LMFDB - Embedded newform 2052.3.bl.a.145.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(145,2052)]

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, 2, 5]))

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

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

chi := DirichletCharacter("2052.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.bl (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			145.15
Character	$\chi$	$=$	2052.145
Dual form			2052.3.bl.a.1585.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.57346 q^{5} +(2.11933 + 3.67079i) q^{7} +O(q^{10})$	$q-3.57346 q^{5} +(2.11933 + 3.67079i) q^{7} +(8.61085 + 14.9144i) q^{11} +(16.7874 - 9.69219i) q^{13} +(16.5029 + 28.5838i) q^{17} +(-0.306057 - 18.9975i) q^{19} +(-16.8016 - 29.1011i) q^{23} -12.2304 q^{25} -0.358227i q^{29} +(43.6395 + 25.1953i) q^{31} +(-7.57335 - 13.1174i) q^{35} +26.4325i q^{37} -39.9163i q^{41} +(-9.92526 + 17.1911i) q^{43} -5.15777 q^{47} +(15.5169 - 26.8760i) q^{49} +(-20.3465 - 11.7470i) q^{53} +(-30.7706 - 53.2962i) q^{55} -26.5702i q^{59} -45.0911 q^{61} +(-59.9890 + 34.6347i) q^{65} +(-68.9204 + 39.7912i) q^{67} +(1.79086 - 1.03395i) q^{71} +(18.0979 + 31.3466i) q^{73} +(-36.4985 + 63.2173i) q^{77} +(62.1349 + 35.8736i) q^{79} +(63.8891 + 110.659i) q^{83} +(-58.9724 - 102.143i) q^{85} +(35.6570 + 20.5866i) q^{89} +(71.1560 + 41.0819i) q^{91} +(1.09368 + 67.8869i) q^{95} +(43.8518 + 25.3178i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - q^{7}+O(q^{10}) $ 80 * q - q^7	$ 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) $ 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−3.57346			−0.714692			−0.357346	−	0.933972i	$-0.616318\pi$
$5$	−3.57346			−0.714692			−0.357346	+	0.933972i	$0.616318\pi$
$6$		0			0
$7$	2.11933	+	3.67079i	0.302762	+	0.524399i	0.976760	−	0.214334i	$-0.0687580\pi$
$7$	2.11933	+	3.67079i	0.302762	+	0.524399i	−0.673999	+	0.738733i	$0.735425\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.61085	+	14.9144i	0.782805	+	1.35586i	0.930302	+	0.366795i	$0.119545\pi$
$11$	8.61085	+	14.9144i	0.782805	+	1.35586i	−0.147497	+	0.989063i	$0.547122\pi$
$12$		0			0
$13$	16.7874	−	9.69219i	1.29134	−	0.745553i	0.312444	−	0.949936i	$-0.398852\pi$
$13$	16.7874	−	9.69219i	1.29134	−	0.745553i	0.978891	+	0.204383i	$0.0655188\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	16.5029	+	28.5838i	0.970758	+	1.68140i	0.693277	+	0.720671i	$0.256166\pi$
$17$	16.5029	+	28.5838i	0.970758	+	1.68140i	0.277481	+	0.960731i	$0.410500\pi$
$18$		0			0
$19$	−0.306057	−	18.9975i	−0.0161083	−	0.999870i
$19$	−0.306057	−	18.9975i	−0.0161083	−	0.999870i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−16.8016	−	29.1011i	−0.730502	−	1.26527i	−0.956669	−	0.291179i	$-0.905953\pi$
$23$	−16.8016	−	29.1011i	−0.730502	−	1.26527i	0.226166	−	0.974089i	$-0.427381\pi$
$24$		0			0
$25$	−12.2304			−0.489215
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	0.358227i		−	0.0123526i	−0.999981	−	0.00617632i	$-0.998034\pi$
$29$		−	0.358227i		−	0.0123526i	0.999981	−	0.00617632i	$-0.00196600\pi$
$30$		0			0
$31$	43.6395	+	25.1953i	1.40773	+	0.812751i	0.995169	−	0.0981804i	$-0.0313022\pi$
$31$	43.6395	+	25.1953i	1.40773	+	0.812751i	0.412558	+	0.910931i	$0.364636\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.57335	−	13.1174i	−0.216381	−	0.374784i
$36$		0			0
$37$			26.4325i			0.714391i	0.934030	+	0.357196i	$0.116267\pi$
$37$			26.4325i			0.714391i	−0.934030	+	0.357196i	$0.883733\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	39.9163i		−	0.973568i	−0.873522	−	0.486784i	$-0.838170\pi$
$41$		−	39.9163i		−	0.973568i	0.873522	−	0.486784i	$-0.161830\pi$
$42$		0			0
$43$	−9.92526	+	17.1911i	−0.230820	+	0.399792i	−0.958050	−	0.286602i	$-0.907474\pi$
$43$	−9.92526	+	17.1911i	−0.230820	+	0.399792i	0.727230	+	0.686394i	$0.240808\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.15777			−0.109740			−0.0548699	−	0.998494i	$-0.517474\pi$
$47$	−5.15777			−0.109740			−0.0548699	+	0.998494i	$0.517474\pi$
$48$		0			0
$49$	15.5169	−	26.8760i	0.316671	−	0.548490i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−20.3465	−	11.7470i	−0.383896	−	0.221642i	0.295616	−	0.955307i	$-0.404475\pi$
$53$	−20.3465	−	11.7470i	−0.383896	−	0.221642i	−0.679512	+	0.733664i	$0.737808\pi$
$54$		0			0
$55$	−30.7706	−	53.2962i	−0.559465	−	0.969021i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	26.5702i		−	0.450342i	−0.974319	−	0.225171i	$-0.927706\pi$
$59$		−	26.5702i		−	0.450342i	0.974319	−	0.225171i	$-0.0722941\pi$
$60$		0			0
$61$	−45.0911			−0.739198			−0.369599	−	0.929191i	$-0.620505\pi$
$61$	−45.0911			−0.739198			−0.369599	+	0.929191i	$0.620505\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−59.9890	+	34.6347i	−0.922907	+	0.532841i
$66$		0			0
$67$	−68.9204	+	39.7912i	−1.02866	+	0.593899i	−0.916601	−	0.399803i	$-0.869079\pi$
$67$	−68.9204	+	39.7912i	−1.02866	+	0.593899i	−0.112061	+	0.993701i	$0.535745\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.79086	−	1.03395i	0.0252234	−	0.0145627i	−0.487335	−	0.873215i	$-0.662031\pi$
$71$	1.79086	−	1.03395i	0.0252234	−	0.0145627i	0.512559	+	0.858652i	$0.328698\pi$
$72$		0			0
$73$	18.0979	+	31.3466i	0.247917	+	0.429405i	0.962948	−	0.269688i	$-0.0869206\pi$
$73$	18.0979	+	31.3466i	0.247917	+	0.429405i	−0.715031	+	0.699093i	$0.753587\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−36.4985	+	63.2173i	−0.474007	+	0.821004i
$78$		0			0
$79$	62.1349	+	35.8736i	0.786517	+	0.454096i	0.838735	−	0.544540i	$-0.183296\pi$
$79$	62.1349	+	35.8736i	0.786517	+	0.454096i	−0.0522178	+	0.998636i	$0.516629\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	63.8891	+	110.659i	0.769748	+	1.33324i	0.937699	+	0.347447i	$0.112951\pi$
