LMFDB - Embedded newform 2052.3.be.a.125.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(125,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2052.125");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.be (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.14
Character	$\chi$	$=$	2052.125
Dual form			2052.3.be.a.197.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.53393 + 1.46297i) q^{5} +(-2.71227 - 4.69779i) q^{7} +O(q^{10})$	$q+(-2.53393 + 1.46297i) q^{5} +(-2.71227 - 4.69779i) q^{7} +(-9.00863 + 5.20114i) q^{11} -24.4777 q^{13} +(-19.6840 - 11.3645i) q^{17} +(-5.11536 + 18.2984i) q^{19} -29.3354i q^{23} +(-8.21946 + 14.2365i) q^{25} +(26.6384 + 15.3797i) q^{29} +(-8.67884 + 15.0322i) q^{31} +(13.7454 + 7.93593i) q^{35} +47.8168 q^{37} +(66.1461 - 38.1895i) q^{41} -6.25697 q^{43} +(15.3230 + 8.84672i) q^{47} +(9.78715 - 16.9518i) q^{49} +(12.9368 - 7.46908i) q^{53} +(15.2182 - 26.3586i) q^{55} +(18.8486 - 10.8822i) q^{59} +(-56.0166 + 97.0236i) q^{61} +(62.0248 - 35.8100i) q^{65} +21.1644 q^{67} +(-43.5246 - 25.1289i) q^{71} +(-6.91001 + 11.9685i) q^{73} +(48.8677 + 28.2138i) q^{77} +1.30487 q^{79} +(-66.2341 + 38.2403i) q^{83} +66.5038 q^{85} +(123.051 - 71.0438i) q^{89} +(66.3902 + 114.991i) q^{91} +(-13.8080 - 53.8506i) q^{95} -184.984 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−2.53393	+	1.46297i	−0.506786	+	0.292593i	−0.731512	−	0.681829i	$-0.761185\pi$
$5$	−2.53393	+	1.46297i	−0.506786	+	0.292593i	0.224725	+	0.974422i	$0.427851\pi$
$6$		0			0
$7$	−2.71227	−	4.69779i	−0.387468	−	0.671114i	0.604641	−	0.796498i	$-0.293317\pi$
$7$	−2.71227	−	4.69779i	−0.387468	−	0.671114i	−0.992108	+	0.125385i	$0.959983\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.00863	+	5.20114i	−0.818966	+	0.472830i	−0.850060	−	0.526686i	$-0.823434\pi$
$11$	−9.00863	+	5.20114i	−0.818966	+	0.472830i	0.0310935	+	0.999516i	$0.490101\pi$
$12$		0			0
$13$	−24.4777			−1.88290			−0.941450	−	0.337153i	$-0.890536\pi$
$13$	−24.4777			−1.88290			−0.941450	+	0.337153i	$0.890536\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.6840	−	11.3645i	−1.15788	−	0.668503i	−0.207086	−	0.978323i	$-0.566398\pi$
$17$	−19.6840	−	11.3645i	−1.15788	−	0.668503i	−0.950795	+	0.309820i	$0.899731\pi$
$18$		0			0
$19$	−5.11536	+	18.2984i	−0.269230	+	0.963076i
$19$	−5.11536	+	18.2984i	−0.269230	+	0.963076i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	29.3354i		−	1.27545i	−0.770263	−	0.637727i	$-0.779875\pi$
$23$		−	29.3354i		−	1.27545i	0.770263	−	0.637727i	$-0.220125\pi$
$24$		0			0
$25$	−8.21946	+	14.2365i	−0.328778	+	0.569461i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	26.6384	+	15.3797i	0.918566	+	0.530334i	0.883177	−	0.469040i	$-0.155400\pi$
$29$	26.6384	+	15.3797i	0.918566	+	0.530334i	0.0353884	+	0.999374i	$0.488733\pi$
$30$		0			0
$31$	−8.67884	+	15.0322i	−0.279962	+	0.484909i	−0.971375	−	0.237551i	$-0.923655\pi$
$31$	−8.67884	+	15.0322i	−0.279962	+	0.484909i	0.691413	+	0.722460i	$0.256989\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	13.7454	+	7.93593i	0.392726	+	0.226741i
$36$		0			0
$37$	47.8168			1.29234			0.646172	−	0.763192i	$-0.276369\pi$
$37$	47.8168			1.29234			0.646172	+	0.763192i	$0.276369\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	66.1461	−	38.1895i	1.61332	−	0.931451i	0.624727	−	0.780843i	$-0.285210\pi$
$41$	66.1461	−	38.1895i	1.61332	−	0.931451i	0.988594	−	0.150608i	$-0.0481231\pi$
$42$		0			0
$43$	−6.25697			−0.145511			−0.0727554	−	0.997350i	$-0.523179\pi$
$43$	−6.25697			−0.145511			−0.0727554	+	0.997350i	$0.523179\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	15.3230	+	8.84672i	0.326021	+	0.188228i	0.654073	−	0.756431i	$-0.273059\pi$
$47$	15.3230	+	8.84672i	0.326021	+	0.188228i	−0.328052	+	0.944660i	$0.606392\pi$
$48$		0			0
$49$	9.78715	−	16.9518i	0.199738	−	0.345956i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.9368	−	7.46908i	0.244091	−	0.140926i	−0.372964	−	0.927846i	$-0.621659\pi$
$53$	12.9368	−	7.46908i	0.244091	−	0.140926i	0.617056	+	0.786920i	$0.288325\pi$
$54$		0			0
$55$	15.2182	−	26.3586i	0.276694	−	0.479248i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	18.8486	−	10.8822i	0.319468	−	0.184445i	−0.331688	−	0.943389i	$-0.607618\pi$
$59$	18.8486	−	10.8822i	0.319468	−	0.184445i	0.651155	+	0.758945i	$0.274285\pi$
$60$		0			0
$61$	−56.0166	+	97.0236i	−0.918305	+	1.59055i	−0.116316	+	0.993212i	$0.537109\pi$
$61$	−56.0166	+	97.0236i	−0.918305	+	1.59055i	−0.801989	+	0.597339i	$0.796225\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	62.0248	−	35.8100i	0.954228	−	0.550924i
$66$		0			0
$67$	21.1644			0.315887			0.157943	−	0.987448i	$-0.449514\pi$
$67$	21.1644			0.315887			0.157943	+	0.987448i	$0.449514\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−43.5246	−	25.1289i	−0.613023	−	0.353929i	0.161125	−	0.986934i	$-0.448488\pi$
$71$	−43.5246	−	25.1289i	−0.613023	−	0.353929i	−0.774147	+	0.633005i	$0.781821\pi$
$72$		0			0
$73$	−6.91001	+	11.9685i	−0.0946577	+	0.163952i	−0.909466	−	0.415779i	$-0.863509\pi$
$73$	−6.91001	+	11.9685i	−0.0946577	+	0.163952i	0.814808	+	0.579731i	$0.196842\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	48.8677	+	28.2138i	0.634646	+	0.366413i
$78$		0			0
$79$	1.30487			0.0165174			0.00825868	−	0.999966i	$-0.497371\pi$
$79$	1.30487			0.0165174			0.00825868	+	0.999966i	$0.497371\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−66.2341	+	38.2403i	−0.798002	+	0.460726i	−0.842772	−	0.538271i	$-0.819078\pi$
$83$	−66.2341	+	38.2403i	−0.798002	+	0.460726i	0.0447703	+	0.998997i	$0.485744\pi$
$84$		0			0
$85$	66.5038			0.782398
$86$		0			0
$87$		0			0
$88$		0			0
$89$	123.051	−	71.0438i	1.38260	−	0.798244i	0.390133	−	0.920758i	$-0.372429\pi$
