LMFDB - Embedded newform 2672.2.a.j.1.2

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:	$5$
Coefficient field:	5.5.826865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 1 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 1
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			1.2
Root			$1.66287$ 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)

$f(q)$	$=$	$q-1.89773 q^{3} +2.00000 q^{5} -4.92082 q^{7} +0.601369 q^{9} +O(q^{10})$	$q-1.89773 q^{3} +2.00000 q^{5} -4.92082 q^{7} +0.601369 q^{9} +2.15568 q^{11} -3.32575 q^{13} -3.79545 q^{15} +5.91846 q^{17} +0.927114 q^{19} +9.33838 q^{21} +7.63710 q^{23} -1.00000 q^{25} +4.55195 q^{27} -1.59508 q^{29} +5.18270 q^{31} -4.09089 q^{33} -9.84165 q^{35} +1.53029 q^{37} +6.31136 q^{39} -9.71392 q^{41} -6.65149 q^{43} +1.20274 q^{45} -5.72438 q^{47} +17.2145 q^{49} -11.2316 q^{51} -9.24421 q^{53} +4.31136 q^{55} -1.75941 q^{57} -5.78287 q^{59} +14.8606 q^{61} -2.95923 q^{63} -6.65149 q^{65} -10.5747 q^{67} -14.4931 q^{69} -5.51410 q^{71} -13.5556 q^{73} +1.89773 q^{75} -10.6077 q^{77} +8.23454 q^{79} -10.4425 q^{81} -7.97724 q^{83} +11.8369 q^{85} +3.02702 q^{87} -8.41363 q^{89} +16.3654 q^{91} -9.83536 q^{93} +1.85423 q^{95} +5.34111 q^{97} +1.29636 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9	$ 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9} - 5 q^{11} - 4 q^{13} - 6 q^{15} - 2 q^{17} - 5 q^{19} - 4 q^{21} - 6 q^{23} - 5 q^{25} - 12 q^{27} - 5 q^{29} - 9 q^{31} - 8 q^{33} - 18 q^{35} + 8 q^{37} - 4 q^{41} - 8 q^{43} + 12 q^{45} - 13 q^{47} + 14 q^{49} - 2 q^{53} - 10 q^{55} - 12 q^{57} - 4 q^{59} + 11 q^{61} + q^{63} - 8 q^{65} - 28 q^{67} - 16 q^{69} - 2 q^{71} + 8 q^{73} + 3 q^{75} - 12 q^{77} + 10 q^{79} - 15 q^{81} - 2 q^{83} - 4 q^{85} - 4 q^{87} - 17 q^{89} + 12 q^{91} - 12 q^{93} - 10 q^{95} - 27 q^{97} - 3 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9 - 5 * q^11 - 4 * q^13 - 6 * q^15 - 2 * q^17 - 5 * q^19 - 4 * q^21 - 6 * q^23 - 5 * q^25 - 12 * q^27 - 5 * q^29 - 9 * q^31 - 8 * q^33 - 18 * q^35 + 8 * q^37 - 4 * q^41 - 8 * q^43 + 12 * q^45 - 13 * q^47 + 14 * q^49 - 2 * q^53 - 10 * q^55 - 12 * q^57 - 4 * q^59 + 11 * q^61 + q^63 - 8 * q^65 - 28 * q^67 - 16 * q^69 - 2 * q^71 + 8 * q^73 + 3 * q^75 - 12 * q^77 + 10 * q^79 - 15 * q^81 - 2 * q^83 - 4 * q^85 - 4 * q^87 - 17 * q^89 + 12 * q^91 - 12 * q^93 - 10 * q^95 - 27 * q^97 - 3 * q^99

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
$3$	−1.89773	−1.09565	−0.547827	−	0.836592i	$-0.684545\pi$
$3$	−1.89773	−1.09565	−0.547827	+	0.836592i	$0.684545\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.92082	−1.85990	−0.929948	−	0.367690i	$-0.880149\pi$
$7$	−4.92082	−1.85990	−0.929948	+	0.367690i	$0.880149\pi$
$8$	0	0
$9$	0.601369	0.200456
$10$	0	0
$11$	2.15568	0.649962	0.324981	−	0.945721i	$-0.394642\pi$
$11$	2.15568	0.649962	0.324981	+	0.945721i	$0.394642\pi$
$12$	0	0
$13$	−3.32575	−0.922396	−0.461198	−	0.887297i	$-0.652580\pi$
$13$	−3.32575	−0.922396	−0.461198	+	0.887297i	$0.652580\pi$
$14$	0	0
$15$	−3.79545	−0.979982
$16$	0	0
$17$	5.91846	1.43544	0.717719	−	0.696333i	$-0.245186\pi$
$17$	5.91846	1.43544	0.717719	+	0.696333i	$0.245186\pi$
$18$	0	0
$19$	0.927114	0.212695	0.106347	−	0.994329i	$-0.466084\pi$
$19$	0.927114	0.212695	0.106347	+	0.994329i	$0.466084\pi$
$20$	0	0
$21$	9.33838	2.03780
$22$	0	0
$23$	7.63710	1.59245	0.796223	−	0.605003i	$-0.206828\pi$
$23$	7.63710	1.59245	0.796223	+	0.605003i	$0.206828\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.55195	0.876023
$28$	0	0
$29$	−1.59508	−0.296199	−0.148099	−	0.988972i	$-0.547316\pi$
$29$	−1.59508	−0.296199	−0.148099	+	0.988972i	$0.547316\pi$
$30$	0	0
$31$	5.18270	0.930841	0.465421	−	0.885090i	$-0.345903\pi$
$31$	5.18270	0.930841	0.465421	+	0.885090i	$0.345903\pi$
$32$	0	0
$33$	−4.09089	−0.712133
$34$	0	0
$35$	−9.84165	−1.66354
$36$	0	0
$37$	1.53029	0.251578	0.125789	−	0.992057i	$-0.459854\pi$
$37$	1.53029	0.251578	0.125789	+	0.992057i	$0.459854\pi$
$38$	0	0
$39$	6.31136	1.01063
$40$	0	0
$41$	−9.71392	−1.51706	−0.758529	−	0.651639i	$-0.774082\pi$
$41$	−9.71392	−1.51706	−0.758529	+	0.651639i	$0.774082\pi$
$42$	0	0
$43$	−6.65149	−1.01434	−0.507171	−	0.861845i	$-0.669309\pi$
$43$	−6.65149	−1.01434	−0.507171	+	0.861845i	$0.669309\pi$
$44$	0	0
$45$	1.20274	0.179294
$46$	0	0
$47$	−5.72438	−0.834986	−0.417493	−	0.908680i	$-0.637091\pi$
$47$	−5.72438	−0.834986	−0.417493	+	0.908680i	$0.637091\pi$
$48$	0	0
$49$	17.2145	2.45922
$50$	0	0
$51$	−11.2316	−1.57274
$52$	0	0
$53$	−9.24421	−1.26979	−0.634895	−	0.772599i	$-0.718957\pi$
$53$	−9.24421	−1.26979	−0.634895	+	0.772599i	$0.718957\pi$
$54$	0	0
$55$	4.31136	0.581343
$56$	0	0
$57$	−1.75941	−0.233040
$58$	0	0
$59$	−5.78287	−0.752866	−0.376433	−	0.926444i	$-0.622850\pi$