$83$	63.8891	+	110.659i	0.769748	+	1.33324i	−0.167951	+	0.985795i	$0.553715\pi$
$84$		0			0
$85$	−58.9724	−	102.143i	−0.693793	−	1.20168i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	35.6570	+	20.5866i	0.400640	+	0.231310i	0.686760	−	0.726884i	$-0.259032\pi$
$89$	35.6570	+	20.5866i	0.400640	+	0.231310i	−0.286120	+	0.958194i	$0.592366\pi$
$90$		0			0
$91$	71.1560	+	41.0819i	0.781934	+	0.451450i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.09368	+	67.8869i	0.0115125	+	0.714599i
$96$		0			0
$97$	43.8518	+	25.3178i	0.452080	+	0.261009i	0.708708	−	0.705502i	$-0.249278\pi$
$97$	43.8518	+	25.3178i	0.452080	+	0.261009i	−0.256628	+	0.966510i	$0.582612\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	145.036			1.43600			0.717998	−	0.696045i	$-0.245059\pi$
$101$	145.036			1.43600			0.717998	+	0.696045i	$0.245059\pi$
$102$		0			0
$103$	120.942	+	69.8258i	1.17419	+	0.677920i	0.954664	−	0.297686i	$-0.0962149\pi$
$103$	120.942	+	69.8258i	1.17419	+	0.677920i	0.219528	+	0.975606i	$0.429548\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	24.8950i		−	0.232664i	−0.993210	−	0.116332i	$-0.962886\pi$
$107$		−	24.8950i		−	0.232664i	0.993210	−	0.116332i	$-0.0371136\pi$
$108$		0			0
$109$	−35.6302	+	20.5711i	−0.326882	+	0.188726i	−0.654456	−	0.756100i	$-0.727102\pi$
$109$	−35.6302	+	20.5711i	−0.326882	+	0.188726i	0.327574	+	0.944826i	$0.393769\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	65.6579	+	37.9076i	0.581043	+	0.335465i	0.761548	−	0.648109i	$-0.224440\pi$
$113$	65.6579	+	37.9076i	0.581043	+	0.335465i	−0.180505	+	0.983574i	$0.557773\pi$
$114$		0			0
$115$	60.0397	+	103.992i	0.522084	+	0.904277i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−69.9502	+	121.157i	−0.587817	+	1.01813i
$120$		0			0
$121$	−87.7936	+	152.063i	−0.725567	+	1.25672i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	133.041			1.06433
$126$		0			0
$127$	−6.68080	−	3.85716i	−0.0526047	−	0.0303714i	0.473467	−	0.880812i	$-0.343002\pi$
$127$	−6.68080	−	3.85716i	−0.0526047	−	0.0303714i	−0.526072	+	0.850440i	$0.676336\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	104.641			0.798784			0.399392	−	0.916780i	$-0.369221\pi$
$131$	104.641			0.798784			0.399392	+	0.916780i	$0.369221\pi$
$132$		0			0
$133$	69.0873	−	41.3856i	0.519454	−	0.311170i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−226.092			−1.65031			−0.825154	−	0.564907i	$-0.808912\pi$
$137$	−226.092			−1.65031			−0.825154	+	0.564907i	$0.808912\pi$
$138$		0			0
$139$	54.3637	+	94.1607i	0.391106	+	0.677415i	0.992596	−	0.121465i	$-0.0387593\pi$
$139$	54.3637	+	94.1607i	0.391106	+	0.677415i	−0.601490	+	0.798880i	$0.705426\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	289.107	+	166.916i	2.02173	+	1.16725i
$144$		0			0
$145$			1.28011i			0.00882834i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−84.5164			−0.567224			−0.283612	−	0.958939i	$-0.591533\pi$
$149$	−84.5164			−0.567224			−0.283612	+	0.958939i	$0.591533\pi$
$150$		0			0
$151$	−129.120	+	74.5477i	−0.855102	+	0.493694i	−0.862369	−	0.506280i	$-0.831020\pi$
$151$	−129.120	+	74.5477i	−0.855102	+	0.493694i	0.00726680	+	0.999974i	$0.497687\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−155.944	−	90.0344i	−1.00609	−	0.580867i
$156$		0			0
$157$	−223.781			−1.42536			−0.712680	−	0.701490i	$-0.752519\pi$
$157$	−223.781			−1.42536			−0.712680	+	0.701490i	$0.752519\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	71.2162	−	123.350i	0.442336	−	0.766149i
$162$		0			0
$163$	280.953			1.72364			0.861818	−	0.507218i	$-0.169326\pi$
$163$	280.953			1.72364			0.861818	+	0.507218i	$0.169326\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−78.8061	+	45.4987i	−0.471893	+	0.272448i	−0.717032	−	0.697040i	$-0.754500\pi$
$167$	−78.8061	+	45.4987i	−0.471893	+	0.272448i	0.245139	+	0.969488i	$0.421166\pi$
$168$		0			0
$169$	103.377	−	179.054i	0.611698	−	1.05949i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	157.254	+	90.7909i	0.908985	+	0.524803i	0.880105	−	0.474780i	$-0.157472\pi$
$173$	157.254	+	90.7909i	0.908985	+	0.524803i	0.0288808	+	0.999583i	$0.490806\pi$
$174$		0			0
$175$	−25.9202	−	44.8952i	−0.148116	−	0.256544i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			285.526i			1.59512i	0.603241	+	0.797559i	$0.293876\pi$
$179$			285.526i			1.59512i	−0.603241	+	0.797559i	$0.706124\pi$
$180$		0			0
$181$	−44.5028	−	25.6937i	−0.245872	−	0.141954i	0.372001	−	0.928232i	$-0.378672\pi$
$181$	−44.5028	−	25.6937i	−0.245872	−	0.141954i	−0.617873	+	0.786278i	$0.712005\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	94.4554i		−	0.510570i
$186$		0			0
$187$	−284.208	+	492.263i	−1.51983	+	2.63242i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−186.020	−	322.197i	−0.973929	−	1.68689i	−0.683424	−	0.730021i	$-0.739510\pi$
$191$	−186.020	−	322.197i	−0.973929	−	1.68689i	−0.290505	−	0.956874i	$-0.593823\pi$
$192$		0			0
$193$		−	83.2048i		−	0.431113i	−0.976491	−	0.215556i	$-0.930843\pi$
$193$		−	83.2048i		−	0.431113i	0.976491	−	0.215556i	$-0.0691565\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	211.557			1.07390			0.536948	−	0.843615i	$-0.319577\pi$
$197$	211.557			1.07390			0.536948	+	0.843615i	$0.319577\pi$
$198$		0			0
$199$	137.799	−	238.674i	0.692456	−	1.19937i	−0.278575	−	0.960415i	$-0.589862\pi$
$199$	137.799	−	238.674i	0.692456	−	1.19937i	0.971031	−	0.238954i	$-0.0768046\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.31498	−	0.759201i	0.00647771	−	0.00373991i
$204$		0			0
$205$			142.639i			0.695801i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	280.702	−	168.150i	1.34307	−	0.804544i
$210$		0			0
$211$			347.624i			1.64751i	0.566949	+	0.823753i	$0.308124\pi$