$89$	123.051	−	71.0438i	1.38260	−	0.798244i	0.992467	+	0.122514i	$0.0390957\pi$
$90$		0			0
$91$	66.3902	+	114.991i	0.729563	+	1.26364i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.8080	−	53.8506i	−0.145348	−	0.566848i
$96$		0			0
$97$	−184.984			−1.90705			−0.953524	−	0.301317i	$-0.902574\pi$
$97$	−184.984			−1.90705			−0.953524	+	0.301317i	$0.902574\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	51.2365	+	29.5814i	0.507292	+	0.292885i	0.731720	−	0.681605i	$-0.238718\pi$
$101$	51.2365	+	29.5814i	0.507292	+	0.292885i	−0.224428	+	0.974491i	$0.572051\pi$
$102$		0			0
$103$	84.6622	−	146.639i	0.821963	−	1.42368i	−0.0822557	−	0.996611i	$-0.526212\pi$
$103$	84.6622	−	146.639i	0.821963	−	1.42368i	0.904218	−	0.427070i	$-0.140454\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	131.165i		−	1.22584i	−0.790144	−	0.612921i	$-0.789994\pi$
$107$		−	131.165i		−	1.22584i	0.790144	−	0.612921i	$-0.210006\pi$
$108$		0			0
$109$	−7.82050	+	13.5455i	−0.0717477	+	0.124271i	−0.899667	−	0.436576i	$-0.856191\pi$
$109$	−7.82050	+	13.5455i	−0.0717477	+	0.124271i	0.827920	+	0.560847i	$0.189524\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	20.4851	+	11.8271i	0.181284	+	0.104665i	0.587896	−	0.808937i	$-0.299956\pi$
$113$	20.4851	+	11.8271i	0.181284	+	0.104665i	−0.406612	+	0.913601i	$0.633290\pi$
$114$		0			0
$115$	42.9167	+	74.3339i	0.373189	+	0.646382i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			123.295i			1.03609i
$120$		0			0
$121$	−6.39638	+	11.0789i	−0.0528627	+	0.0915608i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	121.247i		−	0.969980i
$126$		0			0
$127$	−41.1013	−	71.1895i	−0.323632	−	0.560547i	0.657603	−	0.753365i	$-0.271571\pi$
$127$	−41.1013	−	71.1895i	−0.323632	−	0.560547i	−0.981235	+	0.192818i	$0.938237\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	68.7184	−	39.6746i	0.524568	−	0.302860i	−0.214233	−	0.976782i	$-0.568725\pi$
$131$	68.7184	−	39.6746i	0.524568	−	0.302860i	0.738802	+	0.673923i	$0.235392\pi$
$132$		0			0
$133$	99.8366	−	25.5995i	0.750651	−	0.192477i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	40.3121	+	23.2742i	0.294249	+	0.169885i	0.639856	−	0.768495i	$-0.278994\pi$
$137$	40.3121	+	23.2742i	0.294249	+	0.169885i	−0.345608	+	0.938379i	$0.612327\pi$
$138$		0			0
$139$	73.9559			0.532057			0.266028	−	0.963965i	$-0.414289\pi$
$139$	73.9559			0.532057			0.266028	+	0.963965i	$0.414289\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	220.511	−	127.312i	1.54203	−	0.890292i
$144$		0			0
$145$	−89.9998			−0.620689
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−170.929	+	98.6858i	−1.14717	+	0.662321i	−0.948197	−	0.317683i	$-0.897095\pi$
$149$	−170.929	+	98.6858i	−1.14717	+	0.662321i	−0.198976	+	0.980004i	$0.563762\pi$
$150$		0			0
$151$	104.491	+	180.983i	0.691992	+	1.19856i	0.971184	+	0.238330i	$0.0765998\pi$
$151$	104.491	+	180.983i	0.691992	+	1.19856i	−0.279193	+	0.960235i	$0.590067\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	50.7874i		−	0.327660i
$156$		0			0
$157$	21.7121	+	37.6065i	0.138294	+	0.239532i	0.926851	−	0.375430i	$-0.122505\pi$
$157$	21.7121	+	37.6065i	0.138294	+	0.239532i	−0.788557	+	0.614962i	$0.789171\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−137.812	+	79.5657i	−0.855974	+	0.494197i
$162$		0			0
$163$	268.887			1.64962			0.824808	−	0.565413i	$-0.191283\pi$
$163$	268.887			1.64962			0.824808	+	0.565413i	$0.191283\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			322.085i			1.92865i	0.264713	+	0.964327i	$0.414723\pi$
$167$			322.085i			1.92865i	−0.264713	+	0.964327i	$0.585277\pi$
$168$		0			0
$169$	430.158			2.54531
$170$		0			0
$171$		0			0
$172$		0			0
$173$			35.5104i			0.205262i	0.994719	+	0.102631i	$0.0327261\pi$
$173$			35.5104i			0.205262i	−0.994719	+	0.102631i	$0.967274\pi$
$174$		0			0
$175$	89.1737			0.509564
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	214.059i		−	1.19586i	−0.801549	−	0.597930i	$-0.795990\pi$
$179$		−	214.059i		−	1.19586i	0.801549	−	0.597930i	$-0.204010\pi$
$180$		0			0
$181$	40.8818	+	70.8094i	0.225866	+	0.391212i	0.956579	−	0.291473i	$-0.0941454\pi$
$181$	40.8818	+	70.8094i	0.225866	+	0.391212i	−0.730713	+	0.682685i	$0.760812\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−121.164	+	69.9543i	−0.654943	+	0.378131i
$186$		0			0
$187$	236.434			1.26435
$188$		0			0
$189$		0			0
$190$		0			0
$191$	284.070	−	164.008i	1.48728	−	0.858681i	0.487384	−	0.873188i	$-0.337951\pi$
$191$	284.070	−	164.008i	1.48728	−	0.858681i	0.999895	+	0.0145071i	$0.00461792\pi$
$192$		0			0
$193$	−109.833	−	190.236i	−0.569083	−	0.985680i	−0.996657	−	0.0817000i	$-0.973965\pi$
$193$	−109.833	−	190.236i	−0.569083	−	0.985680i	0.427574	−	0.903980i	$-0.359368\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			161.830i			0.821470i	0.911755	+	0.410735i	$0.134728\pi$
$197$			161.830i			0.821470i	−0.911755	+	0.410735i	$0.865272\pi$
$198$		0			0
$199$	37.0576	+	64.1857i	0.186219	+	0.322541i	0.943987	−	0.329984i	$-0.107043\pi$
$199$	37.0576	+	64.1857i	0.186219	+	0.322541i	−0.757767	+	0.652525i	$0.773710\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	166.856i		−	0.821949i
$204$		0			0
$205$	−111.740	+	193.539i	−0.545072	+	0.944093i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−49.0903	−	191.450i	−0.234882	−	0.916027i
$210$		0			0
$211$	62.2388	+	107.801i	0.294970	+	0.510904i	0.974978	−	0.222301i	$-0.0713568\pi$
$211$	62.2388	+	107.801i	0.294970	+	0.510904i	−0.680008	+	0.733205i	$0.738024\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.8547	−	9.15373i	0.0737429	−	0.0425755i
$216$		0			0
$217$	94.1575			0.433906
$218$		0			0
$219$		0			0