$59$	−5.78287	−0.752866	−0.376433	+	0.926444i	$0.622850\pi$
$60$	0	0
$61$	14.8606	1.90270	0.951351	−	0.308110i	$-0.0996964\pi$
$61$	14.8606	1.90270	0.951351	+	0.308110i	$0.0996964\pi$
$62$	0	0
$63$	−2.95923	−0.372828
$64$	0	0
$65$	−6.65149	−0.825016
$66$	0	0
$67$	−10.5747	−1.29190	−0.645951	−	0.763379i	$-0.723539\pi$
$67$	−10.5747	−1.29190	−0.645951	+	0.763379i	$0.723539\pi$
$68$	0	0
$69$	−14.4931	−1.74477
$70$	0	0
$71$	−5.51410	−0.654403	−0.327201	−	0.944955i	$-0.606106\pi$
$71$	−5.51410	−0.654403	−0.327201	+	0.944955i	$0.606106\pi$
$72$	0	0
$73$	−13.5556	−1.58656	−0.793279	−	0.608858i	$-0.791628\pi$
$73$	−13.5556	−1.58656	−0.793279	+	0.608858i	$0.791628\pi$
$74$	0	0
$75$	1.89773	0.219131
$76$	0	0
$77$	−10.6077	−1.20886
$78$	0	0
$79$	8.23454	0.926459	0.463229	−	0.886238i	$-0.346691\pi$
$79$	8.23454	0.926459	0.463229	+	0.886238i	$0.346691\pi$
$80$	0	0
$81$	−10.4425	−1.16027
$82$	0	0
$83$	−7.97724	−0.875615	−0.437808	−	0.899069i	$-0.644245\pi$
$83$	−7.97724	−0.875615	−0.437808	+	0.899069i	$0.644245\pi$
$84$	0	0
$85$	11.8369	1.28389
$86$	0	0
$87$	3.02702	0.324531
$88$	0	0
$89$	−8.41363	−0.891843	−0.445922	−	0.895072i	$-0.647124\pi$
$89$	−8.41363	−0.891843	−0.445922	+	0.895072i	$0.647124\pi$
$90$	0	0
$91$	16.3654	1.71556
$92$	0	0
$93$	−9.83536	−1.01988
$94$	0	0
$95$	1.85423	0.190240
$96$	0	0
$97$	5.34111	0.542308	0.271154	−	0.962536i	$-0.412595\pi$
$97$	5.34111	0.542308	0.271154	+	0.962536i	$0.412595\pi$
$98$	0	0
$99$	1.29636	0.130289
$100$	0	0
$101$	9.76011	0.971167	0.485584	−	0.874190i	$-0.338607\pi$
$101$	9.76011	0.971167	0.485584	+	0.874190i	$0.338607\pi$
$102$	0	0
$103$	−9.38636	−0.924866	−0.462433	−	0.886654i	$-0.653023\pi$
$103$	−9.38636	−0.924866	−0.462433	+	0.886654i	$0.653023\pi$
$104$	0	0
$105$	18.6768	1.82267
$106$	0	0
$107$	10.9161	1.05530	0.527650	−	0.849462i	$-0.323073\pi$
$107$	10.9161	1.05530	0.527650	+	0.849462i	$0.323073\pi$
$108$	0	0
$109$	9.57287	0.916915	0.458457	−	0.888716i	$-0.348402\pi$
$109$	9.57287	0.916915	0.458457	+	0.888716i	$0.348402\pi$
$110$	0	0
$111$	−2.90407	−0.275643
$112$	0	0
$113$	−8.84346	−0.831922	−0.415961	−	0.909382i	$-0.636555\pi$
$113$	−8.84346	−0.831922	−0.415961	+	0.909382i	$0.636555\pi$
$114$	0	0
$115$	15.2742	1.42433
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−29.1237	−2.66977
$120$	0	0
$121$	−6.35305	−0.577550
$122$	0	0
$123$	18.4344	1.66217
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	9.52762	0.845439	0.422720	−	0.906260i	$-0.361075\pi$
$127$	9.52762	0.845439	0.422720	+	0.906260i	$0.361075\pi$
$128$	0	0
$129$	12.6227	1.11137
$130$	0	0
$131$	−3.18543	−0.278313	−0.139156	−	0.990270i	$-0.544439\pi$
$131$	−3.18543	−0.278313	−0.139156	+	0.990270i	$0.544439\pi$
$132$	0	0
$133$	−4.56217	−0.395590
$134$	0	0
$135$	9.10390	0.783539
$136$	0	0
$137$	−8.62066	−0.736513	−0.368257	−	0.929724i	$-0.620045\pi$
$137$	−8.62066	−0.736513	−0.368257	+	0.929724i	$0.620045\pi$
$138$	0	0
$139$	−16.7457	−1.42035	−0.710177	−	0.704023i	$-0.751385\pi$
$139$	−16.7457	−1.42035	−0.710177	+	0.704023i	$0.751385\pi$
$140$	0	0
$141$	10.8633	0.914855
$142$	0	0
$143$	−7.16924	−0.599522
$144$	0	0
$145$	−3.19016	−0.264928
$146$	0	0
$147$	−32.6684	−2.69445
$148$	0	0
$149$	1.18543	0.0971145	0.0485572	−	0.998820i	$-0.484538\pi$
$149$	1.18543	0.0971145	0.0485572	+	0.998820i	$0.484538\pi$
$150$	0	0
$151$	−8.43075	−0.686085	−0.343042	−	0.939320i	$-0.611457\pi$
$151$	−8.43075	−0.686085	−0.343042	+	0.939320i	$0.611457\pi$
$152$	0	0
$153$	3.55918	0.287743
$154$	0	0
$155$	10.3654	0.832570
$156$	0	0
$157$	−22.9854	−1.83443	−0.917217	−	0.398388i	$-0.869570\pi$
$157$	−22.9854	−1.83443	−0.917217	+	0.398388i	$0.869570\pi$
$158$	0	0
$159$	17.5430	1.39125
$160$	0	0
$161$	−37.5808	−2.96178
$162$	0	0
$163$	5.16267	0.404371	0.202186	−	0.979347i	$-0.435196\pi$
$163$	5.16267	0.404371	0.202186	+	0.979347i	$0.435196\pi$
$164$	0	0
$165$	−8.18178	−0.636951
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−1.93942	−0.149186
$170$	0	0
$171$	0.557538	0.0426360
$172$	0	0
$173$	−0.472383	−0.0359146	−0.0179573	−	0.999839i	$-0.505716\pi$
$173$	−0.472383	−0.0359146	−0.0179573	+	0.999839i	$0.505716\pi$
$174$	0	0
$175$	4.92082	0.371979
$176$	0	0
$177$	10.9743	0.824880
$178$	0	0
$179$	24.2013	1.80889	0.904446	−	0.426589i	$-0.140285\pi$
$179$	24.2013	1.80889	0.904446	+	0.426589i	$0.140285\pi$
$180$	0	0
$181$	8.25286	0.613430	0.306715	−	0.951801i	$-0.400770\pi$
$181$	8.25286	0.613430	0.306715	+	0.951801i	$0.400770\pi$
$182$	0	0
$183$	−28.2013	−2.08470
$184$	0	0
$185$	3.06058	0.225018
$186$	0	0
$187$	12.7583	0.932979
$188$	0	0
$189$	−22.3993	−1.62931
$190$	0	0
$191$	−19.6806	−1.42404	−0.712018	−	0.702161i	$-0.752219\pi$
$191$	−19.6806	−1.42404	−0.712018	+	0.702161i	$0.752219\pi$
