LMFDB - Embedded newform 2052.3.bl.a.145.6

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			145.6
Character	$\chi$	$=$	2052.145
Dual form			2052.3.bl.a.1585.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.81841 q^{5} +(3.70019 + 6.40891i) q^{7} +O(q^{10})$	$q-6.81841 q^{5} +(3.70019 + 6.40891i) q^{7} +(-2.17728 - 3.77116i) q^{11} +(-4.41327 + 2.54800i) q^{13} +(-7.11551 - 12.3244i) q^{17} +(17.8833 - 6.41783i) q^{19} +(-4.25671 - 7.37283i) q^{23} +21.4907 q^{25} -11.3235i q^{29} +(-8.16773 - 4.71564i) q^{31} +(-25.2294 - 43.6986i) q^{35} -33.2038i q^{37} -0.431133i q^{41} +(-38.1888 + 66.1449i) q^{43} +47.6996 q^{47} +(-2.88279 + 4.99314i) q^{49} +(7.51673 + 4.33978i) q^{53} +(14.8456 + 25.7133i) q^{55} +65.1347i q^{59} +7.37572 q^{61} +(30.0915 - 17.3733i) q^{65} +(47.8702 - 27.6378i) q^{67} +(-42.1958 + 24.3618i) q^{71} +(-41.5917 - 72.0390i) q^{73} +(16.1127 - 27.9080i) q^{77} +(91.3114 + 52.7187i) q^{79} +(42.1532 + 73.0115i) q^{83} +(48.5165 + 84.0330i) q^{85} +(55.8744 + 32.2591i) q^{89} +(-32.6599 - 18.8562i) q^{91} +(-121.936 + 43.7594i) q^{95} +(-89.3395 - 51.5802i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−6.81841			−1.36368			−0.681841	−	0.731500i	$-0.738820\pi$
$5$	−6.81841			−1.36368			−0.681841	+	0.731500i	$0.738820\pi$
$6$		0			0
$7$	3.70019	+	6.40891i	0.528598	+	0.915559i	0.999444	+	0.0333436i	$0.0106156\pi$
$7$	3.70019	+	6.40891i	0.528598	+	0.915559i	−0.470846	+	0.882216i	$0.656051\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.17728	−	3.77116i	−0.197935	−	0.342833i	0.749924	−	0.661524i	$-0.230090\pi$
$11$	−2.17728	−	3.77116i	−0.197935	−	0.342833i	−0.947859	+	0.318691i	$0.896757\pi$
$12$		0			0
$13$	−4.41327	+	2.54800i	−0.339483	+	0.196000i	−0.660043	−	0.751228i	$-0.729462\pi$
$13$	−4.41327	+	2.54800i	−0.339483	+	0.196000i	0.320561	+	0.947228i	$0.396129\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.11551	−	12.3244i	−0.418559	−	0.724966i	0.577235	−	0.816578i	$-0.304132\pi$
$17$	−7.11551	−	12.3244i	−0.418559	−	0.724966i	−0.995795	+	0.0916116i	$0.970798\pi$
$18$		0			0
$19$	17.8833	−	6.41783i	0.941225	−	0.337781i
$19$	17.8833	−	6.41783i	0.941225	−	0.337781i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.25671	−	7.37283i	−0.185074	−	0.320558i	0.758527	−	0.651641i	$-0.225919\pi$
$23$	−4.25671	−	7.37283i	−0.185074	−	0.320558i	−0.943602	+	0.331083i	$0.892586\pi$
$24$		0			0
$25$	21.4907			0.859630
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	11.3235i		−	0.390464i	−0.980757	−	0.195232i	$-0.937454\pi$
$29$		−	11.3235i		−	0.390464i	0.980757	−	0.195232i	$-0.0625460\pi$
$30$		0			0
$31$	−8.16773	−	4.71564i	−0.263475	−	0.152117i	0.362444	−	0.932006i	$-0.381943\pi$
$31$	−8.16773	−	4.71564i	−0.263475	−	0.152117i	−0.625919	+	0.779888i	$0.715276\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−25.2294	−	43.6986i	−0.720840	−	1.24853i
$36$		0			0
$37$		−	33.2038i		−	0.897400i	−0.893683	−	0.448700i	$-0.851887\pi$
$37$		−	33.2038i		−	0.897400i	0.893683	−	0.448700i	$-0.148113\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.431133i		−	0.0105154i	−0.999986	−	0.00525772i	$-0.998326\pi$
$41$		−	0.431133i		−	0.0105154i	0.999986	−	0.00525772i	$-0.00167359\pi$
$42$		0			0