$220$		0			0
$221$	481.818	+	278.178i	2.18017	+	1.25872i
$222$		0			0
$223$	−320.431			−1.43691			−0.718456	−	0.695572i	$-0.755151\pi$
$223$	−320.431			−1.43691			−0.718456	+	0.695572i	$0.755151\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	137.264	−	79.2492i	0.604685	−	0.349115i	−0.166197	−	0.986093i	$-0.553149\pi$
$227$	137.264	−	79.2492i	0.604685	−	0.349115i	0.770883	+	0.636977i	$0.219816\pi$
$228$		0			0
$229$	−124.132	+	215.003i	−0.542061	+	0.938876i	0.456725	+	0.889608i	$0.349022\pi$
$229$	−124.132	+	215.003i	−0.542061	+	0.938876i	−0.998786	+	0.0492684i	$0.984311\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−277.122	−	159.996i	−1.18936	−	0.686680i	−0.231202	−	0.972906i	$-0.574266\pi$
$233$	−277.122	−	159.996i	−1.18936	−	0.686680i	−0.958162	+	0.286226i	$0.907599\pi$
$234$		0			0
$235$	−51.7698			−0.220297
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.0028	−	7.50716i	−0.0544049	−	0.0314107i	0.472551	−	0.881303i	$-0.343333\pi$
$239$	−13.0028	−	7.50716i	−0.0544049	−	0.0314107i	−0.526956	+	0.849893i	$0.676667\pi$
$240$		0			0
$241$	22.8860	−	39.6396i	0.0949625	−	0.164480i	−0.814630	−	0.579980i	$-0.803060\pi$
$241$	22.8860	−	39.6396i	0.0949625	−	0.164480i	0.909593	+	0.415501i	$0.136394\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			57.2731i			0.233768i
$246$		0			0
$247$	125.212	−	447.904i	0.506932	−	1.81338i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−306.214	+	176.793i	−1.21997	+	0.704353i	−0.964912	−	0.262572i	$-0.915429\pi$
$251$	−306.214	+	176.793i	−1.21997	+	0.704353i	−0.255062	+	0.966925i	$0.582096\pi$
$252$		0			0
$253$	152.578	+	264.272i	0.603073	+	1.04455i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	157.838i		−	0.614157i	−0.951684	−	0.307079i	$-0.900649\pi$
$257$		−	157.838i		−	0.614157i	0.951684	−	0.307079i	$-0.0993515\pi$
$258$		0			0
$259$	−129.692	−	224.633i	−0.500742	−	0.867310i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			280.562i			1.06678i	0.845870	+	0.533389i	$0.179082\pi$
$263$			280.562i			1.06678i	−0.845870	+	0.533389i	$0.820918\pi$
$264$		0			0
$265$	−21.8540	+	37.8523i	−0.0824680	+	0.142839i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	293.137	+	169.243i	1.08973	+	0.629156i	0.933505	−	0.358566i	$-0.116734\pi$
$269$	293.137	+	169.243i	1.08973	+	0.629156i	0.156225	+	0.987721i	$0.450067\pi$
$270$		0			0
$271$	144.466	−	250.222i	0.533084	−	0.923329i	−0.466169	−	0.884696i	$-0.654366\pi$
$271$	144.466	−	250.222i	0.533084	−	0.923329i	0.999253	−	0.0386333i	$-0.0123004\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	171.002i		−	0.621826i
$276$		0			0
$277$	−254.832	−	441.382i	−0.919971	−	1.59344i	−0.799456	−	0.600725i	$-0.794879\pi$
$277$	−254.832	−	441.382i	−0.919971	−	1.59344i	−0.120515	−	0.992712i	$-0.538454\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	350.515	+	202.370i	1.24738	+	0.720178i	0.970587	−	0.240751i	$-0.0773936\pi$
$281$	350.515	+	202.370i	1.24738	+	0.720178i	0.276797	+	0.960928i	$0.410727\pi$
$282$		0			0
$283$	184.711	+	319.929i	0.652689	+	1.13049i	0.982468	+	0.186432i	$0.0596925\pi$
$283$	184.711	+	319.929i	0.652689	+	1.13049i	−0.329779	+	0.944058i	$0.606974\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−358.813	−	207.161i	−1.25022	−	0.721814i
$288$		0			0
$289$	113.806	+	197.118i	0.393792	+	0.682068i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−115.170	−	66.4934i	−0.393071	−	0.226940i	0.290419	−	0.956900i	$-0.406205\pi$
$293$	−115.170	−	66.4934i	−0.393071	−	0.226940i	−0.683490	+	0.729960i	$0.739539\pi$
$294$		0			0
$295$	−31.8407	+	55.1497i	−0.107935	+	0.186948i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			718.064i			2.40155i
$300$		0			0
$301$	16.9706	+	29.3939i	0.0563807	+	0.0976543i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	327.802i		−	1.07476i
$306$		0			0
$307$	−49.3725	+	85.5156i	−0.160822	+	0.278552i	−0.935164	−	0.354215i	$-0.884748\pi$
$307$	−49.3725	+	85.5156i	−0.160822	+	0.278552i	0.774341	+	0.632768i	$0.218081\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−102.641	−	59.2596i	−0.330034	−	0.190545i	0.325822	−	0.945431i	$-0.394359\pi$
$311$	−102.641	−	59.2596i	−0.330034	−	0.190545i	−0.655856	+	0.754886i	$0.727692\pi$
$312$		0			0
$313$	19.5446	−	33.8523i	0.0624429	−	0.108154i	−0.833114	−	0.553101i	$-0.813444\pi$
$313$	19.5446	−	33.8523i	0.0624429	−	0.108154i	0.895557	+	0.444947i	$0.146778\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−487.762	−	281.609i	−1.53868	−	0.888358i	−0.998916	−	0.0465417i	$-0.985180\pi$
$317$	−487.762	−	281.609i	−1.53868	−	0.888358i	−0.539764	−	0.841816i	$-0.681487\pi$
$318$		0			0
$319$	−319.967			−1.00303
$320$		0			0
$321$		0			0
$322$		0			0
$323$	308.644	−	302.052i	0.955555	−	0.935147i
$324$		0			0
$325$	201.193	−	348.477i	0.619057	−	1.07224i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	95.9789i		−	0.291729i
$330$		0			0
$331$	275.829	+	477.751i	0.833322	+	1.44336i	0.895390	+	0.445284i	$0.146897\pi$
$331$	275.829	+	477.751i	0.833322	+	1.44336i	−0.0620679	+	0.998072i	$0.519770\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−53.6292	+	30.9628i	−0.160087	+	0.0924263i
$336$		0			0
$337$	−94.2874	−	163.311i	−0.279785	−	0.484601i	0.691546	−	0.722332i	$-0.256930\pi$
$337$	−94.2874	−	163.311i	−0.279785	−	0.484601i	−0.971331	+	0.237731i	$0.923596\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	180.559i		−	0.529499i
$342$		0			0
$343$	−371.984			−1.08450
$344$		0			0
$345$		0			0
$346$		0			0
$347$	578.131	−	333.784i	1.66608	−	0.961914i	0.696365	−	0.717688i	$-0.254799\pi$
$347$	578.131	−	333.784i	1.66608	−	0.961914i	0.969718	−	0.244226i	$-0.0785339\pi$
$348$		0			0