$211$			347.624i			1.64751i	−0.566949	+	0.823753i	$0.691876\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	35.4675	−	61.4316i	0.164965	−	0.285728i
$216$		0			0
$217$			213.589i			0.984280i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	554.080	+	319.898i	2.50715	+	1.44750i
$222$		0			0
$223$	−114.827	−	66.2954i	−0.514919	−	0.297289i	0.219934	−	0.975515i	$-0.429416\pi$
$223$	−114.827	−	66.2954i	−0.514919	−	0.297289i	−0.734853	+	0.678226i	$0.762749\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	146.854	−	84.7865i	0.646936	−	0.373509i	−0.140345	−	0.990103i	$-0.544821\pi$
$227$	146.854	−	84.7865i	0.646936	−	0.373509i	0.787281	+	0.616594i	$0.211488\pi$
$228$		0			0
$229$	−195.308	+	338.283i	−0.852872	+	1.47722i	0.0257340	+	0.999669i	$0.491808\pi$
$229$	−195.308	+	338.283i	−0.852872	+	1.47722i	−0.878606	+	0.477548i	$0.841526\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	55.9875	+	96.9731i	0.240290	+	0.416194i	0.960797	−	0.277254i	$-0.0894243\pi$
$233$	55.9875	+	96.9731i	0.240290	+	0.416194i	−0.720507	+	0.693447i	$0.756091\pi$
$234$		0			0
$235$	18.4311			0.0784302
$236$		0			0
$237$		0			0
$238$		0			0
$239$	78.8759	−	136.617i	0.330025	−	0.571619i	−0.652492	−	0.757796i	$-0.726276\pi$
$239$	78.8759	−	136.617i	0.330025	−	0.571619i	0.982516	+	0.186177i	$0.0596096\pi$
$240$		0			0
$241$			360.926i			1.49762i	0.662787	+	0.748808i	$0.269374\pi$
$241$			360.926i			1.49762i	−0.662787	+	0.748808i	$0.730626\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−55.4489	+	96.0403i	−0.226322	+	0.392001i
$246$		0			0
$247$	−189.266	−	315.952i	−0.766257	−	1.27916i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	95.8755	−	166.061i	0.381974	−	0.661598i	−0.609370	−	0.792886i	$-0.708578\pi$
$251$	95.8755	−	166.061i	0.381974	−	0.661598i	0.991344	+	0.131287i	$0.0419111\pi$
$252$		0			0
$253$	289.352	−	501.172i	1.14368	−	1.98092i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−251.132	+	144.991i	−0.977169	+	0.564169i	−0.901414	−	0.432958i	$-0.857470\pi$
$257$	−251.132	+	144.991i	−0.977169	+	0.564169i	−0.0757546	+	0.997126i	$0.524137\pi$
$258$		0			0
$259$	−97.0281	+	56.0192i	−0.374626	+	0.216290i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	29.4850	−	51.0695i	0.112110	−	0.194181i	−0.804511	−	0.593938i	$-0.797572\pi$
$263$	29.4850	−	51.0695i	0.112110	−	0.194181i	0.916621	+	0.399757i	$0.130906\pi$
$264$		0			0
$265$	72.7073	+	41.9776i	0.274367	+	0.158406i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	58.1708	−	33.5849i	0.216248	−	0.124851i	−0.387964	−	0.921675i	$-0.626821\pi$
$269$	58.1708	−	33.5849i	0.216248	−	0.124851i	0.604212	+	0.796824i	$0.293488\pi$
$270$		0			0
$271$	−20.1126	−	34.8360i	−0.0742162	−	0.128546i	0.826529	−	0.562894i	$-0.190312\pi$
$271$	−20.1126	−	34.8360i	−0.0742162	−	0.128546i	−0.900745	+	0.434348i	$0.856979\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−105.314	−	182.409i	−0.382960	−	0.663306i
$276$		0			0
$277$	−62.3160	−	107.935i	−0.224968	−	0.389655i	0.731342	−	0.682011i	$-0.238894\pi$
$277$	−62.3160	−	107.935i	−0.224968	−	0.389655i	−0.956310	+	0.292355i	$0.905561\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			502.268i			1.78743i	0.448636	+	0.893715i	$0.351910\pi$
$281$			502.268i			1.78743i	−0.448636	+	0.893715i	$0.648090\pi$
$282$		0			0
$283$	−250.984			−0.886870			−0.443435	−	0.896306i	$-0.646240\pi$
$283$	−250.984			−0.886870			−0.443435	+	0.896306i	$0.646240\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	146.524	−	84.5958i	0.510538	−	0.294759i
$288$		0			0
$289$	−400.191	+	693.150i	−1.38474	+	2.39844i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	29.5654	+	17.0696i	0.100906	+	0.0582580i	0.549604	−	0.835426i	$-0.314779\pi$
$293$	29.5654	+	17.0696i	0.100906	+	0.0582580i	−0.448698	+	0.893684i	$0.648112\pi$
$294$		0			0
$295$			94.9476i			0.321856i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−564.108	−	325.688i	−1.88665	−	1.08926i
$300$		0			0
$301$	−84.1397			−0.279534
$302$		0			0
$303$		0			0
$304$		0			0
$305$	161.131			0.528299
$306$		0			0
$307$	−174.403	+	100.692i	−0.568088	+	0.327986i	−0.756385	−	0.654126i	$-0.773036\pi$
$307$	−174.403	+	100.692i	−0.568088	+	0.327986i	0.188297	+	0.982112i	$0.439703\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.4402	+	26.7432i	−0.0496470	+	0.0859911i	−0.889781	−	0.456388i	$-0.849143\pi$
$311$	−15.4402	+	26.7432i	−0.0496470	+	0.0859911i	0.840134	+	0.542379i	$0.182476\pi$
$312$		0			0
$313$	−63.7986			−0.203830			−0.101915	−	0.994793i	$-0.532497\pi$
$313$	−63.7986			−0.203830			−0.101915	+	0.994793i	$0.532497\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	315.547i		−	0.995417i	−0.867344	−	0.497708i	$-0.834175\pi$
$317$		−	315.547i		−	0.995417i	0.867344	−	0.497708i	$-0.165825\pi$
$318$		0			0
$319$	5.34275	−	3.08464i	0.0167484	−	0.00966971i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	537.972	−	322.262i	1.66555	−	0.997717i
$324$		0			0
$325$	−205.316	+	118.539i	−0.631741	+	0.364736i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.9310	−	18.9331i	−0.0332250	−	0.0575474i
$330$		0			0
$331$	67.2074	−	38.8022i	0.203044	−	0.117227i	−0.395031	−	0.918668i	$-0.629266\pi$
$331$	67.2074	−	38.8022i	0.203044	−	0.117227i	0.598074	+	0.801441i	$0.295933\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	246.284	−	142.192i	0.735177	−	0.424455i
$336$		0			0
$337$		−	282.135i		−	0.837196i	−0.908172	−	0.418598i	$-0.862522\pi$
$337$		−	282.135i		−	0.837196i	0.908172	−	0.418598i	$-0.137478\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			867.812i			2.54490i
$342$		0			0
$343$	339.236			0.989027
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−290.908			−0.838351			−0.419175	−	0.907905i	$-0.637681\pi$
$347$	−290.908			−0.838351			−0.419175	+	0.907905i	$0.637681\pi$