$43$	−38.1888	+	66.1449i	−0.888111	+	1.53825i	−0.0460053	+	0.998941i	$0.514649\pi$
$43$	−38.1888	+	66.1449i	−0.888111	+	1.53825i	−0.842106	+	0.539312i	$0.818684\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	47.6996			1.01488			0.507442	−	0.861686i	$-0.330591\pi$
$47$	47.6996			1.01488			0.507442	+	0.861686i	$0.330591\pi$
$48$		0			0
$49$	−2.88279	+	4.99314i	−0.0588325	+	0.101901i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.51673	+	4.33978i	0.141825	+	0.0818827i	0.569234	−	0.822176i	$-0.307240\pi$
$53$	7.51673	+	4.33978i	0.141825	+	0.0818827i	−0.427408	+	0.904059i	$0.640573\pi$
$54$		0			0
$55$	14.8456	+	25.7133i	0.269920	+	0.467515i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			65.1347i			1.10398i	0.833852	+	0.551989i	$0.186131\pi$
$59$			65.1347i			1.10398i	−0.833852	+	0.551989i	$0.813869\pi$
$60$		0			0
$61$	7.37572			0.120913			0.0604567	−	0.998171i	$-0.480744\pi$
$61$	7.37572			0.120913			0.0604567	+	0.998171i	$0.480744\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	30.0915	−	17.3733i	0.462946	−	0.267282i
$66$		0			0
$67$	47.8702	−	27.6378i	0.714480	−	0.412505i	−0.0982377	−	0.995163i	$-0.531321\pi$
$67$	47.8702	−	27.6378i	0.714480	−	0.412505i	0.812718	+	0.582658i	$0.197987\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−42.1958	+	24.3618i	−0.594307	+	0.343123i	−0.766799	−	0.641888i	$-0.778152\pi$
$71$	−42.1958	+	24.3618i	−0.594307	+	0.343123i	0.172492	+	0.985011i	$0.444818\pi$
$72$		0			0
$73$	−41.5917	−	72.0390i	−0.569750	−	0.986836i	−0.996590	−	0.0825086i	$-0.973707\pi$
$73$	−41.5917	−	72.0390i	−0.569750	−	0.986836i	0.426841	−	0.904327i	$-0.359627\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.1127	−	27.9080i	0.209256	−	0.362442i
$78$		0			0
$79$	91.3114	+	52.7187i	1.15584	+	0.667325i	0.950304	−	0.311324i	$-0.100773\pi$
$79$	91.3114	+	52.7187i	1.15584	+	0.667325i	0.205537	+	0.978649i	$0.434106\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	42.1532	+	73.0115i	0.507870	+	0.879657i	0.999958	+	0.00911142i	$0.00290030\pi$
$83$	42.1532	+	73.0115i	0.507870	+	0.879657i	−0.492089	+	0.870545i	$0.663766\pi$
$84$		0			0
$85$	48.5165	+	84.0330i	0.570782	+	0.988624i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	55.8744	+	32.2591i	0.627803	+	0.362462i	0.779901	−	0.625903i	$-0.215270\pi$
$89$	55.8744	+	32.2591i	0.627803	+	0.362462i	−0.152098	+	0.988365i	$0.548603\pi$
$90$		0			0
$91$	−32.6599	−	18.8562i	−0.358900	−	0.207211i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−121.936	+	43.7594i	−1.28353	+	0.460626i
$96$		0			0
$97$	−89.3395	−	51.5802i	−0.921026	−	0.531755i	−0.0370639	−	0.999313i	$-0.511801\pi$
$97$	−89.3395	−	51.5802i	−0.921026	−	0.531755i	−0.883962	+	0.467558i	$0.845134\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−96.0507			−0.950997			−0.475498	−	0.879717i	$-0.657732\pi$
$101$	−96.0507			−0.950997			−0.475498	+	0.879717i	$0.657732\pi$
$102$		0			0
$103$	99.9028	+	57.6789i	0.969930	+	0.559990i	0.899215	−	0.437507i	$-0.144138\pi$
$103$	99.9028	+	57.6789i	0.969930	+	0.559990i	0.0707153	+	0.997497i	$0.477472\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			108.224i			1.01144i	0.862699	+	0.505718i	$0.168772\pi$
$107$			108.224i			1.01144i	−0.862699	+	0.505718i	$0.831228\pi$
$108$		0			0