$192$	0	0
$193$	0.980888	0.0706059	0.0353029	−	0.999377i	$-0.488760\pi$
$193$	0.980888	0.0706059	0.0353029	+	0.999377i	$0.488760\pi$
$194$	0	0
$195$	12.6227	0.903931
$196$	0	0
$197$	−16.0379	−1.14265	−0.571325	−	0.820724i	$-0.693570\pi$
$197$	−16.0379	−1.14265	−0.571325	+	0.820724i	$0.693570\pi$
$198$	0	0
$199$	−18.6207	−1.31998	−0.659992	−	0.751273i	$-0.729440\pi$
$199$	−18.6207	−1.31998	−0.659992	+	0.751273i	$0.729440\pi$
$200$	0	0
$201$	20.0679	1.41548
$202$	0	0
$203$	7.84910	0.550899
$204$	0	0
$205$	−19.4278	−1.35690
$206$	0	0
$207$	4.59272	0.319216
$208$	0	0
$209$	1.99856	0.138243
$210$	0	0
$211$	−10.2451	−0.705304	−0.352652	−	0.935755i	$-0.614720\pi$
$211$	−10.2451	−0.705304	−0.352652	+	0.935755i	$0.614720\pi$
$212$	0	0
$213$	10.4642	0.716999
$214$	0	0
$215$	−13.3030	−0.907256
$216$	0	0
$217$	−25.5032	−1.73127
$218$	0	0
$219$	25.7248	1.73832
$220$	0	0
$221$	−19.6833	−1.32404
$222$	0	0
$223$	−2.05135	−0.137369	−0.0686844	−	0.997638i	$-0.521880\pi$
$223$	−2.05135	−0.137369	−0.0686844	+	0.997638i	$0.521880\pi$
$224$	0	0
$225$	−0.601369	−0.0400913
$226$	0	0
$227$	2.13743	0.141866	0.0709332	−	0.997481i	$-0.477402\pi$
$227$	2.13743	0.141866	0.0709332	+	0.997481i	$0.477402\pi$
$228$	0	0
$229$	9.18599	0.607027	0.303514	−	0.952827i	$-0.401840\pi$
$229$	9.18599	0.607027	0.303514	+	0.952827i	$0.401840\pi$
$230$	0	0
$231$	20.1306	1.32449
$232$	0	0
$233$	16.5799	1.08619	0.543093	−	0.839672i	$-0.317253\pi$
$233$	16.5799	1.08619	0.543093	+	0.839672i	$0.317253\pi$
$234$	0	0
$235$	−11.4488	−0.746834
$236$	0	0
$237$	−15.6269	−1.01508
$238$	0	0
$239$	−9.13046	−0.590601	−0.295300	−	0.955404i	$-0.595420\pi$
$239$	−9.13046	−0.590601	−0.295300	+	0.955404i	$0.595420\pi$
$240$	0	0
$241$	−16.9683	−1.09302	−0.546512	−	0.837451i	$-0.684045\pi$
$241$	−16.9683	−1.09302	−0.546512	+	0.837451i	$0.684045\pi$
$242$	0	0
$243$	6.16110	0.395235
$244$	0	0
$245$	34.4290	2.19959
$246$	0	0
$247$	−3.08335	−0.196189
$248$	0	0
$249$	15.1386	0.959371
$250$	0	0
$251$	5.45949	0.344600	0.172300	−	0.985045i	$-0.444880\pi$
$251$	5.45949	0.344600	0.172300	+	0.985045i	$0.444880\pi$
$252$	0	0
$253$	16.4631	1.03503
$254$	0	0
$255$	−22.4633	−1.40670
$256$	0	0
$257$	5.76122	0.359375	0.179687	−	0.983724i	$-0.442491\pi$
$257$	5.76122	0.359375	0.179687	+	0.983724i	$0.442491\pi$
$258$	0	0
$259$	−7.53029	−0.467909
$260$	0	0
$261$	−0.959231	−0.0593749
$262$	0	0
$263$	−0.636216	−0.0392307	−0.0196154	−	0.999808i	$-0.506244\pi$
$263$	−0.636216	−0.0392307	−0.0196154	+	0.999808i	$0.506244\pi$
$264$	0	0
$265$	−18.4884	−1.13573
$266$	0	0
$267$	15.9668	0.977151
$268$	0	0
$269$	31.5970	1.92651	0.963253	−	0.268597i	$-0.0865601\pi$
$269$	31.5970	1.92651	0.963253	+	0.268597i	$0.0865601\pi$
$270$	0	0
$271$	0.651490	0.0395752	0.0197876	−	0.999804i	$-0.493701\pi$
$271$	0.651490	0.0395752	0.0197876	+	0.999804i	$0.493701\pi$
$272$	0	0
$273$	−31.0571	−1.87966
$274$	0	0
$275$	−2.15568	−0.129992
$276$	0	0
$277$	19.3510	1.16269	0.581345	−	0.813657i	$-0.302527\pi$
$277$	19.3510	1.16269	0.581345	+	0.813657i	$0.302527\pi$
$278$	0	0
$279$	3.11672	0.186593
$280$	0	0
$281$	−25.5121	−1.52192	−0.760961	−	0.648797i	$-0.775272\pi$
$281$	−25.5121	−1.52192	−0.760961	+	0.648797i	$0.775272\pi$
$282$	0	0
$283$	−11.6217	−0.690840	−0.345420	−	0.938448i	$-0.612264\pi$
$283$	−11.6217	−0.690840	−0.345420	+	0.938448i	$0.612264\pi$
$284$	0	0
$285$	−3.51882	−0.208437
$286$	0	0
$287$	47.8005	2.82157
$288$	0	0
$289$	18.0282	1.06048
$290$	0	0
$291$	−10.1360	−0.594181
$292$	0	0
$293$	−13.6824	−0.799335	−0.399668	−	0.916660i	$-0.630874\pi$
$293$	−13.6824	−0.799335	−0.399668	+	0.916660i	$0.630874\pi$
$294$	0	0
$295$	−11.5657	−0.673384
$296$	0	0
$297$	9.81254	0.569381
$298$	0	0
$299$	−25.3991	−1.46887
$300$	0	0
$301$	32.7308	1.88657
$302$	0	0
$303$	−18.5220	−1.06406
$304$	0	0
$305$	29.7212	1.70183
$306$	0	0
$307$	−19.1003	−1.09011	−0.545055	−	0.838400i	$-0.683491\pi$
$307$	−19.1003	−1.09011	−0.545055	+	0.838400i	$0.683491\pi$
$308$	0	0
$309$	17.8128	1.01333
$310$	0	0
$311$	23.1206	1.31105	0.655524	−	0.755174i	$-0.272448\pi$
$311$	23.1206	1.31105	0.655524	+	0.755174i	$0.272448\pi$
$312$	0	0
$313$	−5.63526	−0.318524	−0.159262	−	0.987236i	$-0.550911\pi$
$313$	−5.63526	−0.318524	−0.159262	+	0.987236i	$0.550911\pi$
$314$	0	0
$315$	−5.91846	−0.333468
$316$	0	0
$317$	−9.44544	−0.530509	−0.265254	−	0.964178i	$-0.585456\pi$
$317$	−9.44544	−0.530509	−0.265254	+	0.964178i	$0.585456\pi$
$318$	0	0
$319$	−3.43848	−0.192518
$320$	0	0
$321$	−20.7158	−1.15624
$322$	0	0
$323$	5.48709	0.305310
$324$	0	0
$325$	3.32575	0.184479
$326$	0	0
$327$	−18.1667	−1.00462
$328$	0	0
$329$	28.1686	1.55299
$330$	0	0
$331$	−12.7565	−0.701158	−0.350579	−	0.936533i	$-0.614015\pi$
$331$	−12.7565	−0.701158	−0.350579	+	0.936533i	$0.614015\pi$