$348$		0			0
$349$	−195.221	−	338.133i	−0.559373	−	0.968863i	−0.997549	−	0.0699734i	$-0.977709\pi$
$349$	−195.221	−	338.133i	−0.559373	−	0.968863i	0.438176	−	0.898889i	$-0.355625\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	229.262	+	397.093i	0.649467	+	1.12491i	0.983250	+	0.182260i	$0.0583412\pi$
$353$	229.262	+	397.093i	0.649467	+	1.12491i	−0.333784	+	0.942650i	$0.608325\pi$
$354$		0			0
$355$	−6.39956	+	3.69479i	−0.0180269	+	0.0104079i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	34.1410	+	59.1339i	0.0951002	+	0.164718i	0.909650	−	0.415375i	$-0.136350\pi$
$359$	34.1410	+	59.1339i	0.0951002	+	0.164718i	−0.814550	+	0.580093i	$0.803016\pi$
$360$		0			0
$361$	−360.813	+	11.6287i	−0.999481	+	0.0322123i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−64.6723	−	112.016i	−0.177184	−	0.306892i
$366$		0			0
$367$	452.642			1.23336			0.616678	−	0.787216i	$-0.288478\pi$
$367$	452.642			1.23336			0.616678	+	0.787216i	$0.288478\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	99.5836i		−	0.268419i
$372$		0			0
$373$	180.571	+	104.253i	0.484105	+	0.279498i	0.722126	−	0.691762i	$-0.243165\pi$
$373$	180.571	+	104.253i	0.484105	+	0.279498i	−0.238021	+	0.971260i	$0.576499\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.47200	−	6.01368i	−0.00920955	−	0.0159514i
$378$		0			0
$379$		−	26.5246i		−	0.0699858i	−0.999388	−	0.0349929i	$-0.988859\pi$
$379$		−	26.5246i		−	0.0699858i	0.999388	−	0.0349929i	$-0.0111409\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			114.469i			0.298874i	0.988771	+	0.149437i	$0.0477461\pi$
$383$			114.469i			0.298874i	−0.988771	+	0.149437i	$0.952254\pi$
$384$		0			0
$385$	130.426	−	225.905i	0.338769	−	0.586765i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−117.495			−0.302045			−0.151022	−	0.988530i	$-0.548257\pi$
$389$	−117.495			−0.302045			−0.151022	+	0.988530i	$0.548257\pi$
$390$		0			0
$391$	554.548	−	960.506i	1.41828	−	2.45654i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−222.036	−	128.193i	−0.562118	−	0.324539i
$396$		0			0
$397$	281.926	+	488.311i	0.710142	+	1.23000i	0.964804	+	0.262971i	$0.0847024\pi$
$397$	281.926	+	488.311i	0.710142	+	1.23000i	−0.254662	+	0.967030i	$0.581964\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			567.108i			1.41423i	0.707097	+	0.707117i	$0.250005\pi$
$401$			567.108i			1.41423i	−0.707097	+	0.707117i	$0.749995\pi$
$402$		0			0
$403$	976.790			2.42380
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−394.226	+	227.606i	−0.968613	+	0.559229i
$408$		0			0
$409$	151.753	−	87.6148i	0.371035	−	0.214217i	−0.302876	−	0.953030i	$-0.597947\pi$
$409$	151.753	−	87.6148i	0.371035	−	0.214217i	0.673910	+	0.738813i	$0.264613\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	97.5337	−	56.3111i	0.236159	−	0.136346i
$414$		0			0
$415$	−228.305	−	395.436i	−0.550133	−	0.952858i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−37.2462	+	64.5122i	−0.0888930	+	0.153967i	−0.907043	−	0.421037i	$-0.861666\pi$
$419$	−37.2462	+	64.5122i	−0.0888930	+	0.153967i	0.818150	+	0.575004i	$0.195000\pi$
$420$		0			0
$421$	53.0298	+	30.6168i	0.125962	+	0.0727240i	0.561657	−	0.827370i	$-0.310164\pi$
$421$	53.0298	+	30.6168i	0.125962	+	0.0727240i	−0.435695	+	0.900094i	$0.643497\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−201.837	−	349.591i	−0.474910	−	0.822567i
$426$		0			0
$427$	−95.5630	−	165.520i	−0.223801	−	0.387635i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−462.163	−	266.830i	−1.07230	−	0.619095i	−0.143494	−	0.989651i	$-0.545834\pi$
$431$	−462.163	−	266.830i	−1.07230	−	0.619095i	−0.928810	+	0.370557i	$0.879167\pi$
$432$		0			0
$433$	−338.448	−	195.403i	−0.781635	−	0.451277i	0.0553741	−	0.998466i	$-0.482365\pi$
$433$	−338.448	−	195.403i	−0.781635	−	0.451277i	−0.837010	+	0.547188i	$0.815698\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−547.708	+	328.095i	−1.25334	+	0.750789i
$438$		0			0
$439$	682.447	+	394.011i	1.55455	+	0.897519i	0.997762	+	0.0668625i	$0.0212989\pi$
$439$	682.447	+	394.011i	1.55455	+	0.897519i	0.556786	+	0.830656i	$0.312034\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−510.089			−1.15144			−0.575721	−	0.817646i	$-0.695279\pi$
$443$	−510.089			−1.15144			−0.575721	+	0.817646i	$0.695279\pi$
$444$		0			0
$445$	−127.419	−	73.5652i	−0.286334	−	0.165315i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3.02812i		−	0.00674414i	−0.999994	−	0.00337207i	$-0.998927\pi$
$449$		−	3.02812i		−	0.00674414i	0.999994	−	0.00337207i	$-0.00107336\pi$
$450$		0			0
$451$	595.329	−	343.713i	1.32002	−	0.762114i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−254.273	−	146.805i	−0.558842	−	0.322648i
$456$		0			0
$457$	−431.036	−	746.576i	−0.943186	−	1.63365i	−0.759343	−	0.650691i	$-0.774479\pi$
$457$	−431.036	−	746.576i	−0.943186	−	1.63365i	−0.183843	−	0.982956i	$-0.558854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	197.275	−	341.690i	0.427928	−	0.741193i	−0.568761	−	0.822503i	$-0.692577\pi$
$461$	197.275	−	341.690i	0.427928	−	0.741193i	0.996689	+	0.0813098i	$0.0259103\pi$
$462$		0			0
$463$	−64.8349	+	112.297i	−0.140032	+	0.242543i	−0.927509	−	0.373802i	$-0.878054\pi$
$463$	−64.8349	+	112.297i	−0.140032	+	0.242543i	0.787476	+	0.616345i	$0.211387\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−126.290			−0.270428			−0.135214	−	0.990816i	$-0.543172\pi$
$467$	−126.290			−0.270428			−0.135214	+	0.990816i	$0.543172\pi$
$468$		0			0
$469$	−292.130	−	168.662i	−0.622879	−	0.359620i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−341.860			−0.722748
$474$		0			0
$475$	3.74319	+	232.347i	0.00788041	+	0.489152i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	280.513			0.585623			0.292811	−	0.956170i	$-0.405409\pi$