$109$	−90.8195	+	52.4347i	−0.833207	+	0.481052i	−0.854949	−	0.518712i	$-0.826412\pi$
$109$	−90.8195	+	52.4347i	−0.833207	+	0.481052i	0.0217427	+	0.999764i	$0.493079\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	64.6628	+	37.3331i	0.572237	+	0.330381i	0.758042	−	0.652205i	$-0.226156\pi$
$113$	64.6628	+	37.3331i	0.572237	+	0.330381i	−0.185805	+	0.982587i	$0.559489\pi$
$114$		0			0
$115$	29.0240	+	50.2710i	0.252382	+	0.437139i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	52.6575	−	91.2054i	0.442500	−	0.766432i
$120$		0			0
$121$	51.0189	−	88.3673i	0.421644	−	0.730308i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	23.9276			0.191421
$126$		0			0
$127$	55.6956	+	32.1558i	0.438548	+	0.253196i	0.702981	−	0.711208i	$-0.251852\pi$
$127$	55.6956	+	32.1558i	0.438548	+	0.253196i	−0.264434	+	0.964404i	$0.585185\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	76.1706			0.581455			0.290728	−	0.956806i	$-0.406103\pi$
$131$	76.1706			0.581455			0.290728	+	0.956806i	$0.406103\pi$
$132$		0			0
$133$	107.303	+	90.8652i	0.806788	+	0.683197i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	110.111			0.803731			0.401865	−	0.915699i	$-0.368362\pi$
$137$	110.111			0.803731			0.401865	+	0.915699i	$0.368362\pi$
$138$		0			0
$139$	113.863	+	197.217i	0.819159	+	1.41882i	0.906303	+	0.422629i	$0.138893\pi$
$139$	113.863	+	197.217i	0.819159	+	1.41882i	−0.0871438	+	0.996196i	$0.527774\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	19.2179	+	11.0955i	0.134391	+	0.0775906i
$144$		0			0
$145$			77.2080i			0.532469i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	114.036			0.765343			0.382672	−	0.923884i	$-0.375004\pi$
$149$	114.036			0.765343			0.382672	+	0.923884i	$0.375004\pi$
$150$		0			0
$151$	174.049	−	100.487i	1.15264	−	0.665480i	0.203114	−	0.979155i	$-0.434894\pi$
$151$	174.049	−	100.487i	1.15264	−	0.665480i	0.949530	+	0.313675i	$0.101560\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	55.6909	+	32.1532i	0.359296	+	0.207440i
$156$		0			0
$157$	55.1038			0.350980			0.175490	−	0.984481i	$-0.443849\pi$
$157$	55.1038			0.350980			0.175490	+	0.984481i	$0.443849\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	31.5012	−	54.5617i	0.195660	−	0.338893i
$162$		0			0
$163$	−124.653			−0.764745			−0.382373	−	0.924008i	$-0.624893\pi$
$163$	−124.653			−0.764745			−0.382373	+	0.924008i	$0.624893\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	70.6021	−	40.7621i	0.422767	−	0.244085i	−0.273493	−	0.961874i	$-0.588179\pi$
$167$	70.6021	−	40.7621i	0.422767	−	0.244085i	0.696260	+	0.717789i	$0.254846\pi$
$168$		0			0
$169$	−71.5153	+	123.868i	−0.423168	+	0.732948i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	213.017	+	122.985i	1.23131	+	0.710899i	0.967304	−	0.253621i	$-0.0816218\pi$
$173$	213.017	+	122.985i	1.23131	+	0.710899i	0.264009	+	0.964520i	$0.414955\pi$
$174$		0			0
$175$	79.5198	+	137.732i	0.454399	+	0.787042i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	311.071i		−	1.73783i	−0.494964	−	0.868914i	$-0.664819\pi$
$179$		−	311.071i		−	1.73783i	0.494964	−	0.868914i	$-0.335181\pi$
$180$		0			0
$181$	−50.1888	−	28.9765i	−0.277286	−	0.160091i	0.354908	−	0.934901i	$-0.384512\pi$