$332$	0	0
$333$	0.920269	0.0504304
$334$	0	0
$335$	−21.1494	−1.15551
$336$	0	0
$337$	−6.10285	−0.332443	−0.166222	−	0.986088i	$-0.553157\pi$
$337$	−6.10285	−0.332443	−0.166222	+	0.986088i	$0.553157\pi$
$338$	0	0
$339$	16.7825	0.911498
$340$	0	0
$341$	11.1722	0.605011
$342$	0	0
$343$	−50.2638	−2.71399
$344$	0	0
$345$	−28.9863	−1.56057
$346$	0	0
$347$	9.85312	0.528943	0.264472	−	0.964393i	$-0.414802\pi$
$347$	9.85312	0.528943	0.264472	+	0.964393i	$0.414802\pi$
$348$	0	0
$349$	6.75107	0.361376	0.180688	−	0.983540i	$-0.442168\pi$
$349$	6.75107	0.361376	0.180688	+	0.983540i	$0.442168\pi$
$350$	0	0
$351$	−15.1386	−0.808040
$352$	0	0
$353$	−21.7715	−1.15878	−0.579390	−	0.815050i	$-0.696709\pi$
$353$	−21.7715	−1.15878	−0.579390	+	0.815050i	$0.696709\pi$
$354$	0	0
$355$	−11.0282	−0.585316
$356$	0	0
$357$	55.2689	2.92514
$358$	0	0
$359$	−0.773613	−0.0408297	−0.0204149	−	0.999792i	$-0.506499\pi$
$359$	−0.773613	−0.0408297	−0.0204149	+	0.999792i	$0.506499\pi$
$360$	0	0
$361$	−18.1405	−0.954761
$362$	0	0
$363$	12.0564	0.632795
$364$	0	0
$365$	−27.1111	−1.41906
$366$	0	0
$367$	−34.9813	−1.82601	−0.913004	−	0.407951i	$-0.866243\pi$
$367$	−34.9813	−1.82601	−0.913004	+	0.407951i	$0.866243\pi$
$368$	0	0
$369$	−5.84165	−0.304104
$370$	0	0
$371$	45.4891	2.36168
$372$	0	0
$373$	26.3114	1.36235	0.681175	−	0.732120i	$-0.261469\pi$
$373$	26.3114	1.36235	0.681175	+	0.732120i	$0.261469\pi$
$374$	0	0
$375$	22.7727	1.17598
$376$	0	0
$377$	5.30483	0.273212
$378$	0	0
$379$	−5.49668	−0.282345	−0.141173	−	0.989985i	$-0.545087\pi$
$379$	−5.49668	−0.282345	−0.141173	+	0.989985i	$0.545087\pi$
$380$	0	0
$381$	−18.0808	−0.926308
$382$	0	0
$383$	−17.4512	−0.891712	−0.445856	−	0.895105i	$-0.647101\pi$
$383$	−17.4512	−0.891712	−0.445856	+	0.895105i	$0.647101\pi$
$384$	0	0
$385$	−21.2154	−1.08124
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	0.763649	0.0387185	0.0193593	−	0.999813i	$-0.493837\pi$
$389$	0.763649	0.0387185	0.0193593	+	0.999813i	$0.493837\pi$
$390$	0	0
$391$	45.1999	2.28586
$392$	0	0
$393$	6.04508	0.304934
$394$	0	0
$395$	16.4691	0.828650
$396$	0	0
$397$	33.8451	1.69864	0.849318	−	0.527882i	$-0.177014\pi$
$397$	33.8451	1.69864	0.849318	+	0.527882i	$0.177014\pi$
$398$	0	0
$399$	8.65775	0.433429
$400$	0	0
$401$	−32.0451	−1.60026	−0.800128	−	0.599830i	$-0.795235\pi$
$401$	−32.0451	−1.60026	−0.800128	+	0.599830i	$0.795235\pi$
$402$	0	0
$403$	−17.2364	−0.858604
$404$	0	0
$405$	−20.8849	−1.03778
$406$	0	0
$407$	3.29881	0.163516
$408$	0	0
$409$	27.2109	1.34549	0.672747	−	0.739873i	$-0.265114\pi$
$409$	27.2109	1.34549	0.672747	+	0.739873i	$0.265114\pi$
$410$	0	0
$411$	16.3597	0.806963
$412$	0	0
$413$	28.4565	1.40025
$414$	0	0
$415$	−15.9545	−0.783174
$416$	0	0
$417$	31.7788	1.55622
$418$	0	0
$419$	−8.65061	−0.422610	−0.211305	−	0.977420i	$-0.567771\pi$
$419$	−8.65061	−0.422610	−0.211305	+	0.977420i	$0.567771\pi$
$420$	0	0
$421$	−14.8729	−0.724861	−0.362430	−	0.932011i	$-0.618053\pi$
$421$	−14.8729	−0.724861	−0.362430	+	0.932011i	$0.618053\pi$
$422$	0	0
$423$	−3.44246	−0.167378
$424$	0	0
$425$	−5.91846	−0.287088
$426$	0	0
$427$	−73.1263	−3.53883
$428$	0	0
$429$	13.6053	0.656868
$430$	0	0
$431$	3.60441	0.173618	0.0868092	−	0.996225i	$-0.472333\pi$
$431$	3.60441	0.173618	0.0868092	+	0.996225i	$0.472333\pi$
$432$	0	0
$433$	22.3165	1.07246	0.536231	−	0.844072i	$-0.319848\pi$
$433$	22.3165	1.07246	0.536231	+	0.844072i	$0.319848\pi$
$434$	0	0
$435$	6.05405	0.290269
$436$	0	0
$437$	7.08047	0.338705
$438$	0	0
$439$	37.6145	1.79524	0.897620	−	0.440770i	$-0.145295\pi$
$439$	37.6145	1.79524	0.897620	+	0.440770i	$0.145295\pi$
$440$	0	0
$441$	10.3523	0.492965
$442$	0	0
$443$	−14.8525	−0.705665	−0.352833	−	0.935686i	$-0.614781\pi$
$443$	−14.8525	−0.705665	−0.352833	+	0.935686i	$0.614781\pi$
$444$	0	0
$445$	−16.8273	−0.797689
$446$	0	0
$447$	−2.24963	−0.106404
$448$	0	0
$449$	−17.5558	−0.828511	−0.414256	−	0.910161i	$-0.635958\pi$
$449$	−17.5558	−0.828511	−0.414256	+	0.910161i	$0.635958\pi$
$450$	0	0
$451$	−20.9401	−0.986030
$452$	0	0
$453$	15.9993	0.751711
$454$	0	0
$455$	32.7308	1.53444
$456$	0	0
$457$	−25.3798	−1.18722	−0.593608	−	0.804754i	$-0.702297\pi$
$457$	−25.3798	−1.18722	−0.593608	+	0.804754i	$0.702297\pi$
$458$	0	0
$459$	26.9405	1.25748
$460$	0	0
$461$	15.0213	0.699614	0.349807	−	0.936822i	$-0.386247\pi$
$461$	15.0213	0.699614	0.349807	+	0.936822i	$0.386247\pi$
$462$	0	0
$463$	−26.6335	−1.23776	−0.618881	−	0.785485i	$-0.712414\pi$
$463$	−26.6335	−1.23776	−0.618881	+	0.785485i	$0.712414\pi$
$464$	0	0
$465$	−19.6707	−0.912208
$466$	0	0
$467$	−5.42230	−0.250914	−0.125457	−	0.992099i	$-0.540040\pi$
$467$	−5.42230	−0.250914	−0.125457	+	0.992099i	$0.540040\pi$
$468$	0	0
$469$	52.0361	2.40280
$470$	0	0
$471$	43.6200	2.00990