$479$	280.513			0.585623			0.292811	+	0.956170i	$0.405409\pi$
$480$		0			0
$481$	256.189	+	443.732i	0.532616	+	0.922519i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−156.703	−	90.4723i	−0.323098	−	0.186541i
$486$		0			0
$487$			783.264i			1.60835i	0.594396	+	0.804173i	$0.297391\pi$
$487$			783.264i			1.60835i	−0.594396	+	0.804173i	$0.702609\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	204.495			0.416486			0.208243	−	0.978077i	$-0.433225\pi$
$491$	204.495			0.416486			0.208243	+	0.978077i	$0.433225\pi$
$492$		0			0
$493$	10.2395	−	5.91177i	0.0207698	−	0.0119914i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.59085	+	4.38258i	0.0152733	+	0.00881807i
$498$		0			0
$499$	505.238			1.01250			0.506251	−	0.862386i	$-0.331031\pi$
$499$	505.238			1.01250			0.506251	+	0.862386i	$0.331031\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	340.083	−	589.040i	0.676108	−	1.17105i	−0.300035	−	0.953928i	$-0.596998\pi$
$503$	340.083	−	589.040i	0.676108	−	1.17105i	0.976144	−	0.217126i	$-0.0696683\pi$
$504$		0			0
$505$	−518.279			−1.02630
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−389.356	+	224.795i	−0.764943	+	0.441640i	−0.831068	−	0.556171i	$-0.812270\pi$
$509$	−389.356	+	224.795i	−0.764943	+	0.441640i	0.0661247	+	0.997811i	$0.478936\pi$
$510$		0			0
$511$	−76.7111	+	132.868i	−0.150120	+	0.260015i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−432.181	−	249.520i	−0.839186	−	0.484504i
$516$		0			0
$517$	−44.4128	−	76.9253i	−0.0859049	−	0.148792i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	725.604i		−	1.39271i	−0.717696	−	0.696357i	$-0.754803\pi$
$521$		−	725.604i		−	1.39271i	0.717696	−	0.696357i	$-0.245197\pi$
$522$		0			0
$523$	−456.195	−	263.384i	−0.872265	−	0.503602i	−0.00416462	−	0.999991i	$-0.501326\pi$
$523$	−456.195	−	263.384i	−0.872265	−	0.503602i	−0.868100	+	0.496389i	$0.834659\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1663.18i			3.15594i
$528$		0			0
$529$	−300.085	+	519.762i	−0.567268	+	0.982536i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−386.876	−	670.089i	−0.725846	−	1.25720i
$534$		0			0
$535$			88.9614i			0.166283i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	534.454			0.991565
$540$		0			0
$541$	−241.845	+	418.888i	−0.447034	+	0.774286i	−0.998191	−	0.0601157i	$-0.980853\pi$
$541$	−241.845	+	418.888i	−0.447034	+	0.774286i	0.551157	+	0.834401i	$0.314186\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	127.323	−	73.5100i	0.233620	−	0.134881i
$546$		0			0
$547$		−	1040.41i		−	1.90203i	−0.309151	−	0.951013i	$-0.600045\pi$
$547$		−	1040.41i		−	1.90203i	0.309151	−	0.951013i	$-0.399955\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.80542	+	0.109638i	−0.0123510	+	0.000198980i
$552$		0			0
$553$			304.112i			0.549932i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−321.105	+	556.170i	−0.576490	+	0.998510i	0.419388	+	0.907807i	$0.362245\pi$
$557$	−321.105	+	556.170i	−0.576490	+	0.998510i	−0.995878	+	0.0907030i	$0.971089\pi$
$558$		0			0
$559$			384.790i			0.688354i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.8501	−	10.3058i	−0.0317053	−	0.0183051i	0.484064	−	0.875033i	$-0.339160\pi$
$563$	−17.8501	−	10.3058i	−0.0317053	−	0.0183051i	−0.515769	+	0.856728i	$0.672494\pi$
$564$		0			0
$565$	−234.626	−	135.461i	−0.415267	−	0.239755i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−414.099	+	239.080i	−0.727767	+	0.420176i	−0.817605	−	0.575780i	$-0.804698\pi$
$569$	−414.099	+	239.080i	−0.727767	+	0.420176i	0.0898379	+	0.995956i	$0.471365\pi$
$570$		0			0
$571$	−154.725	+	267.991i	−0.270972	+	0.469337i	−0.969111	−	0.246625i	$-0.920678\pi$
$571$	−154.725	+	267.991i	−0.270972	+	0.469337i	0.698139	+	0.715962i	$0.254012\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	205.489	+	355.918i	0.357373	+	0.618988i
$576$		0			0
$577$	684.624			1.18652			0.593262	−	0.805010i	$-0.297840\pi$
$577$	684.624			1.18652			0.593262	+	0.805010i	$0.297840\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−270.804	+	469.047i	−0.466100	+	0.807310i
$582$		0			0
$583$		−	404.608i		−	0.694011i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	110.635	−	191.625i	0.188475	−	0.326447i	−0.756267	−	0.654263i	$-0.772979\pi$
$587$	110.635	−	191.625i	0.188475	−	0.326447i	0.944742	+	0.327815i	$0.106312\pi$
$588$		0			0
$589$	465.292	−	836.754i	0.789970	−	1.42064i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	173.199	−	299.989i	0.292072	−	0.505883i	−0.682228	−	0.731140i	$-0.738989\pi$
$593$	173.199	−	299.989i	0.292072	−	0.505883i	0.974299	+	0.225257i	$0.0723220\pi$
$594$		0			0
$595$	249.964	−	432.951i	0.420108	−	0.727649i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	975.236	−	563.053i	1.62811	−	0.939988i	0.643451	−	0.765488i	$-0.277502\pi$
$599$	975.236	−	563.053i	1.62811	−	0.939988i	0.984657	−	0.174501i	$-0.0558311\pi$
$600$		0			0
$601$	14.5616	−	8.40717i	0.0242290	−	0.0139886i	−0.487837	−	0.872935i	$-0.662214\pi$
$601$	14.5616	−	8.40717i	0.0242290	−	0.0139886i	0.512066	+	0.858946i	$0.328880\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	313.727	−	543.391i	0.518557	−	0.898167i
$606$		0			0
$607$	542.819	+	313.396i	0.894265	+	0.516304i	0.875335	−	0.483517i	$-0.160641\pi$
$607$	542.819	+	313.396i	0.894265	+	0.516304i	0.0189295	+	0.999821i	$0.493974\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−86.5854	+	49.9901i	−0.141711	+	0.0818169i
$612$		0			0
$613$	−127.095	−	220.136i	−0.207333	−	0.359112i	0.743540	−	0.668691i	$-0.233145\pi$
$613$	−127.095	−	220.136i	−0.207333	−	0.359112i	−0.950874	+	0.309579i	$0.899812\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−308.509	−	534.353i	−0.500015	−	0.866051i	−1.00000		1.70857e-5i	$-0.999995\pi$