$181$	−50.1888	−	28.9765i	−0.277286	−	0.160091i	−0.632194	+	0.774810i	$0.717845\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			226.397i			1.22377i
$186$		0			0
$187$	−30.9850	+	53.6675i	−0.165695	+	0.286992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	147.359	+	255.233i	0.771513	+	1.33630i	0.936734	+	0.350043i	$0.113833\pi$
$191$	147.359	+	255.233i	0.771513	+	1.33630i	−0.165221	+	0.986257i	$0.552834\pi$
$192$		0			0
$193$			36.1000i			0.187046i	0.995617	+	0.0935232i	$0.0298129\pi$
$193$			36.1000i			0.187046i	−0.995617	+	0.0935232i	$0.970187\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	230.429			1.16969			0.584844	−	0.811146i	$-0.301156\pi$
$197$	230.429			1.16969			0.584844	+	0.811146i	$0.301156\pi$
$198$		0			0
$199$	−38.8365	+	67.2667i	−0.195158	+	0.338024i	−0.946952	−	0.321374i	$-0.895855\pi$
$199$	−38.8365	+	67.2667i	−0.195158	+	0.338024i	0.751794	+	0.659398i	$0.229189\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	72.5711	−	41.8989i	0.357493	−	0.206399i
$204$		0			0
$205$			2.93964i			0.0143397i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−63.1396	−	53.4673i	−0.302104	−	0.255824i
$210$		0			0
$211$			117.518i			0.556959i	0.960442	+	0.278479i	$0.0898304\pi$
$211$			117.518i			0.556959i	−0.960442	+	0.278479i	$0.910170\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	260.387	−	451.003i	1.21110	−	2.09769i
$216$		0			0
$217$		−	69.7950i		−	0.321636i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	62.8054	+	36.2607i	0.284187	+	0.164076i
$222$		0			0
$223$	209.178	+	120.769i	0.938019	+	0.541565i	0.889339	−	0.457249i	$-0.151165\pi$
$223$	209.178	+	120.769i	0.938019	+	0.541565i	0.0486800	+	0.998814i	$0.484499\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−319.299	+	184.347i	−1.40660	+	0.812103i	−0.995059	−	0.0992858i	$-0.968344\pi$
$227$	−319.299	+	184.347i	−1.40660	+	0.812103i	−0.411545	+	0.911389i	$0.635011\pi$
$228$		0			0
$229$	−110.036	+	190.588i	−0.480506	+	0.832261i	−0.999750	−	0.0223649i	$-0.992880\pi$
$229$	−110.036	+	190.588i	−0.480506	+	0.832261i	0.519244	+	0.854626i	$0.326214\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.3618	+	31.8035i	0.0788059	+	0.136496i	0.902735	−	0.430197i	$-0.141556\pi$
$233$	18.3618	+	31.8035i	0.0788059	+	0.136496i	−0.823929	+	0.566693i	$0.808223\pi$
$234$		0			0
$235$	−325.235			−1.38398
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−34.5559	+	59.8525i	−0.144585	+	0.250429i	−0.929218	−	0.369532i	$-0.879518\pi$
$239$	−34.5559	+	59.8525i	−0.144585	+	0.250429i	0.784633	+	0.619961i	$0.212851\pi$
$240$		0			0
$241$		−	241.792i		−	1.00329i	−0.865075	−	0.501643i	$-0.832729\pi$
$241$		−	241.792i		−	1.00329i	0.865075	−	0.501643i	$-0.167271\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	19.6561	−	34.0453i	0.0802288	−	0.138960i
$246$		0			0
$247$	−62.5711	+	73.8903i	−0.253324	+	0.299151i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	205.737	−	356.346i	0.819668	−	1.41971i	−0.0862599	−	0.996273i	$-0.527492\pi$
$251$	205.737	−	356.346i	0.819668	−	1.41971i	0.905927	−	0.423433i	$-0.139175\pi$
$252$		0			0
$253$	−18.5361	+	32.1055i	−0.0732652	+	0.126899i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.1942	+	13.9685i	−0.0941409	+	0.0543523i	−0.546331	−	0.837569i	$-0.683976\pi$