$349$	41.5364	+	71.9431i	0.119015	+	0.206141i	0.919378	−	0.393376i	$-0.128693\pi$
$349$	41.5364	+	71.9431i	0.119015	+	0.206141i	−0.800362	+	0.599517i	$0.795360\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	220.283	−	127.181i	0.624032	−	0.360285i	−0.154405	−	0.988008i	$-0.549346\pi$
$353$	220.283	−	127.181i	0.624032	−	0.360285i	0.778437	+	0.627722i	$0.216013\pi$
$354$		0			0
$355$	147.051			0.414229
$356$		0			0
$357$		0			0
$358$		0			0
$359$	165.360	+	95.4709i	0.460614	+	0.265936i	0.712302	−	0.701873i	$-0.247653\pi$
$359$	165.360	+	95.4709i	0.460614	+	0.265936i	−0.251688	+	0.967808i	$0.580986\pi$
$360$		0			0
$361$	−308.666	−	187.206i	−0.855031	−	0.518577i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	40.4364i		−	0.110785i
$366$		0			0
$367$	−84.5427	+	146.432i	−0.230362	+	0.398998i	−0.957915	−	0.287053i	$-0.907324\pi$
$367$	−84.5427	+	146.432i	−0.230362	+	0.398998i	0.727553	+	0.686052i	$0.240658\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−70.1765	−	40.5164i	−0.189155	−	0.109209i
$372$		0			0
$373$	188.822	−	327.049i	0.506225	−	0.876808i	−0.493749	−	0.869605i	$-0.664374\pi$
$373$	188.822	−	327.049i	0.506225	−	0.876808i	0.999974	−	0.00720342i	$-0.00229294\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−652.047	−	376.459i	−1.72957	−	0.998566i
$378$		0			0
$379$	673.545			1.77716			0.888581	−	0.458719i	$-0.151692\pi$
$379$	673.545			1.77716			0.888581	+	0.458719i	$0.151692\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−45.6149	+	26.3358i	−0.119099	+	0.0687618i	−0.558366	−	0.829595i	$-0.688572\pi$
$383$	−45.6149	+	26.3358i	−0.119099	+	0.0687618i	0.439267	+	0.898357i	$0.355238\pi$
$384$		0			0
$385$	−165.103			−0.428840
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−76.4930	−	44.1633i	−0.196640	−	0.113530i	0.398447	−	0.917191i	$-0.369549\pi$
$389$	−76.4930	−	44.1633i	−0.196640	−	0.113530i	−0.595087	+	0.803661i	$0.702883\pi$
$390$		0			0
$391$	−333.384	+	577.438i	−0.852644	+	1.47682i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.30645	+	1.90898i	−0.00837077	+	0.00483287i
$396$		0			0
$397$	50.6046	−	87.6498i	0.127468	−	0.220780i	−0.795227	−	0.606312i	$-0.792648\pi$
$397$	50.6046	−	87.6498i	0.127468	−	0.220780i	0.922695	+	0.385531i	$0.125982\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	34.5487	−	19.9467i	0.0861563	−	0.0497423i	−0.456303	−	0.889825i	$-0.650827\pi$
$401$	34.5487	−	19.9467i	0.0861563	−	0.0497423i	0.542459	+	0.840082i	$0.317493\pi$
$402$		0			0
$403$	212.438	−	367.953i	0.527141	−	0.913035i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−430.763	+	248.701i	−1.05839	+	0.611060i
$408$		0			0
$409$	136.057			0.332657			0.166329	−	0.986070i	$-0.446809\pi$
$409$	136.057			0.332657			0.166329	+	0.986070i	$0.446809\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−102.245	−	59.0312i	−0.247567	−	0.142933i
$414$		0			0
$415$	111.888	−	193.797i	0.269611	−	0.466980i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	434.800	+	251.032i	1.03771	+	0.599122i	0.919184	−	0.393830i	$-0.128850\pi$
$419$	434.800	+	251.032i	1.03771	+	0.599122i	0.118525	+	0.992951i	$0.462183\pi$
$420$		0			0
$421$	−176.166			−0.418446			−0.209223	−	0.977868i	$-0.567093\pi$
$421$	−176.166			−0.418446			−0.209223	+	0.977868i	$0.567093\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	323.583	−	186.821i	0.761373	−	0.439579i
$426$		0			0
$427$	607.729			1.42325
$428$		0			0
$429$		0			0
$430$		0			0
$431$	168.235	−	97.1304i	0.390336	−	0.225361i	−0.291970	−	0.956428i	$-0.594311\pi$
$431$	168.235	−	97.1304i	0.390336	−	0.225361i	0.682306	+	0.731067i	$0.260977\pi$
$432$		0			0
$433$	306.183	+	530.324i	0.707120	+	1.22477i	0.965921	+	0.258837i	$0.0833391\pi$
$433$	306.183	+	530.324i	0.707120	+	1.22477i	−0.258802	+	0.965931i	$0.583328\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	536.793	+	150.061i	1.22836	+	0.343390i
$438$		0			0
$439$	−171.777			−0.391292			−0.195646	−	0.980675i	$-0.562680\pi$
$439$	−171.777			−0.391292			−0.195646	+	0.980675i	$0.562680\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−199.252	−	115.038i	−0.449780	−	0.259680i	0.257957	−	0.966156i	$-0.416951\pi$
$443$	−199.252	−	115.038i	−0.449780	−	0.259680i	−0.707737	+	0.706476i	$0.750284\pi$
$444$		0			0
$445$	−207.869	+	360.040i	−0.467122	+	0.809079i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			443.639i			0.988061i	0.869445	+	0.494031i	$0.164477\pi$
$449$			443.639i			0.988061i	−0.869445	+	0.494031i	$0.835523\pi$
$450$		0			0
$451$	−397.257	+	688.070i	−0.880837	+	1.52565i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−336.456	−	194.253i	−0.739465	−	0.426930i
$456$		0			0
$457$	372.974	+	646.009i	0.816135	+	1.41359i	0.908510	+	0.417863i	$0.137221\pi$
$457$	372.974	+	646.009i	0.816135	+	1.41359i	−0.0923753	+	0.995724i	$0.529446\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			250.804i			0.544043i	0.962291	+	0.272021i	$0.0876921\pi$
$461$			250.804i			0.544043i	−0.962291	+	0.272021i	$0.912308\pi$
$462$		0			0
$463$	83.5338	−	144.685i	0.180419	−	0.312494i	−0.761605	−	0.648042i	$-0.775588\pi$
$463$	83.5338	−	144.685i	0.180419	−	0.312494i	0.942023	+	0.335548i	$0.108921\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			438.133i			0.938186i	0.883149	+	0.469093i	$0.155419\pi$
$467$			438.133i			0.938186i	−0.883149	+	0.469093i	$0.844581\pi$
$468$		0			0
$469$	−57.4037	−	99.4261i	−0.122396	−	0.211996i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	56.3667	−	32.5433i	0.119168	−	0.0688020i
$474$		0			0
$475$	−218.461	−	223.228i	−0.459917	−	0.469954i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.0713	+	11.5882i	0.0419025	+	0.0241924i	0.520805	−	0.853676i	$-0.325632\pi$