$617$	−308.509	−	534.353i	−0.500015	−	0.866051i	0.499985	−	0.866034i	$-0.333339\pi$
$618$		0			0
$619$	−373.018	−	646.086i	−0.602614	−	1.04376i	−0.992424	−	0.122863i	$-0.960793\pi$
$619$	−373.018	−	646.086i	−0.602614	−	1.04376i	0.389810	−	0.920895i	$-0.372541\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			174.519i			0.280127i
$624$		0			0
$625$	−169.658			−0.271453
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−755.542	+	436.212i	−1.20118	+	0.693501i
$630$		0			0
$631$	423.895	−	734.207i	0.671782	−	1.16356i	−0.305616	−	0.952155i	$-0.598862\pi$
$631$	423.895	−	734.207i	0.671782	−	1.16356i	0.977398	−	0.211406i	$-0.0678043\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.8736	+	13.7834i	0.0375962	+	0.0217062i
$636$		0			0
$637$		−	601.569i		−	0.944379i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	637.670	+	368.159i	0.994804	+	0.574351i	0.906707	−	0.421761i	$-0.138588\pi$
$641$	637.670	+	368.159i	0.994804	+	0.574351i	0.0880974	+	0.996112i	$0.471921\pi$
$642$		0			0
$643$	−260.391			−0.404963			−0.202481	−	0.979286i	$-0.564901\pi$
$643$	−260.391			−0.404963			−0.202481	+	0.979286i	$0.564901\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	692.664			1.07058			0.535289	−	0.844669i	$-0.320203\pi$
$647$	692.664			1.07058			0.535289	+	0.844669i	$0.320203\pi$
$648$		0			0
$649$	396.280	−	228.792i	0.610600	−	0.352530i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−316.724	+	548.582i	−0.485029	+	0.840095i	−0.999852	−	0.0172016i	$-0.994524\pi$
$653$	−316.724	+	548.582i	−0.485029	+	0.840095i	0.514823	+	0.857296i	$0.327858\pi$
$654$		0			0
$655$	−373.930			−0.570885
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	666.288i		−	1.01106i	−0.862809	−	0.505529i	$-0.831297\pi$
$659$		−	666.288i		−	1.01106i	0.862809	−	0.505529i	$-0.168703\pi$
$660$		0			0
$661$	40.3823	−	23.3147i	0.0610927	−	0.0352719i	−0.469143	−	0.883122i	$-0.655437\pi$
$661$	40.3823	−	23.3147i	0.0610927	−	0.0352719i	0.530235	+	0.847851i	$0.322104\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−246.881	+	147.890i	−0.371249	+	0.222390i
$666$		0			0
$667$	−10.4248	+	6.01876i	−0.0156294	+	0.00902364i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−388.273	−	672.509i	−0.578648	−	1.00225i
$672$		0			0
$673$	−17.1127	+	9.88002i	−0.0254275	+	0.0146806i	−0.512660	−	0.858592i	$-0.671340\pi$
$673$	−17.1127	+	9.88002i	−0.0254275	+	0.0146806i	0.487232	+	0.873272i	$0.338006\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.4947	+	21.6476i	−0.0553836	+	0.0319757i	−0.527436	−	0.849595i	$-0.676847\pi$
$677$	−37.4947	+	21.6476i	−0.0553836	+	0.0319757i	0.472053	+	0.881570i	$0.343513\pi$
$678$		0			0
$679$			214.628i			0.316094i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1317.95i		−	1.92965i	−0.262895	−	0.964825i	$-0.584677\pi$
$683$		−	1317.95i		−	1.92965i	0.262895	−	0.964825i	$-0.415323\pi$
$684$		0			0
$685$	807.932			1.17946
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−455.418			−0.660984
$690$		0			0
$691$	−556.438	−	963.778i	−0.805264	−	1.39476i	−0.916113	−	0.400921i	$-0.868690\pi$
$691$	−556.438	−	963.778i	−0.805264	−	1.39476i	0.110848	−	0.993837i	$-0.464643\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−194.267	−	336.480i	−0.279520	−	0.484143i
$696$		0			0
$697$	1140.96	−	658.734i	1.63696	−	0.945099i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	312.796	+	541.779i	0.446214	+	0.772866i	0.998136	−	0.0610303i	$-0.0194386\pi$
$701$	312.796	+	541.779i	0.446214	+	0.772866i	−0.551922	+	0.833896i	$0.686105\pi$
$702$		0			0
$703$	502.152	−	8.08985i	0.714299	−	0.0115076i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	307.379	+	532.395i	0.434765	+	0.753035i
$708$		0			0
$709$	−1317.13			−1.85773			−0.928863	−	0.370424i	$-0.879212\pi$
$709$	−1317.13			−1.85773			−0.928863	+	0.370424i	$0.879212\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1693.28i		−	2.37487i
$714$		0			0
$715$	−1033.11	−	596.468i	−1.44491	−	0.834221i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−189.686	−	328.545i	−0.263819	−	0.456947i	0.703435	−	0.710760i	$-0.251649\pi$
$719$	−189.686	−	328.545i	−0.263819	−	0.456947i	−0.967253	+	0.253812i	$0.918315\pi$
$720$		0			0
$721$			591.936i			0.820993i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			4.38125i			0.00604310i
$726$		0			0
$727$	279.535	−	484.169i	0.384505	−	0.665982i	−0.607196	−	0.794552i	$-0.707706\pi$
$727$	279.535	−	484.169i	0.384505	−	0.665982i	0.991700	+	0.128571i	$0.0410389\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−655.182			−0.896282
$732$		0			0
$733$	−700.738	+	1213.71i	−0.955986	+	1.65582i	−0.223891	+	0.974614i	$0.571876\pi$
$733$	−700.738	+	1213.71i	−0.955986	+	1.65582i	−0.732095	+	0.681202i	$0.761457\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1186.93	−	685.273i	−1.61048	−	0.929814i
$738$		0			0
$739$	−149.513	−	258.964i	−0.202318	−	0.350425i	0.746957	−	0.664872i	$-0.231514\pi$
$739$	−149.513	−	258.964i	−0.202318	−	0.350425i	−0.949275	+	0.314448i	$0.898181\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			205.062i			0.275992i	0.990433	+	0.137996i	$0.0440661\pi$
$743$			205.062i			0.275992i	−0.990433	+	0.137996i	$0.955934\pi$
$744$		0			0
$745$	302.016			0.405391
$746$		0			0
$747$		0			0
$748$		0			0
$749$	91.3844	−	52.7608i	0.122009	−	0.0704417i
$750$		0			0
$751$	264.425	−	152.666i	0.352097	−	0.203284i	−0.313511	−	0.949584i	$-0.601505\pi$
$751$	264.425	−	152.666i	0.352097	−	0.203284i	0.665609	+	0.746301i	$0.268172\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	461.407	−	266.393i	0.611135	−	0.352839i
$756$		0			0
$757$	−178.879	−	309.827i	−0.236300	−	0.409283i	0.723350	−	0.690482i	$-0.242601\pi$