$472$	0	0
$473$	−14.3385	−0.659284
$474$	0	0
$475$	−0.927114	−0.0425389
$476$	0	0
$477$	−5.55918	−0.254537
$478$	0	0
$479$	−14.3701	−0.656588	−0.328294	−	0.944576i	$-0.606474\pi$
$479$	−14.3701	−0.656588	−0.328294	+	0.944576i	$0.606474\pi$
$480$	0	0
$481$	−5.08936	−0.232055
$482$	0	0
$483$	71.3182	3.24509
$484$	0	0
$485$	10.6822	0.485055
$486$	0	0
$487$	9.60533	0.435259	0.217630	−	0.976031i	$-0.430168\pi$
$487$	9.60533	0.435259	0.217630	+	0.976031i	$0.430168\pi$
$488$	0	0
$489$	−9.79734	−0.443051
$490$	0	0
$491$	3.60036	0.162482	0.0812409	−	0.996694i	$-0.474112\pi$
$491$	3.60036	0.162482	0.0812409	+	0.996694i	$0.474112\pi$
$492$	0	0
$493$	−9.44041	−0.425175
$494$	0	0
$495$	2.59272	0.116534
$496$	0	0
$497$	27.1339	1.21712
$498$	0	0
$499$	32.5772	1.45836	0.729178	−	0.684325i	$-0.239903\pi$
$499$	32.5772	1.45836	0.729178	+	0.684325i	$0.239903\pi$
$500$	0	0
$501$	−1.89773	−0.0847842
$502$	0	0
$503$	−5.05070	−0.225199	−0.112600	−	0.993640i	$-0.535918\pi$
$503$	−5.05070	−0.225199	−0.112600	+	0.993640i	$0.535918\pi$
$504$	0	0
$505$	19.5202	0.868638
$506$	0	0
$507$	3.68049	0.163456
$508$	0	0
$509$	10.9284	0.484393	0.242197	−	0.970227i	$-0.422132\pi$
$509$	10.9284	0.484393	0.242197	+	0.970227i	$0.422132\pi$
$510$	0	0
$511$	66.7045	2.95084
$512$	0	0
$513$	4.22018	0.186325
$514$	0	0
$515$	−18.7727	−0.827225
$516$	0	0
$517$	−12.3399	−0.542709
$518$	0	0
$519$	0.896453	0.0393499
$520$	0	0
$521$	39.4092	1.72655	0.863274	−	0.504736i	$-0.168410\pi$
$521$	39.4092	1.72655	0.863274	+	0.504736i	$0.168410\pi$
$522$	0	0
$523$	38.5511	1.68572	0.842862	−	0.538130i	$-0.180869\pi$
$523$	38.5511	1.68572	0.842862	+	0.538130i	$0.180869\pi$
$524$	0	0
$525$	−9.33838	−0.407560
$526$	0	0
$527$	30.6736	1.33616
$528$	0	0
$529$	35.3253	1.53588
$530$	0	0
$531$	−3.47764	−0.150917
$532$	0	0
$533$	32.3060	1.39933
$534$	0	0
$535$	21.8322	0.943888
$536$	0	0
$537$	−45.9275	−1.98192
$538$	0	0
$539$	37.1090	1.59840
$540$	0	0
$541$	17.8850	0.768935	0.384467	−	0.923139i	$-0.374385\pi$
$541$	17.8850	0.768935	0.384467	+	0.923139i	$0.374385\pi$
$542$	0	0
$543$	−15.6617	−0.672107
$544$	0	0
$545$	19.1457	0.820113
$546$	0	0
$547$	6.73915	0.288145	0.144073	−	0.989567i	$-0.453980\pi$
$547$	6.73915	0.288145	0.144073	+	0.989567i	$0.453980\pi$
$548$	0	0
$549$	8.93669	0.381409
$550$	0	0
$551$	−1.47882	−0.0629999
$552$	0	0
$553$	−40.5207	−1.72312
$554$	0	0
$555$	−5.80815	−0.246542
$556$	0	0
$557$	−11.2470	−0.476550	−0.238275	−	0.971198i	$-0.576582\pi$
$557$	−11.2470	−0.476550	−0.238275	+	0.971198i	$0.576582\pi$
$558$	0	0
$559$	22.1212	0.935625
$560$	0	0
$561$	−24.2118	−1.02222
$562$	0	0
$563$	28.1919	1.18815	0.594073	−	0.804411i	$-0.297519\pi$
$563$	28.1919	1.18815	0.594073	+	0.804411i	$0.297519\pi$
$564$	0	0
$565$	−17.6869	−0.744094
$566$	0	0
$567$	51.3855	2.15799
$568$	0	0
$569$	−15.2335	−0.638620	−0.319310	−	0.947650i	$-0.603451\pi$
$569$	−15.2335	−0.638620	−0.319310	+	0.947650i	$0.603451\pi$
$570$	0	0
$571$	9.42060	0.394240	0.197120	−	0.980379i	$-0.436841\pi$
$571$	9.42060	0.394240	0.197120	+	0.980379i	$0.436841\pi$
$572$	0	0
$573$	37.3483	1.56025
$574$	0	0
$575$	−7.63710	−0.318489
$576$	0	0
$577$	−14.3375	−0.596879	−0.298439	−	0.954429i	$-0.596466\pi$
$577$	−14.3375	−0.596879	−0.298439	+	0.954429i	$0.596466\pi$
$578$	0	0
$579$	−1.86146	−0.0773596
$580$	0	0
$581$	39.2546	1.62855
$582$	0	0
$583$	−19.9275	−0.825314
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−27.1224	−1.11946	−0.559731	−	0.828674i	$-0.689096\pi$
$587$	−27.1224	−1.11946	−0.559731	+	0.828674i	$0.689096\pi$
$588$	0	0
$589$	4.80496	0.197985
$590$	0	0
$591$	30.4355	1.25195
$592$	0	0
$593$	−6.05331	−0.248580	−0.124290	−	0.992246i	$-0.539665\pi$
$593$	−6.05331	−0.248580	−0.124290	+	0.992246i	$0.539665\pi$
$594$	0	0
$595$	−58.2474	−2.38791
$596$	0	0
$597$	35.3369	1.44624
$598$	0	0
$599$	−35.9024	−1.46693	−0.733466	−	0.679726i	$-0.762098\pi$
$599$	−35.9024	−1.46693	−0.733466	+	0.679726i	$0.762098\pi$
$600$	0	0
$601$	19.1106	0.779537	0.389768	−	0.920913i	$-0.372555\pi$
$601$	19.1106	0.779537	0.389768	+	0.920913i	$0.372555\pi$
$602$	0	0
$603$	−6.35928	−0.258970
$604$	0	0
$605$	−12.7061	−0.516576
$606$	0	0
$607$	20.1464	0.817719	0.408859	−	0.912597i	$-0.365927\pi$
$607$	20.1464	0.817719	0.408859	+	0.912597i	$0.365927\pi$
$608$	0	0
$609$	−14.8955	−0.603594
$610$	0	0
$611$	19.0378	0.770188
$612$	0	0
$613$	−35.5775	−1.43696	−0.718481	−	0.695547i	$-0.755162\pi$
$613$	−35.5775	−1.43696	−0.718481	+	0.695547i	$0.755162\pi$
$614$	0	0
$615$	36.8687	1.48669
$616$	0	0
$617$	45.6697	1.83859	0.919296	−	0.393567i	$-0.128759\pi$
$617$	45.6697	1.83859	0.919296	+	0.393567i	$0.128759\pi$
$618$	0	0
$619$	45.1754	1.81575	0.907876	−	0.419240i	$-0.137703\pi$
$619$	45.1754	1.81575	0.907876	+	0.419240i	$0.137703\pi$