$257$	−24.1942	+	13.9685i	−0.0941409	+	0.0543523i	0.452190	+	0.891921i	$0.350643\pi$
$258$		0			0
$259$	212.800	−	122.860i	0.821623	−	0.474364i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	17.2971	−	29.9595i	0.0657686	−	0.113915i	−0.831266	−	0.555875i	$-0.812383\pi$
$263$	17.2971	−	29.9595i	0.0657686	−	0.113915i	0.897035	+	0.441960i	$0.145717\pi$
$264$		0			0
$265$	−51.2521	−	29.5904i	−0.193404	−	0.111662i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	332.317	−	191.863i	1.23538	−	0.713247i	0.267233	−	0.963632i	$-0.413891\pi$
$269$	332.317	−	191.863i	1.23538	−	0.713247i	0.968146	+	0.250385i	$0.0805572\pi$
$270$		0			0
$271$	125.316	+	217.054i	0.462422	+	0.800939i	0.999081	−	0.0428603i	$-0.0136471\pi$
$271$	125.316	+	217.054i	0.462422	+	0.800939i	−0.536659	+	0.843799i	$0.680314\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−46.7914	−	81.0451i	−0.170151	−	0.294709i
$276$		0			0
$277$	97.8217	+	169.432i	0.353147	+	0.611668i	0.986799	−	0.161950i	$-0.0517782\pi$
$277$	97.8217	+	169.432i	0.353147	+	0.611668i	−0.633652	+	0.773618i	$0.718445\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	361.510i		−	1.28651i	−0.765651	−	0.643256i	$-0.777583\pi$
$281$		−	361.510i		−	1.28651i	0.765651	−	0.643256i	$-0.222417\pi$
$282$		0			0
$283$	−226.506			−0.800374			−0.400187	−	0.916433i	$-0.631055\pi$
$283$	−226.506			−0.800374			−0.400187	+	0.916433i	$0.631055\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.76310	−	1.59527i	0.00962752	−	0.00555845i
$288$		0			0
$289$	43.2390	−	74.8922i	0.149616	−	0.259142i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	200.162	+	115.563i	0.683146	+	0.394415i	0.801039	−	0.598612i	$-0.204281\pi$
$293$	200.162	+	115.563i	0.683146	+	0.394415i	−0.117893	+	0.993026i	$0.537614\pi$
$294$		0			0
$295$		−	444.115i		−	1.50547i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	37.5720	+	21.6922i	0.125659	+	0.0725492i
$300$		0			0
$301$	−565.223			−1.87782
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−50.2907			−0.164887
$306$		0			0
$307$	−150.342	+	86.8000i	−0.489714	+	0.282736i	−0.724456	−	0.689321i	$-0.757909\pi$
$307$	−150.342	+	86.8000i	−0.489714	+	0.282736i	0.234742	+	0.972058i	$0.424575\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−180.905	+	313.337i	−0.581688	+	1.00751i	0.413592	+	0.910462i	$0.364274\pi$
$311$	−180.905	+	313.337i	−0.581688	+	1.00751i	−0.995279	+	0.0970504i	$0.969059\pi$
$312$		0			0
$313$	−594.669			−1.89990			−0.949951	−	0.312398i	$-0.898868\pi$
$313$	−594.669			−1.89990			−0.949951	+	0.312398i	$0.898868\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			416.809i			1.31485i	0.753518	+	0.657427i	$0.228355\pi$
$317$			416.809i			1.31485i	−0.753518	+	0.657427i	$0.771645\pi$
$318$		0			0
$319$	−42.7026	+	24.6544i	−0.133864	+	0.0772864i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−206.345	−	174.735i	−0.638838	−	0.540975i
$324$		0			0
$325$	−94.8445	+	54.7585i	−0.291829	+	0.168488i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	176.497	+	305.702i	0.536466	+	0.929187i
$330$		0			0
$331$	−282.303	+	162.988i	−0.852880	+	0.492410i	−0.861621	−	0.507552i	$-0.830551\pi$
$331$	−282.303	+	162.988i	−0.852880	+	0.492410i	0.00874183	+	0.999962i	$0.497217\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−326.398	+	188.446i	−0.974324	+	0.562526i