$479$	20.0713	+	11.5882i	0.0419025	+	0.0241924i	−0.478902	+	0.877868i	$0.658965\pi$
$480$		0			0
$481$	−1170.44			−2.43336
$482$		0			0
$483$		0			0
$484$		0			0
$485$	468.736	−	270.625i	0.966466	−	0.557989i
$486$		0			0
$487$	−412.683			−0.847399			−0.423700	−	0.905803i	$-0.639269\pi$
$487$	−412.683			−0.847399			−0.423700	+	0.905803i	$0.639269\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−218.449	+	126.121i	−0.444905	+	0.256866i	−0.705676	−	0.708535i	$-0.749357\pi$
$491$	−218.449	+	126.121i	−0.444905	+	0.256866i	0.260771	+	0.965401i	$0.416023\pi$
$492$		0			0
$493$	−349.566	−	605.467i	−0.709060	−	1.22813i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			272.626i			0.548544i
$498$		0			0
$499$	−139.364	−	241.385i	−0.279286	−	0.483738i	0.691921	−	0.721973i	$-0.256764\pi$
$499$	−139.364	−	241.385i	−0.279286	−	0.483738i	−0.971207	+	0.238235i	$0.923431\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	397.715	−	229.621i	0.790685	−	0.456502i	−0.0495184	−	0.998773i	$-0.515769\pi$
$503$	397.715	−	229.621i	0.790685	−	0.456502i	0.840204	+	0.542271i	$0.182435\pi$
$504$		0			0
$505$	−173.106			−0.342785
$506$		0			0
$507$		0			0
$508$		0			0
$509$			116.649i			0.229173i	0.993413	+	0.114586i	$0.0365543\pi$
$509$			116.649i			0.229173i	−0.993413	+	0.114586i	$0.963446\pi$
$510$		0			0
$511$	74.9674			0.146707
$512$		0			0
$513$		0			0
$514$		0			0
$515$			495.431i			0.962003i
$516$		0			0
$517$	−184.052			−0.356000
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	280.254i		−	0.537916i	−0.963152	−	0.268958i	$-0.913321\pi$
$521$		−	280.254i		−	0.537916i	0.963152	−	0.268958i	$-0.0866793\pi$
$522$		0			0
$523$	51.0370	+	88.3987i	0.0975851	+	0.169022i	0.910685	−	0.413102i	$-0.135555\pi$
$523$	51.0370	+	88.3987i	0.0975851	+	0.169022i	−0.813099	+	0.582125i	$0.802222\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	341.668	−	197.262i	0.648326	−	0.374311i
$528$		0			0
$529$	−331.567			−0.626781
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1619.11	+	934.791i	−3.03772	+	1.75383i
$534$		0			0
$535$	191.890	+	332.363i	0.358673	+	0.621240i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			203.617i			0.377768i
$540$		0			0
$541$	−376.134	−	651.483i	−0.695257	−	1.20422i	−0.970094	−	0.242729i	$-0.921957\pi$
$541$	−376.134	−	651.483i	−0.695257	−	1.20422i	0.274837	−	0.961491i	$-0.411376\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	45.7645i		−	0.0839715i
$546$		0			0
$547$	327.909	−	567.955i	0.599468	−	1.03831i	−0.393431	−	0.919354i	$-0.628712\pi$
$547$	327.909	−	567.955i	0.599468	−	1.03831i	0.992900	−	0.118955i	$-0.0379545\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−417.689	+	408.769i	−0.758057	+	0.741867i
$552$		0			0
$553$	−3.53917	−	6.13002i	−0.00639994	−	0.0110850i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−497.328	+	287.133i	−0.892870	+	0.515498i	−0.874880	−	0.484340i	$-0.839060\pi$
$557$	−497.328	+	287.133i	−0.892870	+	0.515498i	−0.0179895	+	0.999838i	$0.505727\pi$
$558$		0			0
$559$	153.156			0.273982
$560$		0			0
$561$		0			0
$562$		0			0
$563$	773.883	+	446.801i	1.37457	+	0.793608i	0.991499	−	0.130111i	$-0.0415334\pi$
$563$	773.883	+	446.801i	1.37457	+	0.793608i	0.383070	+	0.923719i	$0.374867\pi$
$564$		0			0
$565$	−69.2105			−0.122497
$566$		0			0
$567$		0			0
$568$		0			0
$569$	337.331	−	194.758i	0.592848	−	0.342281i	−0.173375	−	0.984856i	$-0.555467\pi$
$569$	337.331	−	194.758i	0.592848	−	0.342281i	0.766223	+	0.642575i	$0.222134\pi$
$570$		0			0
$571$	61.1358	−	105.890i	0.107068	−	0.185447i	−0.807513	−	0.589849i	$-0.799187\pi$
$571$	61.1358	−	105.890i	0.107068	−	0.185447i	0.914581	+	0.404402i	$0.132520\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	417.634	+	241.121i	0.726321	+	0.419342i
$576$		0			0
$577$	−927.013			−1.60661			−0.803304	−	0.595569i	$-0.796926\pi$
$577$	−927.013			−1.60661			−0.803304	+	0.595569i	$0.796926\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	359.290	+	207.436i	0.618399	+	0.357033i
$582$		0			0
$583$	−77.6954	+	134.572i	−0.133268	+	0.230828i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	530.684i		−	0.904062i	−0.892002	−	0.452031i	$-0.850700\pi$
$587$		−	530.684i		−	0.904062i	0.892002	−	0.452031i	$-0.149300\pi$
$588$		0			0
$589$	−230.670	−	235.704i	−0.391630	−	0.400177i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−486.494	+	280.878i	−0.820395	+	0.473655i	−0.850553	−	0.525890i	$-0.823732\pi$
$593$	−486.494	+	280.878i	−0.820395	+	0.473655i	0.0301577	+	0.999545i	$0.490399\pi$
$594$		0			0
$595$	−180.376	−	312.421i	−0.303154	−	0.525078i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	639.542i		−	1.06768i	−0.845584	−	0.533842i	$-0.820748\pi$
$599$		−	639.542i		−	1.06768i	0.845584	−	0.533842i	$-0.179252\pi$
$600$		0			0
$601$	−248.581	−	430.555i	−0.413612	−	0.716397i	0.581670	−	0.813425i	$-0.302400\pi$
$601$	−248.581	−	430.555i	−0.413612	−	0.716397i	−0.995282	+	0.0970281i	$0.969066\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	37.4308i		−	0.0618690i
$606$		0			0
$607$	55.0624	−	95.3708i	0.0907123	−	0.157118i	−0.817099	−	0.576498i	$-0.804419\pi$
$607$	55.0624	−	95.3708i	0.0907123	−	0.157118i	0.907811	+	0.419379i	$0.137752\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−375.071	−	216.547i	−0.613864	−	0.354415i
$612$		0			0
$613$	−51.6325	+	89.4302i	−0.0842293	+	0.145889i	−0.905063	−	0.425279i	$-0.860176\pi$
$613$	−51.6325	+	89.4302i	−0.0842293	+	0.145889i	0.820833	+	0.571168i	$0.193509\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			109.410i			0.177325i	0.996062	+	0.0886627i	$0.0282593\pi$
$617$			109.410i			0.177325i	−0.996062	+	0.0886627i	$0.971741\pi$
$618$		0			0
$619$	−280.995	−	486.697i	−0.453950	−	0.786264i	0.544677	−	0.838646i	$-0.316652\pi$