$620$	0	0
$621$	34.7637	1.39502
$622$	0	0
$623$	41.4020	1.65874
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−3.79272	−0.151467
$628$	0	0
$629$	9.05697	0.361125
$630$	0	0
$631$	−6.77591	−0.269745	−0.134872	−	0.990863i	$-0.543062\pi$
$631$	−6.77591	−0.269745	−0.134872	+	0.990863i	$0.543062\pi$
$632$	0	0
$633$	19.4425	0.772768
$634$	0	0
$635$	19.0552	0.756184
$636$	0	0
$637$	−57.2511	−2.26837
$638$	0	0
$639$	−3.31601	−0.131179
$640$	0	0
$641$	−25.7595	−1.01744	−0.508719	−	0.860932i	$-0.669881\pi$
$641$	−25.7595	−1.01744	−0.508719	+	0.860932i	$0.669881\pi$
$642$	0	0
$643$	−50.3013	−1.98369	−0.991845	−	0.127453i	$-0.959320\pi$
$643$	−50.3013	−1.98369	−0.991845	+	0.127453i	$0.959320\pi$
$644$	0	0
$645$	25.2454	0.994038
$646$	0	0
$647$	−30.0782	−1.18250	−0.591248	−	0.806490i	$-0.701365\pi$
$647$	−30.0782	−1.18250	−0.591248	+	0.806490i	$0.701365\pi$
$648$	0	0
$649$	−12.4660	−0.489334
$650$	0	0
$651$	48.3981	1.89687
$652$	0	0
$653$	−7.05608	−0.276126	−0.138063	−	0.990423i	$-0.544088\pi$
$653$	−7.05608	−0.276126	−0.138063	+	0.990423i	$0.544088\pi$
$654$	0	0
$655$	−6.37087	−0.248930
$656$	0	0
$657$	−8.15190	−0.318036
$658$	0	0
$659$	−13.4326	−0.523258	−0.261629	−	0.965168i	$-0.584260\pi$
$659$	−13.4326	−0.523258	−0.261629	+	0.965168i	$0.584260\pi$
$660$	0	0
$661$	28.6833	1.11565	0.557826	−	0.829958i	$-0.311636\pi$
$661$	28.6833	1.11565	0.557826	+	0.829958i	$0.311636\pi$
$662$	0	0
$663$	37.3535	1.45069
$664$	0	0
$665$	−9.12433	−0.353826
$666$	0	0
$667$	−12.1818	−0.471680
$668$	0	0
$669$	3.89291	0.150509
$670$	0	0
$671$	32.0346	1.23668
$672$	0	0
$673$	20.6474	0.795900	0.397950	−	0.917407i	$-0.369722\pi$
$673$	20.6474	0.795900	0.397950	+	0.917407i	$0.369722\pi$
$674$	0	0
$675$	−4.55195	−0.175205
$676$	0	0
$677$	−6.86873	−0.263987	−0.131993	−	0.991251i	$-0.542138\pi$
$677$	−6.86873	−0.263987	−0.131993	+	0.991251i	$0.542138\pi$
$678$	0	0
$679$	−26.2827	−1.00864
$680$	0	0
$681$	−4.05626	−0.155436
$682$	0	0
$683$	−25.5166	−0.976366	−0.488183	−	0.872741i	$-0.662340\pi$
$683$	−25.5166	−0.976366	−0.488183	+	0.872741i	$0.662340\pi$
$684$	0	0
$685$	−17.2413	−0.658757
$686$	0	0
$687$	−17.4325	−0.665091
$688$	0	0
$689$	30.7439	1.17125
$690$	0	0
$691$	−34.6547	−1.31833	−0.659164	−	0.751999i	$-0.729090\pi$
$691$	−34.6547	−1.31833	−0.659164	+	0.751999i	$0.729090\pi$
$692$	0	0
$693$	−6.37915	−0.242324
$694$	0	0
$695$	−33.4914	−1.27040
$696$	0	0
$697$	−57.4914	−2.17764
$698$	0	0
$699$	−31.4642	−1.19008
$700$	0	0
$701$	−11.5408	−0.435892	−0.217946	−	0.975961i	$-0.569936\pi$
$701$	−11.5408	−0.435892	−0.217946	+	0.975961i	$0.569936\pi$
$702$	0	0
$703$	1.41875	0.0535093
$704$	0	0
$705$	21.7266	0.818271
$706$	0	0
$707$	−48.0278	−1.80627
$708$	0	0
$709$	13.5022	0.507085	0.253543	−	0.967324i	$-0.418404\pi$
$709$	13.5022	0.507085	0.253543	+	0.967324i	$0.418404\pi$
$710$	0	0
$711$	4.95200	0.185714
$712$	0	0
$713$	39.5808	1.48231
$714$	0	0
$715$	−14.3385	−0.536229
$716$	0	0
$717$	17.3271	0.647094
$718$	0	0
$719$	45.3481	1.69120	0.845599	−	0.533819i	$-0.179244\pi$
$719$	45.3481	1.69120	0.845599	+	0.533819i	$0.179244\pi$
$720$	0	0
$721$	46.1886	1.72016
$722$	0	0
$723$	32.2012	1.19758
$724$	0	0
$725$	1.59508	0.0592397
$726$	0	0
$727$	4.92211	0.182551	0.0912756	−	0.995826i	$-0.470906\pi$
$727$	4.92211	0.182551	0.0912756	+	0.995826i	$0.470906\pi$
$728$	0	0
$729$	19.6353	0.727233
$730$	0	0
$731$	−39.3666	−1.45603
$732$	0	0
$733$	−17.4838	−0.645780	−0.322890	−	0.946437i	$-0.604654\pi$
$733$	−17.4838	−0.645780	−0.322890	+	0.946437i	$0.604654\pi$
$734$	0	0
$735$	−65.3369	−2.40999
$736$	0	0
$737$	−22.7956	−0.839687
$738$	0	0
$739$	27.0498	0.995042	0.497521	−	0.867452i	$-0.334244\pi$
$739$	27.0498	0.995042	0.497521	+	0.867452i	$0.334244\pi$
$740$	0	0
$741$	5.85135	0.214955
$742$	0	0
$743$	35.6126	1.30650	0.653250	−	0.757142i	$-0.273405\pi$
$743$	35.6126	1.30650	0.653250	+	0.757142i	$0.273405\pi$
$744$	0	0
$745$	2.37087	0.0868618
$746$	0	0
$747$	−4.79726	−0.175523
$748$	0	0
$749$	−53.7162	−1.96275
$750$	0	0
$751$	22.4268	0.818364	0.409182	−	0.912453i	$-0.365814\pi$
$751$	22.4268	0.818364	0.409182	+	0.912453i	$0.365814\pi$
$752$	0	0
$753$	−10.3606	−0.377562
$754$	0	0
$755$	−16.8615	−0.613653
$756$	0	0
$757$	−16.4941	−0.599488	−0.299744	−	0.954020i	$-0.596901\pi$
$757$	−16.4941	−0.599488	−0.299744	+	0.954020i	$0.596901\pi$
$758$	0	0
$759$	−31.2426	−1.13403
$760$	0	0
$761$	−12.9669	−0.470050	−0.235025	−	0.971989i	$-0.575517\pi$
$761$	−12.9669	−0.470050	−0.235025	+	0.971989i	$0.575517\pi$
$762$	0	0
$763$	−47.1064	−1.70537
$764$	0	0
$765$	7.11836	0.257365
$766$	0	0
$767$	19.2324	0.694441
$768$	0	0
$769$	−36.5712	−1.31879	−0.659395	−	0.751797i	$-0.729188\pi$
$769$	−36.5712	−1.31879	−0.659395	+	0.751797i	$0.729188\pi$