$336$		0			0
$337$			223.270i			0.662521i	0.943539	+	0.331261i	$0.107474\pi$
$337$			223.270i			0.662521i	−0.943539	+	0.331261i	$0.892526\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			41.0691i			0.120437i
$342$		0			0
$343$	319.951			0.932802
$344$		0			0
$345$		0			0
$346$		0			0
$347$	306.233			0.882515			0.441257	−	0.897381i	$-0.354533\pi$
$347$	306.233			0.882515			0.441257	+	0.897381i	$0.354533\pi$
$348$		0			0
$349$	−168.436	−	291.740i	−0.482625	−	0.835931i	0.517176	−	0.855879i	$-0.326983\pi$
$349$	−168.436	−	291.740i	−0.482625	−	0.835931i	−0.999801	+	0.0199480i	$0.993650\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−108.109	−	187.250i	−0.306258	−	0.530454i	0.671283	−	0.741201i	$-0.265744\pi$
$353$	−108.109	−	187.250i	−0.306258	−	0.530454i	−0.977541	+	0.210747i	$0.932410\pi$
$354$		0			0
$355$	287.708	−	166.109i	0.810446	−	0.467911i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−275.951	−	477.961i	−0.768666	−	1.33137i	−0.938286	−	0.345859i	$-0.887587\pi$
$359$	−275.951	−	477.961i	−0.768666	−	1.33137i	0.169620	−	0.985509i	$-0.445746\pi$
$360$		0			0
$361$	278.623	−	229.544i	0.771808	−	0.635855i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	283.590	+	491.192i	0.776958	+	1.34573i
$366$		0			0
$367$	342.258			0.932583			0.466292	−	0.884631i	$-0.345590\pi$
$367$	342.258			0.932583			0.466292	+	0.884631i	$0.345590\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			64.2321i			0.173132i
$372$		0			0
$373$	70.4600	+	40.6801i	0.188901	+	0.109062i	0.591468	−	0.806329i	$-0.298549\pi$
$373$	70.4600	+	40.6801i	0.188901	+	0.109062i	−0.402567	+	0.915390i	$0.631882\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	28.8522	+	49.9735i	0.0765311	+	0.132556i
$378$		0			0
$379$			265.656i			0.700939i	0.936574	+	0.350469i	$0.113978\pi$
$379$			265.656i			0.700939i	−0.936574	+	0.350469i	$0.886022\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	268.022i		−	0.699796i	−0.936788	−	0.349898i	$-0.886216\pi$
$383$		−	268.022i		−	0.699796i	0.936788	−	0.349898i	$-0.113784\pi$
$384$		0			0
$385$	−109.863	+	190.288i	−0.285359	+	0.494256i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−192.040			−0.493675			−0.246838	−	0.969057i	$-0.579391\pi$
$389$	−192.040			−0.493675			−0.246838	+	0.969057i	$0.579391\pi$
$390$		0			0
$391$	−60.5773	+	104.923i	−0.154929	+	0.268345i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−622.599	−	359.458i	−1.57620	−	0.910019i
$396$		0			0
$397$	231.035	+	400.165i	0.581953	+	1.00797i	0.995248	+	0.0973746i	$0.0310445\pi$
$397$	231.035	+	400.165i	0.581953	+	1.00797i	−0.413295	+	0.910597i	$0.635622\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			61.7381i			0.153960i	0.997033	+	0.0769801i	$0.0245278\pi$
$401$			61.7381i			0.153960i	−0.997033	+	0.0769801i	$0.975472\pi$
$402$		0			0
$403$	48.0619			0.119260
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−125.217	+	72.2940i	−0.307658	+	0.177627i
$408$		0			0
$409$	274.398	−	158.424i	0.670899	−	0.387344i	−0.125518	−	0.992091i	$-0.540059\pi$
$409$	274.398	−	158.424i	0.670899	−	0.387344i	0.796417	+	0.604748i	$0.206726\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−417.443	+	241.011i	−1.01076	+	0.583561i