$619$	−280.995	−	486.697i	−0.453950	−	0.786264i	−0.998627	+	0.0523816i	$0.983319\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−667.498	−	385.380i	−1.07143	−	0.618588i
$624$		0			0
$625$	−28.1056	−	48.6804i	−0.0449690	−	0.0778887i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−941.224	−	543.416i	−1.49638	−	0.863936i
$630$		0			0
$631$	−310.083	−	537.079i	−0.491415	−	0.851155i	0.508537	−	0.861040i	$-0.330187\pi$
$631$	−310.083	−	537.079i	−0.491415	−	0.851155i	−0.999951	+	0.00988537i	$0.996853\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	208.296	+	120.259i	0.328024	+	0.189385i
$636$		0			0
$637$	−239.567	+	414.942i	−0.376086	+	0.651400i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	150.952i		−	0.235495i	−0.993044	−	0.117748i	$-0.962433\pi$
$641$		−	150.952i		−	0.235495i	0.993044	−	0.117748i	$-0.0375673\pi$
$642$		0			0
$643$	23.7325	+	41.1058i	0.0369089	+	0.0639282i	0.883890	−	0.467695i	$-0.154916\pi$
$643$	23.7325	+	41.1058i	0.0369089	+	0.0639282i	−0.846981	+	0.531623i	$0.821582\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.8284i			0.0569218i	0.999595	+	0.0284609i	$0.00906061\pi$
$647$			36.8284i			0.0569218i	−0.999595	+	0.0284609i	$0.990939\pi$
$648$		0			0
$649$	−113.200	+	196.068i	−0.174422	+	0.302108i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	336.741	+	194.418i	0.515684	+	0.297730i	0.735167	−	0.677886i	$-0.237104\pi$
$653$	336.741	+	194.418i	0.515684	+	0.297730i	−0.219483	+	0.975616i	$0.570437\pi$
$654$		0			0
$655$	−116.085	+	201.065i	−0.177229	+	0.306970i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	27.7225	+	16.0056i	0.0420676	+	0.0242877i	0.520886	−	0.853626i	$-0.325602\pi$
$659$	27.7225	+	16.0056i	0.0420676	+	0.0242877i	−0.478819	+	0.877914i	$0.658935\pi$
$660$		0			0
$661$	49.2240			0.0744690			0.0372345	−	0.999307i	$-0.488145\pi$
$661$	49.2240			0.0744690			0.0372345	+	0.999307i	$0.488145\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−215.528	+	210.925i	−0.324102	+	0.317180i
$666$		0			0
$667$	451.170	−	781.449i	0.676416	−	1.17159i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1165.40i		−	1.73681i
$672$		0			0
$673$	386.780	+	669.922i	0.574710	+	0.995426i	0.996073	+	0.0885345i	$0.0282184\pi$
$673$	386.780	+	669.922i	0.574710	+	0.995426i	−0.421363	+	0.906892i	$0.638448\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1071.11	+	618.403i	−1.58214	+	0.913446i	−0.587588	+	0.809160i	$0.699922\pi$
$677$	−1071.11	+	618.403i	−1.58214	+	0.913446i	−0.994547	+	0.104286i	$0.966744\pi$
$678$		0			0
$679$	501.726	+	869.015i	0.738919	+	1.27985i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	234.238i		−	0.342955i	−0.985188	−	0.171478i	$-0.945146\pi$
$683$		−	234.238i		−	0.342955i	0.985188	−	0.171478i	$-0.0548541\pi$
$684$		0			0
$685$	−136.197			−0.198828
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−316.664	+	182.826i	−0.459599	+	0.265350i
$690$		0			0
$691$	284.313	+	492.445i	0.411452	+	0.712655i	0.995049	−	0.0993883i	$-0.0316886\pi$
$691$	284.313	+	492.445i	0.411452	+	0.712655i	−0.583597	+	0.812043i	$0.698355\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−187.399	+	108.195i	−0.269639	+	0.155676i
$696$		0			0
$697$	−1736.03			−2.49071
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1031.29	−	595.416i	−1.47117	−	0.849381i	−0.471696	−	0.881761i	$-0.656358\pi$
$701$	−1031.29	−	595.416i	−1.47117	−	0.849381i	−0.999476	+	0.0323797i	$0.989691\pi$
$702$		0			0
$703$	−244.600	+	874.972i	−0.347937	+	1.24463i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	320.932i		−	0.453934i
$708$		0			0
$709$	599.530	−	1038.42i	0.845599	−	1.46462i	−0.0395008	−	0.999220i	$-0.512577\pi$
$709$	599.530	−	1038.42i	0.845599	−	1.46462i	0.885100	−	0.465401i	$-0.154090\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	440.976	+	254.597i	0.618479	+	0.357079i
$714$		0			0
$715$	−372.506	+	645.199i	−0.520987	+	0.902376i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	911.986	+	526.535i	1.26841	+	0.732316i	0.974687	−	0.223574i	$-0.0717726\pi$
$719$	911.986	+	526.535i	1.26841	+	0.732316i	0.293722	+	0.955891i	$0.405106\pi$
$720$		0			0
$721$	−918.508			−1.27394
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−437.907	+	252.826i	−0.604009	+	0.348725i
$726$		0			0
$727$	876.971			1.20629			0.603143	−	0.797633i	$-0.293915\pi$
$727$	876.971			1.20629			0.603143	+	0.797633i	$0.293915\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	123.162	+	71.1076i	0.168484	+	0.0972744i
$732$		0			0
$733$	517.161	−	895.748i	0.705540	−	1.22203i	−0.260957	−	0.965351i	$-0.584038\pi$
$733$	517.161	−	895.748i	0.705540	−	1.22203i	0.966496	−	0.256680i	$-0.0826287\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−190.662	+	110.079i	−0.258701	+	0.149361i
$738$		0			0
$739$	−349.864	+	605.982i	−0.473429	+	0.820003i	−0.999537	−	0.0304143i	$-0.990317\pi$
$739$	−349.864	+	605.982i	−0.473429	+	0.820003i	0.526108	+	0.850418i	$0.323651\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	766.665	−	442.634i	1.03185	−	0.595739i	0.114336	−	0.993442i	$-0.463526\pi$
$743$	766.665	−	442.634i	1.03185	−	0.595739i	0.917514	+	0.397703i	$0.130192\pi$
$744$		0			0
$745$	288.748	−	500.126i	0.387581	−	0.671310i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−616.187	+	355.756i	−0.822679	+	0.474974i
$750$		0			0
$751$	1294.48			1.72368			0.861840	−	0.507181i	$-0.169312\pi$
$751$	1294.48			1.72368			0.861840	+	0.507181i	$0.169312\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−529.545	−	305.733i	−0.701384	−	0.404944i
$756$		0			0
$757$	−80.2906	+	139.067i	−0.106064	+	0.183708i	−0.914172	−	0.405325i	$-0.867158\pi$