$770$	0	0
$771$	−10.9332	−0.393750
$772$	0	0
$773$	−33.0199	−1.18764	−0.593821	−	0.804597i	$-0.702381\pi$
$773$	−33.0199	−1.18764	−0.593821	+	0.804597i	$0.702381\pi$
$774$	0	0
$775$	−5.18270	−0.186168
$776$	0	0
$777$	14.2904	0.512667
$778$	0	0
$779$	−9.00591	−0.322670
$780$	0	0
$781$	−11.8866	−0.425337
$782$	0	0
$783$	−7.26072	−0.259477
$784$	0	0
$785$	−45.9708	−1.64077
$786$	0	0
$787$	36.9726	1.31793	0.658965	−	0.752173i	$-0.270994\pi$
$787$	36.9726	1.31793	0.658965	+	0.752173i	$0.270994\pi$
$788$	0	0
$789$	1.20736	0.0429833
$790$	0	0
$791$	43.5171	1.54729
$792$	0	0
$793$	−49.4225	−1.75504
$794$	0	0
$795$	35.0860	1.24437
$796$	0	0
$797$	−5.29826	−0.187674	−0.0938369	−	0.995588i	$-0.529913\pi$
$797$	−5.29826	−0.187674	−0.0938369	+	0.995588i	$0.529913\pi$
$798$	0	0
$799$	−33.8795	−1.19857
$800$	0	0
$801$	−5.05970	−0.178776
$802$	0	0
$803$	−29.2214	−1.03120
$804$	0	0
$805$	−75.1617	−2.64910
$806$	0	0
$807$	−59.9626	−2.11078
$808$	0	0
$809$	−42.7600	−1.50336	−0.751680	−	0.659527i	$-0.770756\pi$
$809$	−42.7600	−1.50336	−0.751680	+	0.659527i	$0.770756\pi$
$810$	0	0
$811$	26.1404	0.917914	0.458957	−	0.888458i	$-0.348223\pi$
$811$	26.1404	0.917914	0.458957	+	0.888458i	$0.348223\pi$
$812$	0	0
$813$	−1.23635	−0.0433607
$814$	0	0
$815$	10.3253	0.361681
$816$	0	0
$817$	−6.16669	−0.215745
$818$	0	0
$819$	9.84165	0.343895
$820$	0	0
$821$	−10.0450	−0.350575	−0.175287	−	0.984517i	$-0.556085\pi$
$821$	−10.0450	−0.350575	−0.175287	+	0.984517i	$0.556085\pi$
$822$	0	0
$823$	−0.690450	−0.0240676	−0.0120338	−	0.999928i	$-0.503831\pi$
$823$	−0.690450	−0.0240676	−0.0120338	+	0.999928i	$0.503831\pi$
$824$	0	0
$825$	4.09089	0.142427
$826$	0	0
$827$	−32.1255	−1.11711	−0.558557	−	0.829466i	$-0.688645\pi$
$827$	−32.1255	−1.11711	−0.558557	+	0.829466i	$0.688645\pi$
$828$	0	0
$829$	1.09009	0.0378605	0.0189302	−	0.999821i	$-0.493974\pi$
$829$	1.09009	0.0378605	0.0189302	+	0.999821i	$0.493974\pi$
$830$	0	0
$831$	−36.7230	−1.27391
$832$	0	0
$833$	101.883	3.53005
$834$	0	0
$835$	2.00000	0.0692129
$836$	0	0
$837$	23.5914	0.815438
$838$	0	0
$839$	23.0574	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$839$	23.0574	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$840$	0	0
$841$	−26.4557	−0.912266
$842$	0	0
$843$	48.4149	1.66750
$844$	0	0
$845$	−3.87884	−0.133436
$846$	0	0
$847$	31.2622	1.07418
$848$	0	0
$849$	22.0549	0.756922
$850$	0	0
$851$	11.6870	0.400625
$852$	0	0
$853$	−16.0903	−0.550923	−0.275461	−	0.961312i	$-0.588831\pi$
$853$	−16.0903	−0.550923	−0.275461	+	0.961312i	$0.588831\pi$
$854$	0	0
$855$	1.11508	0.0381348
$856$	0	0
$857$	20.1916	0.689733	0.344866	−	0.938652i	$-0.387924\pi$
$857$	20.1916	0.689733	0.344866	+	0.938652i	$0.387924\pi$
$858$	0	0
$859$	−48.7326	−1.66273	−0.831367	−	0.555723i	$-0.812441\pi$
$859$	−48.7326	−1.66273	−0.831367	+	0.555723i	$0.812441\pi$
$860$	0	0
$861$	−90.7123	−3.09147
$862$	0	0
$863$	39.3807	1.34054	0.670268	−	0.742119i	$-0.266179\pi$
$863$	39.3807	1.34054	0.670268	+	0.742119i	$0.266179\pi$
$864$	0	0
$865$	−0.944765	−0.0321230
$866$	0	0
$867$	−34.2126	−1.16192
$868$	0	0
$869$	17.7510	0.602162
$870$	0	0
$871$	35.1687	1.19165
$872$	0	0
$873$	3.21198	0.108709
$874$	0	0
$875$	59.0499	1.99625
$876$	0	0
$877$	17.0862	0.576960	0.288480	−	0.957486i	$-0.406850\pi$
$877$	17.0862	0.576960	0.288480	+	0.957486i	$0.406850\pi$
$878$	0	0
$879$	25.9655	0.875794
$880$	0	0
$881$	41.0783	1.38396	0.691981	−	0.721916i	$-0.256738\pi$
$881$	41.0783	1.38396	0.691981	+	0.721916i	$0.256738\pi$
$882$	0	0
$883$	−44.5706	−1.49992	−0.749960	−	0.661483i	$-0.769927\pi$
$883$	−44.5706	−1.49992	−0.749960	+	0.661483i	$0.769927\pi$
$884$	0	0
$885$	21.9486	0.737795
$886$	0	0
$887$	−53.7165	−1.80362	−0.901812	−	0.432128i	$-0.857763\pi$
$887$	−53.7165	−1.80362	−0.901812	+	0.432128i	$0.857763\pi$
$888$	0	0
$889$	−46.8837	−1.57243
$890$	0	0
$891$	−22.5106	−0.754133
$892$	0	0
$893$	−5.30715	−0.177597
$894$	0	0
$895$	48.4026	1.61792
$896$	0	0
$897$	48.2005	1.60937
$898$	0	0
$899$	−8.26682	−0.275714
$900$	0	0
$901$	−54.7115	−1.82270
$902$	0	0
$903$	−62.1142	−2.06703
$904$	0	0
$905$	16.5057	0.548669
$906$	0	0
$907$	21.9320	0.728239	0.364120	−	0.931352i	$-0.381370\pi$
$907$	21.9320	0.728239	0.364120	+	0.931352i	$0.381370\pi$
$908$	0	0
$909$	5.86943	0.194677
$910$	0	0
$911$	−52.9321	−1.75372	−0.876860	−	0.480746i	$-0.840366\pi$
$911$	−52.9321	−1.75372	−0.876860	+	0.480746i	$0.840366\pi$
$912$	0	0
$913$	−17.1964	−0.569116
$914$	0	0
$915$	−56.4026	−1.86461
$916$	0	0
$917$	15.6750	0.517633
$918$	0	0
$919$	33.5195	1.10570	0.552852	−	0.833279i	$-0.313539\pi$
$919$	33.5195	1.10570	0.552852	+	0.833279i	$0.313539\pi$
$920$	0	0
$921$	36.2471	1.19438
$922$	0	0
$923$	18.3385	0.603618
$924$	0	0
$925$	−1.53029	−0.0503156
$926$	0	0