$757$	−178.879	−	309.827i	−0.236300	−	0.409283i	−0.959650	+	0.281199i	$0.909268\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	133.599	−	231.399i	0.175557	−	0.304073i	−0.764797	−	0.644271i	$-0.777161\pi$
$761$	133.599	−	231.399i	0.175557	−	0.304073i	0.940354	+	0.340198i	$0.110494\pi$
$762$		0			0
$763$	−151.024	−	87.1940i	−0.197935	−	0.114278i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−257.523	−	446.044i	−0.335754	−	0.581543i
$768$		0			0
$769$	395.797	+	685.541i	0.514691	+	0.891471i	0.999855	+	0.0170473i	$0.00542659\pi$
$769$	395.797	+	685.541i	0.514691	+	0.891471i	−0.485164	+	0.874423i	$0.661240\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	232.793	+	134.403i	0.301155	+	0.173872i	0.642962	−	0.765898i	$-0.277705\pi$
$773$	232.793	+	134.403i	0.301155	+	0.173872i	−0.341807	+	0.939770i	$0.611039\pi$
$774$		0			0
$775$	−533.728	−	308.148i	−0.688681	−	0.397610i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−758.311	+	12.2167i	−0.973441	+	0.0156825i
$780$		0			0
$781$	30.8416	+	17.8064i	0.0394899	+	0.0227995i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	799.674			1.01869
$786$		0			0
$787$	−659.045	−	380.500i	−0.837414	−	0.483481i	0.0189706	−	0.999820i	$-0.493961\pi$
$787$	−659.045	−	380.500i	−0.837414	−	0.483481i	−0.856384	+	0.516339i	$0.827294\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			321.355i			0.406264i
$792$		0			0
$793$	−756.961	+	437.031i	−0.954553	+	0.551112i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	69.0764	+	39.8813i	0.0866705	+	0.0500392i	0.542709	−	0.839921i	$-0.317399\pi$
$797$	69.0764	+	39.8813i	0.0866705	+	0.0500392i	−0.456038	+	0.889960i	$0.650732\pi$
$798$		0			0
$799$	−85.1182	−	147.429i	−0.106531	−	0.184517i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−311.677	+	539.841i	−0.388141	+	0.672280i
$804$		0			0
$805$	−254.488	+	440.786i	−0.316134	+	0.547561i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−783.760			−0.968800			−0.484400	−	0.874847i	$-0.660962\pi$
$809$	−783.760			−0.968800			−0.484400	+	0.874847i	$0.660962\pi$
$810$		0			0
$811$	−1171.15	−	676.163i	−1.44408	−	0.833740i	−0.445961	−	0.895052i	$-0.647138\pi$
$811$	−1171.15	−	676.163i	−1.44408	−	0.833740i	−0.998119	+	0.0613126i	$0.980471\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1003.97			−1.23187
$816$		0			0
$817$	329.625	+	183.294i	0.403458	+	0.224350i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−710.231			−0.865080			−0.432540	−	0.901615i	$-0.642383\pi$
$821$	−710.231			−0.865080			−0.432540	+	0.901615i	$0.642383\pi$
$822$		0			0
$823$	324.990	+	562.900i	0.394885	+	0.683961i	0.993087	−	0.117385i	$-0.0374511\pi$
$823$	324.990	+	562.900i	0.394885	+	0.683961i	−0.598201	+	0.801346i	$0.704118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−414.480	−	239.300i	−0.501185	−	0.289359i	0.228018	−	0.973657i	$-0.426776\pi$
$827$	−414.480	−	239.300i	−0.501185	−	0.289359i	−0.729203	+	0.684298i	$0.760109\pi$
$828$		0			0
$829$		−	446.712i		−	0.538857i	−0.963020	−	0.269429i	$-0.913165\pi$
$829$		−	446.712i		−	0.538857i	0.963020	−	0.269429i	$-0.0868348\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1024.29			1.22964
$834$		0			0
$835$	281.611	−	162.588i	0.337258	−	0.194716i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−622.238	−	359.249i	−0.741642	−	0.428187i	0.0810237	−	0.996712i	$-0.474181\pi$
$839$	−622.238	−	359.249i	−0.741642	−	0.428187i	−0.822666	+	0.568525i	$0.807514\pi$
$840$		0			0
$841$	840.872			0.999847
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−369.414	+	639.843i	−0.437176	+	0.757211i
$846$		0			0
$847$	−744.256			−0.878696
$848$		0			0
$849$		0			0
$850$		0			0
$851$	769.215	−	444.107i	0.903896	−	0.521865i
$852$		0			0
$853$	8.49178	−	14.7082i	0.00995519	−	0.0172429i	−0.861005	−	0.508597i	$-0.830164\pi$
$853$	8.49178	−	14.7082i	0.00995519	−	0.0172429i	0.870960	+	0.491354i	$0.163498\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−570.153	−	329.178i	−0.665289	−	0.384105i	0.129000	−	0.991645i	$-0.458823\pi$
$857$	−570.153	−	329.178i	−0.665289	−	0.384105i	−0.794289	+	0.607540i	$0.792156\pi$
$858$		0			0
$859$	−397.989	−	689.337i	−0.463317	−	0.802488i	0.535807	−	0.844340i	$-0.320007\pi$
$859$	−397.989	−	689.337i	−0.463317	−	0.802488i	−0.999124	+	0.0418525i	$0.986674\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			540.003i			0.625727i	0.949798	+	0.312864i	$0.101288\pi$
$863$			540.003i			0.625727i	−0.949798	+	0.312864i	$0.898712\pi$
$864$		0			0
$865$	−561.943	−	324.438i	−0.649645	−	0.375073i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1235.61i			1.42187i
$870$		0			0
$871$	−771.328	+	1335.98i	−0.885566	+	1.53384i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	281.959	+	488.367i	0.322239	+	0.558133i
$876$		0			0
$877$		−	1139.15i		−	1.29891i	−0.760399	−	0.649457i	$-0.774996\pi$
$877$		−	1139.15i		−	1.29891i	0.760399	−	0.649457i	$-0.225004\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−168.246			−0.190972			−0.0954861	−	0.995431i	$-0.530441\pi$
$881$	−168.246			−0.190972			−0.0954861	+	0.995431i	$0.530441\pi$
$882$		0			0
$883$	−301.318	+	521.897i	−0.341243	+	0.591050i	−0.984664	−	0.174463i	$-0.944181\pi$
$883$	−301.318	+	521.897i	−0.341243	+	0.591050i	0.643421	+	0.765513i	$0.277515\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1028.20	−	593.629i	1.15918	−	0.669255i	0.208076	−	0.978113i	$-0.433280\pi$
$887$	1028.20	−	593.629i	1.15918	−	0.669255i	0.951108	+	0.308857i	$0.0999466\pi$
$888$		0			0
$889$		−	32.6984i		−	0.0367811i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.57857	+	97.9850i	0.00176772	+	0.109726i
$894$		0			0
$895$		−	1020.32i		−	1.14002i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.02562	−	15.6328i	0.0100396	−	0.0173891i
$900$		0			0