$757$	−80.2906	+	139.067i	−0.106064	+	0.183708i	0.808108	+	0.589034i	$0.200492\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	396.859	+	229.127i	0.521497	+	0.301087i	0.737547	−	0.675296i	$-0.235984\pi$
$761$	396.859	+	229.127i	0.521497	+	0.301087i	−0.216050	+	0.976382i	$0.569317\pi$
$762$		0			0
$763$	84.8453			0.111200
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−461.370	+	266.372i	−0.601526	+	0.347291i
$768$		0			0
$769$	175.710			0.228492			0.114246	−	0.993453i	$-0.463555\pi$
$769$	175.710			0.228492			0.114246	+	0.993453i	$0.463555\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−936.670	+	540.787i	−1.21173	+	0.699595i	−0.963137	−	0.269012i	$-0.913303\pi$
$773$	−936.670	+	540.787i	−1.21173	+	0.699595i	−0.248597	+	0.968607i	$0.579970\pi$
$774$		0			0
$775$	−142.671	−	247.113i	−0.184091	−	0.318855i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	360.447	+	1405.72i	0.462705	+	1.80452i
$780$		0			0
$781$	522.796			0.669393
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−110.034	−	63.5282i	−0.140171	−	0.0809277i
$786$		0			0
$787$	143.087	−	247.834i	0.181813	−	0.314910i	−0.760685	−	0.649121i	$-0.775137\pi$
$787$	143.087	−	247.834i	0.181813	−	0.314910i	0.942498	+	0.334212i	$0.108470\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	128.313i		−	0.162216i
$792$		0			0
$793$	1371.16	−	2374.91i	1.72908	−	2.99485i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−207.738	−	119.938i	−0.260650	−	0.150487i	0.363981	−	0.931406i	$-0.381417\pi$
$797$	−207.738	−	119.938i	−0.260650	−	0.150487i	−0.624631	+	0.780920i	$0.714751\pi$
$798$		0			0
$799$	−201.078	−	348.277i	−0.251662	−	0.435892i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	143.760i		−	0.179028i
$804$		0			0
$805$	232.804	−	403.228i	0.289197	−	0.500904i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			69.4467i			0.0858426i	0.999078	+	0.0429213i	$0.0136665\pi$
$809$			69.4467i			0.0858426i	−0.999078	+	0.0429213i	$0.986334\pi$
$810$		0			0
$811$	−5.39295	−	9.34086i	−0.00664975	−	0.0115177i	0.862681	−	0.505748i	$-0.168783\pi$
$811$	−5.39295	−	9.34086i	−0.00664975	−	0.0115177i	−0.869331	+	0.494230i	$0.835450\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−681.342	+	393.373i	−0.836003	+	0.482666i
$816$		0			0
$817$	32.0066	−	114.493i	0.0391758	−	0.140138i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	383.062	+	221.161i	0.466580	+	0.269380i	0.714807	−	0.699322i	$-0.246515\pi$
$821$	383.062	+	221.161i	0.466580	+	0.269380i	−0.248227	+	0.968702i	$0.579848\pi$
$822$		0			0
$823$	−21.7579			−0.0264374			−0.0132187	−	0.999913i	$-0.504208\pi$
$823$	−21.7579			−0.0264374			−0.0132187	+	0.999913i	$0.504208\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	556.307	−	321.184i	0.672681	−	0.388372i	−0.124411	−	0.992231i	$-0.539704\pi$
$827$	556.307	−	321.184i	0.672681	−	0.388372i	0.797092	+	0.603858i	$0.206371\pi$
$828$		0			0
$829$	−540.967			−0.652554			−0.326277	−	0.945274i	$-0.605794\pi$
$829$	−540.967			−0.652554			−0.326277	+	0.945274i	$0.605794\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−385.300	+	222.453i	−0.462545	+	0.267051i
$834$		0			0
$835$	−471.200	−	816.142i	−0.564311	−	0.977416i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1098.29i		−	1.30905i	−0.756042	−	0.654523i	$-0.772870\pi$
$839$		−	1098.29i		−	1.30905i	0.756042	−	0.654523i	$-0.227130\pi$
$840$		0			0
$841$	52.5696	+	91.0532i	0.0625084	+	0.108268i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1089.99	+	629.306i	−1.28993	+	0.744741i
$846$		0			0
$847$	69.3949			0.0819303
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1402.72i		−	1.64833i
$852$		0			0
$853$	−272.054			−0.318938			−0.159469	−	0.987203i	$-0.550978\pi$
$853$	−272.054			−0.318938			−0.159469	+	0.987203i	$0.550978\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1043.71i			1.21787i	0.793220	+	0.608935i	$0.208403\pi$
$857$			1043.71i			1.21787i	−0.793220	+	0.608935i	$0.791597\pi$
$858$		0			0
$859$	−1078.69			−1.25576			−0.627878	−	0.778312i	$-0.716076\pi$
$859$	−1078.69			−1.25576			−0.627878	+	0.778312i	$0.716076\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	613.165i		−	0.710504i	−0.934771	−	0.355252i	$-0.884395\pi$
$863$		−	613.165i		−	0.710504i	0.934771	−	0.355252i	$-0.115605\pi$
$864$		0			0
$865$	−51.9505	−	89.9809i	−0.0600583	−	0.104024i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.7551	+	6.78681i	−0.0135272	+	0.00780991i
$870$		0			0
$871$	−518.056			−0.594783
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−569.596	+	328.856i	−0.650967	+	0.375836i
$876$		0			0
$877$	−172.712	−	299.146i	−0.196935	−	0.341102i	0.750598	−	0.660759i	$-0.229766\pi$
$877$	−172.712	−	299.146i	−0.196935	−	0.341102i	−0.947533	+	0.319657i	$0.896432\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	452.169i		−	0.513245i	−0.966512	−	0.256623i	$-0.917390\pi$
$881$		−	452.169i		−	0.513245i	0.966512	−	0.256623i	$-0.0826098\pi$
$882$		0			0
$883$	40.2017	+	69.6314i	0.0455285	+	0.0788577i	0.887892	−	0.460053i	$-0.152169\pi$
$883$	40.2017	+	69.6314i	0.0455285	+	0.0788577i	−0.842363	+	0.538910i	$0.818836\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1446.02i			1.63024i	0.579290	+	0.815121i	$0.303330\pi$
$887$			1446.02i			1.63024i	−0.579290	+	0.815121i	$0.696670\pi$
$888$		0			0
$889$	−222.956	+	386.171i	−0.250794	+	0.434388i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−240.264	+	235.132i	−0.269052	+	0.263306i
$894$		0			0
$895$	313.161	+	542.410i	0.349900	+	0.606045i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−462.381	+	266.956i	−0.514328	+	0.296947i
$900$		0			0
$901$	−339.531			−0.376838
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−207.183	−	119.617i	−0.228932	−	0.132174i