$927$	−5.64467	−0.185395
$928$	0	0
$929$	−42.7611	−1.40295	−0.701473	−	0.712696i	$-0.747474\pi$
$929$	−42.7611	−1.40295	−0.701473	+	0.712696i	$0.747474\pi$
$930$	0	0
$931$	15.9598	0.523062
$932$	0	0
$933$	−43.8766	−1.43645
$934$	0	0
$935$	25.5166	0.834482
$936$	0	0
$937$	−28.5004	−0.931067	−0.465533	−	0.885030i	$-0.654137\pi$
$937$	−28.5004	−0.931067	−0.465533	+	0.885030i	$0.654137\pi$
$938$	0	0
$939$	10.6942	0.348991
$940$	0	0
$941$	26.2339	0.855201	0.427601	−	0.903968i	$-0.359359\pi$
$941$	26.2339	0.855201	0.427601	+	0.903968i	$0.359359\pi$
$942$	0	0
$943$	−74.1862	−2.41583
$944$	0	0
$945$	−44.7987	−1.45730
$946$	0	0
$947$	1.11732	0.0363082	0.0181541	−	0.999835i	$-0.494221\pi$
$947$	1.11732	0.0363082	0.0181541	+	0.999835i	$0.494221\pi$
$948$	0	0
$949$	45.0824	1.46344
$950$	0	0
$951$	17.9249	0.581253
$952$	0	0
$953$	−21.6413	−0.701032	−0.350516	−	0.936557i	$-0.613994\pi$
$953$	−21.6413	−0.701032	−0.350516	+	0.936557i	$0.613994\pi$
$954$	0	0
$955$	−39.3611	−1.27370
$956$	0	0
$957$	6.52529	0.210933
$958$	0	0
$959$	42.4208	1.36984
$960$	0	0
$961$	−4.13958	−0.133535
$962$	0	0
$963$	6.56460	0.211541
$964$	0	0
$965$	1.96178	0.0631518
$966$	0	0
$967$	15.0526	0.484060	0.242030	−	0.970269i	$-0.422187\pi$
$967$	15.0526	0.484060	0.242030	+	0.970269i	$0.422187\pi$
$968$	0	0
$969$	−10.4130	−0.334514
$970$	0	0
$971$	−8.32460	−0.267149	−0.133575	−	0.991039i	$-0.542646\pi$
$971$	−8.32460	−0.267149	−0.133575	+	0.991039i	$0.542646\pi$
$972$	0	0
$973$	82.4028	2.64171
$974$	0	0
$975$	−6.31136	−0.202125
$976$	0	0
$977$	45.9701	1.47071	0.735357	−	0.677680i	$-0.237014\pi$
$977$	45.9701	1.47071	0.735357	+	0.677680i	$0.237014\pi$
$978$	0	0
$979$	−18.1371	−0.579664
$980$	0	0
$981$	5.75683	0.183801
$982$	0	0
$983$	−40.8484	−1.30286	−0.651430	−	0.758709i	$-0.725831\pi$
$983$	−40.8484	−1.30286	−0.651430	+	0.758709i	$0.725831\pi$
$984$	0	0
$985$	−32.0757	−1.02202
$986$	0	0
$987$	−53.4564	−1.70154
$988$	0	0
$989$	−50.7981	−1.61529
$990$	0	0
$991$	−28.1825	−0.895247	−0.447624	−	0.894222i	$-0.647730\pi$
$991$	−28.1825	−0.895247	−0.447624	+	0.894222i	$0.647730\pi$
$992$	0	0
$993$	24.2083	0.768226
$994$	0	0
$995$	−37.2413	−1.18063
$996$	0	0
$997$	−18.9832	−0.601205	−0.300602	−	0.953750i	$-0.597188\pi$
$997$	−18.9832	−0.601205	−0.300602	+	0.953750i	$0.597188\pi$
$998$	0	0
$999$	6.96580	0.220388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2672.2.a.j.1.2		5
4.3	odd	2		668.2.a.b.1.4	✓	5
12.11	even	2		6012.2.a.d.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.b.1.4	✓	5	4.3	odd	2
2672.2.a.j.1.2		5	1.1	even	1	trivial
6012.2.a.d.1.5		5	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)

\(f(q)\)	\(=\)	\(q-1.89773 q^{3} +2.00000 q^{5} -4.92082 q^{7} +0.601369 q^{9} +O(q^{10})\)	\(q-1.89773 q^{3} +2.00000 q^{5} -4.92082 q^{7} +0.601369 q^{9} +2.15568 q^{11} -3.32575 q^{13} -3.79545 q^{15} +5.91846 q^{17} +0.927114 q^{19} +9.33838 q^{21} +7.63710 q^{23} -1.00000 q^{25} +4.55195 q^{27} -1.59508 q^{29} +5.18270 q^{31} -4.09089 q^{33} -9.84165 q^{35} +1.53029 q^{37} +6.31136 q^{39} -9.71392 q^{41} -6.65149 q^{43} +1.20274 q^{45} -5.72438 q^{47} +17.2145 q^{49} -11.2316 q^{51} -9.24421 q^{53} +4.31136 q^{55} -1.75941 q^{57} -5.78287 q^{59} +14.8606 q^{61} -2.95923 q^{63} -6.65149 q^{65} -10.5747 q^{67} -14.4931 q^{69} -5.51410 q^{71} -13.5556 q^{73} +1.89773 q^{75} -10.6077 q^{77} +8.23454 q^{79} -10.4425 q^{81} -7.97724 q^{83} +11.8369 q^{85} +3.02702 q^{87} -8.41363 q^{89} +16.3654 q^{91} -9.83536 q^{93} +1.85423 q^{95} +5.34111 q^{97} +1.29636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9	\( 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9} - 5 q^{11} - 4 q^{13} - 6 q^{15} - 2 q^{17} - 5 q^{19} - 4 q^{21} - 6 q^{23} - 5 q^{25} - 12 q^{27} - 5 q^{29} - 9 q^{31} - 8 q^{33} - 18 q^{35} + 8 q^{37} - 4 q^{41} - 8 q^{43} + 12 q^{45} - 13 q^{47} + 14 q^{49} - 2 q^{53} - 10 q^{55} - 12 q^{57} - 4 q^{59} + 11 q^{61} + q^{63} - 8 q^{65} - 28 q^{67} - 16 q^{69} - 2 q^{71} + 8 q^{73} + 3 q^{75} - 12 q^{77} + 10 q^{79} - 15 q^{81} - 2 q^{83} - 4 q^{85} - 4 q^{87} - 17 q^{89} + 12 q^{91} - 12 q^{93} - 10 q^{95} - 27 q^{97} - 3 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9 - 5 * q^11 - 4 * q^13 - 6 * q^15 - 2 * q^17 - 5 * q^19 - 4 * q^21 - 6 * q^23 - 5 * q^25 - 12 * q^27 - 5 * q^29 - 9 * q^31 - 8 * q^33 - 18 * q^35 + 8 * q^37 - 4 * q^41 - 8 * q^43 + 12 * q^45 - 13 * q^47 + 14 * q^49 - 2 * q^53 - 10 * q^55 - 12 * q^57 - 4 * q^59 + 11 * q^61 + q^63 - 8 * q^65 - 28 * q^67 - 16 * q^69 - 2 * q^71 + 8 * q^73 + 3 * q^75 - 12 * q^77 + 10 * q^79 - 15 * q^81 - 2 * q^83 - 4 * q^85 - 4 * q^87 - 17 * q^89 + 12 * q^91 - 12 * q^93 - 10 * q^95 - 27 * q^97 - 3 * q^99

Introduction

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