$901$		−	775.441i		−	0.860644i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	159.029	+	91.8155i	0.175723	+	0.101454i
$906$		0			0
$907$	871.834	+	503.354i	0.961229	+	0.554966i	0.896551	−	0.442941i	$-0.146065\pi$
$907$	871.834	+	503.354i	0.961229	+	0.554966i	0.0646777	+	0.997906i	$0.479398\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1111.24	−	641.574i	1.21980	−	0.704253i	0.254926	−	0.966961i	$-0.417949\pi$
$911$	1111.24	−	641.574i	1.21980	−	0.704253i	0.964875	+	0.262708i	$0.0846156\pi$
$912$		0			0
$913$	−1100.28	+	1905.74i	−1.20513	+	2.08734i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	221.769	+	384.114i	0.241841	+	0.418882i
$918$		0			0
$919$	676.693			0.736336			0.368168	−	0.929759i	$-0.379985\pi$
$919$	676.693			0.736336			0.368168	+	0.929759i	$0.379985\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	20.0425	−	34.7147i	0.0217145	−	0.0376107i
$924$		0			0
$925$		−	323.279i		−	0.349491i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	849.514	−	1471.40i	0.914440	−	1.58386i	0.106720	−	0.994289i	$-0.465965\pi$
$929$	849.514	−	1471.40i	0.914440	−	1.58386i	0.807720	−	0.589567i	$-0.200702\pi$
$930$		0			0
$931$	−515.327	−	286.557i	−0.553519	−	0.307794i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1015.61	−	1759.08i	1.08621	−	1.88137i
$936$		0			0
$937$	−440.942	+	763.734i	−0.470589	+	0.815084i	−0.999434	−	0.0336344i	$-0.989292\pi$
$937$	−440.942	+	763.734i	−0.470589	+	0.815084i	0.528845	+	0.848718i	$0.322625\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−951.258	+	549.209i	−1.01090	+	0.583644i	−0.911455	−	0.411399i	$-0.865040\pi$
$941$	−951.258	+	549.209i	−1.01090	+	0.583644i	−0.0994456	+	0.995043i	$0.531707\pi$
$942$		0			0
$943$	−1161.61	+	670.655i	−1.23182	+	0.711193i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−168.557	+	291.949i	−0.177990	+	0.308288i	−0.941192	−	0.337872i	$-0.890293\pi$
$947$	−168.557	+	291.949i	−0.177990	+	0.308288i	0.763202	+	0.646160i	$0.223626\pi$
$948$		0			0
$949$	607.633	+	350.817i	0.640288	+	0.369670i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	178.565	−	103.095i	0.187372	−	0.108179i	−0.403380	−	0.915033i	$-0.632165\pi$
$953$	178.565	−	103.095i	0.187372	−	0.108179i	0.590752	+	0.806854i	$0.298831\pi$
$954$		0			0
$955$	664.737	+	1151.36i	0.696060	+	1.20561i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−479.165	−	829.938i	−0.499650	−	0.865420i
$960$		0			0
$961$	789.105	+	1366.77i	0.821129	+	1.42224i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			297.329i			0.308113i
$966$		0			0
$967$	790.183			0.817148			0.408574	−	0.912725i	$-0.366026\pi$
$967$	790.183			0.817148			0.408574	+	0.912725i	$0.366026\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	696.373	−	402.051i	0.717171	−	0.414059i	−0.0965395	−	0.995329i	$-0.530777\pi$
$971$	696.373	−	402.051i	0.717171	−	0.414059i	0.813711	+	0.581270i	$0.197444\pi$
$972$		0			0
$973$	−230.429	+	399.116i	−0.236824	+	0.410191i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1248.86	+	721.028i	1.27826	+	0.738002i	0.976528	−	0.215391i	$-0.0691027\pi$
$977$	1248.86	+	721.028i	1.27826	+	0.738002i	0.301730	+	0.953394i	$0.402436\pi$
$978$		0			0
$979$			709.071i			0.724281i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1514.90	+	874.626i	1.54109	+	0.889751i	0.998770	+	0.0495825i	$0.0157891\pi$
$983$	1514.90	+	874.626i	1.54109	+	0.889751i	0.542325	+	0.840169i	$0.317544\pi$
$984$		0			0
$985$	−755.992			−0.767505
$986$		0			0
$987$		0			0
$988$		0			0
$989$	667.039			0.674458
$990$		0			0
$991$	306.985	−	177.238i	0.309773	−	0.178847i	−0.337052	−	0.941486i	$-0.609430\pi$
$991$	306.985	−	177.238i	0.309773	−	0.178847i	0.646825	+	0.762639i	$0.276096\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−492.418	+	852.894i	−0.494893	+	0.857180i
$996$		0			0
$997$	1280.29			1.28415			0.642073	−	0.766643i	$-0.278074\pi$
$997$	1280.29			1.28415			0.642073	+	0.766643i	$0.278074\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.bl.a.145.15		80
3.2	odd	2		684.3.bl.a.373.40	yes	80
9.2	odd	6		684.3.s.a.601.26	yes	80
9.7	even	3		2052.3.s.a.829.26		80
19.8	odd	6		2052.3.s.a.901.26		80
57.8	even	6		684.3.s.a.445.26	✓	80
171.65	even	6		684.3.bl.a.673.40	yes	80
171.160	odd	6	inner	2052.3.bl.a.1585.15		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.26	✓	80	57.8	even	6
684.3.s.a.601.26	yes	80	9.2	odd	6
684.3.bl.a.373.40	yes	80	3.2	odd	2
684.3.bl.a.673.40	yes	80	171.65	even	6
2052.3.s.a.829.26		80	9.7	even	3
2052.3.s.a.901.26		80	19.8	odd	6
2052.3.bl.a.145.15		80	1.1	even	1	trivial
2052.3.bl.a.1585.15		80	171.160	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.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.57346 q^{5} +(2.11933 + 3.67079i) q^{7} +O(q^{10})\)	\(q-3.57346 q^{5} +(2.11933 + 3.67079i) q^{7} +(8.61085 + 14.9144i) q^{11} +(16.7874 - 9.69219i) q^{13} +(16.5029 + 28.5838i) q^{17} +(-0.306057 - 18.9975i) q^{19} +(-16.8016 - 29.1011i) q^{23} -12.2304 q^{25} -0.358227i q^{29} +(43.6395 + 25.1953i) q^{31} +(-7.57335 - 13.1174i) q^{35} +26.4325i q^{37} -39.9163i q^{41} +(-9.92526 + 17.1911i) q^{43} -5.15777 q^{47} +(15.5169 - 26.8760i) q^{49} +(-20.3465 - 11.7470i) q^{53} +(-30.7706 - 53.2962i) q^{55} -26.5702i q^{59} -45.0911 q^{61} +(-59.9890 + 34.6347i) q^{65} +(-68.9204 + 39.7912i) q^{67} +(1.79086 - 1.03395i) q^{71} +(18.0979 + 31.3466i) q^{73} +(-36.4985 + 63.2173i) q^{77} +(62.1349 + 35.8736i) q^{79} +(63.8891 + 110.659i) q^{83} +(-58.9724 - 102.143i) q^{85} +(35.6570 + 20.5866i) q^{89} +(71.1560 + 41.0819i) q^{91} +(1.09368 + 67.8869i) q^{95} +(43.8518 + 25.3178i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 80 * q - q^7	\( 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) \) 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more