$906$		0			0
$907$	1675.42			1.84721			0.923606	−	0.383344i	$-0.125228\pi$
$907$	1675.42			1.84721			0.923606	+	0.383344i	$0.125228\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	51.2037	−	29.5624i	0.0562060	−	0.0324505i	−0.471634	−	0.881795i	$-0.656336\pi$
$911$	51.2037	−	29.5624i	0.0562060	−	0.0324505i	0.527840	+	0.849344i	$0.323002\pi$
$912$		0			0
$913$	397.786	−	688.985i	0.435691	−	0.754639i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−372.766	−	215.217i	−0.406506	−	0.234697i
$918$		0			0
$919$	−811.809			−0.883361			−0.441680	−	0.897172i	$-0.645617\pi$
$919$	−811.809			−0.883361			−0.441680	+	0.897172i	$0.645617\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1065.38	+	615.099i	1.15426	+	0.666412i
$924$		0			0
$925$	−393.028	+	680.744i	−0.424895	+	0.735940i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1471.94i			1.58444i	0.610238	+	0.792218i	$0.291074\pi$
$929$			1471.94i			1.58444i	−0.610238	+	0.792218i	$0.708926\pi$
$930$		0			0
$931$	260.127	+	265.804i	0.279407	+	0.285504i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−599.108	+	345.895i	−0.640757	+	0.369941i
$936$		0			0
$937$	−292.962	−	507.425i	−0.312660	−	0.541543i	0.666277	−	0.745704i	$-0.267887\pi$
$937$	−292.962	−	507.425i	−0.312660	−	0.541543i	−0.978937	+	0.204161i	$0.934553\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	641.387i		−	0.681601i	−0.940136	−	0.340801i	$-0.889302\pi$
$941$		−	641.387i		−	0.681601i	0.940136	−	0.340801i	$-0.110698\pi$
$942$		0			0
$943$	−1120.31	−	1940.43i	−1.18802	−	2.05771i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1190.64i		−	1.25728i	−0.777697	−	0.628639i	$-0.783612\pi$
$947$		−	1190.64i		−	1.25728i	0.777697	−	0.628639i	$-0.216388\pi$
$948$		0			0
$949$	169.141	−	292.961i	0.178231	−	0.308705i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	834.102	+	481.569i	0.875238	+	0.505319i	0.869085	−	0.494662i	$-0.164708\pi$
$953$	834.102	+	481.569i	0.875238	+	0.505319i	0.00615290	+	0.999981i	$0.498041\pi$
$954$		0			0
$955$	−479.876	+	831.170i	−0.502488	+	0.870335i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	252.504i		−	0.263299i
$960$		0			0
$961$	329.856	+	571.327i	0.343242	+	0.594513i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	556.618	+	321.364i	0.576807	+	0.333019i
$966$		0			0
$967$	677.770	+	1173.93i	0.700899	+	1.21399i	0.968151	+	0.250367i	$0.0805512\pi$
$967$	677.770	+	1173.93i	0.700899	+	1.21399i	−0.267252	+	0.963627i	$0.586116\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1057.02	−	610.270i	−1.08859	−	0.628497i	−0.155388	−	0.987853i	$-0.549663\pi$
$971$	−1057.02	−	610.270i	−1.08859	−	0.628497i	−0.933200	+	0.359357i	$0.882996\pi$
$972$		0			0
$973$	−200.588	−	347.429i	−0.206155	−	0.357070i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−659.911	−	381.000i	−0.675446	−	0.389969i	0.122691	−	0.992445i	$-0.460848\pi$
$977$	−659.911	−	381.000i	−0.675446	−	0.389969i	−0.798137	+	0.602476i	$0.794181\pi$
$978$		0			0
$979$	−739.016	+	1280.01i	−0.754869	+	1.30747i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1104.03i			1.12312i	0.827435	+	0.561562i	$0.189799\pi$
$983$			1104.03i			1.12312i	−0.827435	+	0.561562i	$0.810201\pi$
$984$		0			0
$985$	−236.751	−	410.065i	−0.240357	−	0.416310i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			183.551i			0.185592i
$990$		0			0
$991$	−308.321	+	534.027i	−0.311121	+	0.538877i	−0.978605	−	0.205747i	$-0.934038\pi$
$991$	−308.321	+	534.027i	−0.311121	+	0.538877i	0.667485	+	0.744624i	$0.267371\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−187.803	−	108.428i	−0.188747	−	0.108973i
$996$		0			0
$997$	−290.261	+	502.746i	−0.291134	+	0.504259i	−0.974078	−	0.226212i	$-0.927366\pi$
$997$	−290.261	+	502.746i	−0.291134	+	0.504259i	0.682944	+	0.730471i	$0.260699\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.be.a.125.14		80
3.2	odd	2		684.3.be.a.581.12	yes	80
9.2	odd	6		2052.3.m.a.1493.27		80
9.7	even	3		684.3.m.a.353.16	✓	80
19.7	even	3		2052.3.m.a.881.14		80
57.26	odd	6		684.3.m.a.653.16	yes	80
171.7	even	3		684.3.be.a.425.12	yes	80
171.83	odd	6	inner	2052.3.be.a.197.14		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.16	✓	80	9.7	even	3
684.3.m.a.653.16	yes	80	57.26	odd	6
684.3.be.a.425.12	yes	80	171.7	even	3
684.3.be.a.581.12	yes	80	3.2	odd	2
2052.3.m.a.881.14		80	19.7	even	3
2052.3.m.a.1493.27		80	9.2	odd	6
2052.3.be.a.125.14		80	1.1	even	1	trivial
2052.3.be.a.197.14		80	171.83	odd	6	inner

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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.be (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.53393 + 1.46297i) q^{5} +(-2.71227 - 4.69779i) q^{7} +O(q^{10})\)	\(q+(-2.53393 + 1.46297i) q^{5} +(-2.71227 - 4.69779i) q^{7} +(-9.00863 + 5.20114i) q^{11} -24.4777 q^{13} +(-19.6840 - 11.3645i) q^{17} +(-5.11536 + 18.2984i) q^{19} -29.3354i q^{23} +(-8.21946 + 14.2365i) q^{25} +(26.6384 + 15.3797i) q^{29} +(-8.67884 + 15.0322i) q^{31} +(13.7454 + 7.93593i) q^{35} +47.8168 q^{37} +(66.1461 - 38.1895i) q^{41} -6.25697 q^{43} +(15.3230 + 8.84672i) q^{47} +(9.78715 - 16.9518i) q^{49} +(12.9368 - 7.46908i) q^{53} +(15.2182 - 26.3586i) q^{55} +(18.8486 - 10.8822i) q^{59} +(-56.0166 + 97.0236i) q^{61} +(62.0248 - 35.8100i) q^{65} +21.1644 q^{67} +(-43.5246 - 25.1289i) q^{71} +(-6.91001 + 11.9685i) q^{73} +(48.8677 + 28.2138i) q^{77} +1.30487 q^{79} +(-66.2341 + 38.2403i) q^{83} +66.5038 q^{85} +(123.051 - 71.0438i) q^{89} +(66.3902 + 114.991i) q^{91} +(-13.8080 - 53.8506i) q^{95} -184.984 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Introduction

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