$414$		0			0
$415$	−287.418	−	497.822i	−0.692573	−	1.19957i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	196.953	−	341.133i	0.470055	−	0.814160i	−0.529358	−	0.848398i	$-0.677567\pi$
$419$	196.953	−	341.133i	0.470055	−	0.814160i	0.999414	+	0.0342385i	$0.0109006\pi$
$420$		0			0
$421$	−12.7310	−	7.35027i	−0.0302400	−	0.0174591i	0.484804	−	0.874623i	$-0.338891\pi$
$421$	−12.7310	−	7.35027i	−0.0302400	−	0.0174591i	−0.515044	+	0.857164i	$0.672224\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−152.918	−	264.861i	−0.359806	−	0.623202i
$426$		0			0
$427$	27.2915	+	47.2703i	0.0639146	+	0.110703i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	426.955	+	246.503i	0.990615	+	0.571932i	0.905458	−	0.424436i	$-0.139528\pi$
$431$	426.955	+	246.503i	0.990615	+	0.571932i	0.0851566	+	0.996368i	$0.472861\pi$
$432$		0			0
$433$	577.503	+	333.422i	1.33373	+	0.770027i	0.985869	−	0.167521i	$-0.0535761\pi$
$433$	577.503	+	333.422i	1.33373	+	0.770027i	0.347857	+	0.937548i	$0.386909\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−123.441	−	104.532i	−0.282475	−	0.239203i
$438$		0			0
$439$	352.648	+	203.602i	0.803299	+	0.463785i	0.844623	−	0.535361i	$-0.179824\pi$
$439$	352.648	+	203.602i	0.803299	+	0.463785i	−0.0413243	+	0.999146i	$0.513158\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−250.258			−0.564917			−0.282459	−	0.959279i	$-0.591150\pi$
$443$	−250.258			−0.564917			−0.282459	+	0.959279i	$0.591150\pi$
$444$		0			0
$445$	−380.975	−	219.956i	−0.856123	−	0.494283i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			761.855i			1.69678i	0.529371	+	0.848390i	$0.322428\pi$
$449$			761.855i			1.69678i	−0.529371	+	0.848390i	$0.677572\pi$
$450$		0			0
$451$	−1.62587	+	0.938699i	−0.00360504	+	0.00208137i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	222.689	+	128.569i	0.489425	+	0.282570i
$456$		0			0
$457$	228.695	+	396.112i	0.500427	+	0.866766i	1.00000		0.000493522i	$0.000157093\pi$
$457$	228.695	+	396.112i	0.500427	+	0.866766i	−0.499573	+	0.866272i	$0.666510\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−158.925	+	275.266i	−0.344740	+	0.597107i	−0.985306	−	0.170796i	$-0.945366\pi$
$461$	−158.925	+	275.266i	−0.344740	+	0.597107i	0.640567	+	0.767903i	$0.278700\pi$
$462$		0			0
$463$	391.136	−	677.468i	0.844786	−	1.46321i	−0.0410200	−	0.999158i	$-0.513061\pi$
$463$	391.136	−	677.468i	0.844786	−	1.46321i	0.885807	−	0.464055i	$-0.153606\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−687.036			−1.47117			−0.735585	−	0.677433i	$-0.763093\pi$
$467$	−687.036			−1.47117			−0.735585	+	0.677433i	$0.763093\pi$
$468$		0			0
$469$	354.257	+	204.530i	0.755346	+	0.436099i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	332.591			0.703152
$474$		0			0
$475$	384.325	−	137.924i	0.809105	−	0.290366i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−338.225			−0.706106			−0.353053	−	0.935603i	$-0.614857\pi$
$479$	−338.225			−0.706106			−0.353053	+	0.935603i	$0.614857\pi$
$480$		0			0
$481$	84.6034	+	146.537i	0.175891	+	0.304652i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	609.154	+	351.695i	1.25599	+	0.725145i
$486$		0			0
$487$		−	629.561i		−	1.29273i	−0.763027	−	0.646367i	$-0.776288\pi$
