0	0
$997$	−39.2111	−1.24183	−0.620914	−	0.783879i	$-0.713238\pi$
$997$	−39.2111	−1.24183	−0.620914	+	0.783879i	$0.713238\pi$
$998$	0	0
$999$	12.2111	0.386342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2672.2.a.c.1.1		2
4.3	odd	2		668.2.a.a.1.2	✓	2
12.11	even	2		6012.2.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.a.1.2	✓	2	4.3	odd	2
2672.2.a.c.1.1		2	1.1	even	1	trivial
6012.2.a.a.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 2672 = 2^{4} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2672.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{3} -3.00000 q^{5} -3.30278 q^{7} +2.30278 q^{9} +O(q^{10})\)	\(q-2.30278 q^{3} -3.00000 q^{5} -3.30278 q^{7} +2.30278 q^{9} +6.30278 q^{13} +6.90833 q^{15} -1.30278 q^{17} -2.00000 q^{19} +7.60555 q^{21} -1.30278 q^{23} +4.00000 q^{25} +1.60555 q^{27} +0.394449 q^{29} +0.605551 q^{31} +9.90833 q^{35} +7.60555 q^{37} -14.5139 q^{39} +8.21110 q^{41} -2.39445 q^{43} -6.90833 q^{45} +5.60555 q^{47} +3.90833 q^{49} +3.00000 q^{51} +2.60555 q^{53} +4.60555 q^{57} +13.8167 q^{59} -14.8167 q^{61} -7.60555 q^{63} -18.9083 q^{65} +6.21110 q^{67} +3.00000 q^{69} +6.90833 q^{71} -13.9083 q^{73} -9.21110 q^{75} -13.6056 q^{79} -10.6056 q^{81} -5.60555 q^{83} +3.90833 q^{85} -0.908327 q^{87} +7.81665 q^{89} -20.8167 q^{91} -1.39445 q^{93} +6.00000 q^{95} -13.5139 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 6 q^{5} - 3 q^{7} + q^{9}+O(q^{10}) \) 2 * q - q^3 - 6 * q^5 - 3 * q^7 + q^9	\( 2 q - q^{3} - 6 q^{5} - 3 q^{7} + q^{9} + 9 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 8 q^{21} + q^{23} + 8 q^{25} - 4 q^{27} + 8 q^{29} - 6 q^{31} + 9 q^{35} + 8 q^{37} - 11 q^{39} + 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} - 3 q^{49} + 6 q^{51} - 2 q^{53} + 2 q^{57} + 6 q^{59} - 8 q^{61} - 8 q^{63} - 27 q^{65} - 2 q^{67} + 6 q^{69} + 3 q^{71} - 17 q^{73} - 4 q^{75} - 20 q^{79} - 14 q^{81} - 4 q^{83} - 3 q^{85} + 9 q^{87} - 6 q^{89} - 20 q^{91} - 10 q^{93} + 12 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q - q^3 - 6 * q^5 - 3 * q^7 + q^9 + 9 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 8 * q^21 + q^23 + 8 * q^25 - 4 * q^27 + 8 * q^29 - 6 * q^31 + 9 * q^35 + 8 * q^37 - 11 * q^39 + 2 * q^41 - 12 * q^43 - 3 * q^45 + 4 * q^47 - 3 * q^49 + 6 * q^51 - 2 * q^53 + 2 * q^57 + 6 * q^59 - 8 * q^61 - 8 * q^63 - 27 * q^65 - 2 * q^67 + 6 * q^69 + 3 * q^71 - 17 * q^73 - 4 * q^75 - 20 * q^79 - 14 * q^81 - 4 * q^83 - 3 * q^85 + 9 * q^87 - 6 * q^89 - 20 * q^91 - 10 * q^93 + 12 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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