$487$		−	629.561i		−	1.29273i	0.763027	−	0.646367i	$-0.223712\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−491.757			−1.00154			−0.500771	−	0.865580i	$-0.666950\pi$
$491$	−491.757			−1.00154			−0.500771	+	0.865580i	$0.666950\pi$
$492$		0			0
$493$	−139.555	+	80.5722i	−0.283073	+	0.163432i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−312.265	−	180.286i	−0.628300	−	0.362749i
$498$		0			0
$499$	310.002			0.621247			0.310624	−	0.950533i	$-0.399462\pi$
$499$	310.002			0.621247			0.310624	+	0.950533i	$0.399462\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	392.457	−	679.756i	0.780233	−	1.35140i	−0.151573	−	0.988446i	$-0.548434\pi$
$503$	392.457	−	679.756i	0.780233	−	1.35140i	0.931806	−	0.362957i	$-0.118233\pi$
$504$		0			0
$505$	654.913			1.29686
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−709.494	+	409.626i	−1.39390	+	0.804767i	−0.993744	−	0.111681i	$-0.964376\pi$
$509$	−709.494	+	409.626i	−1.39390	+	0.804767i	−0.400153	+	0.916448i	$0.631043\pi$
$510$		0			0
$511$	307.795	−	533.116i	0.602338	−	1.04328i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−681.179	−	393.279i	−1.32268	−	0.763648i
$516$		0			0
$517$	−103.855	−	179.883i	−0.200881	−	0.347936i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			316.315i			0.607130i	0.952811	+	0.303565i	$0.0981770\pi$
$521$			316.315i			0.607130i	−0.952811	+	0.303565i	$0.901823\pi$
$522$		0			0
$523$	−284.438	−	164.220i	−0.543859	−	0.313997i	0.202783	−	0.979224i	$-0.435002\pi$
$523$	−284.438	−	164.220i	−0.543859	−	0.313997i	−0.746641	+	0.665227i	$0.768335\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			134.217i			0.254681i
$528$		0			0
$529$	228.261	−	395.359i	0.431495	−	0.747371i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.09853	+	1.90271i	0.00206103	+	0.00356981i
$534$		0			0
$535$		−	737.914i		−	1.37928i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.1066			0.0465800
$540$		0			0
$541$	193.895	−	335.836i	0.358401	−	0.620768i	−0.629293	−	0.777168i	$-0.716655\pi$
$541$	193.895	−	335.836i	0.358401	−	0.620768i	0.987694	+	0.156400i	$0.0499888\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	619.245	−	357.521i	1.13623	−	0.656002i
$546$		0			0
$547$		−	659.346i		−	1.20539i	−0.797973	−	0.602693i	$-0.794094\pi$
$547$		−	659.346i		−	1.20539i	0.797973	−	0.602693i	$-0.205906\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−72.6721	−	202.501i	−0.131891	−	0.367515i
$552$		0			0
$553$			780.276i			1.41099i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−240.816	+	417.106i	−0.432345	+	0.748844i	−0.997075	−	0.0764324i	$-0.975647\pi$
$557$	−240.816	+	417.106i	−0.432345	+	0.748844i	0.564730	+	0.825276i	$0.308980\pi$
$558$		0			0
$559$		−	389.221i		−	0.696280i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	706.308	+	407.787i	1.25454	+	0.724311i	0.972009	−	0.234946i	$-0.0754912\pi$
$563$	706.308	+	407.787i	1.25454	+	0.724311i	0.282535	+	0.959257i	$0.408825\pi$
$564$		0			0
$565$	−440.897	−	254.552i	−0.780349	−	0.450535i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	362.066	−	209.039i	0.636319	−	0.367379i	−0.146876	−	0.989155i	$-0.546922\pi$
$569$	362.066	−	209.039i	0.636319	−	0.367379i	0.783195	+	0.621776i	$0.213589\pi$
$570$		0			0
$571$	78.0813	−	135.241i	0.136745	−	0.236849i	−0.789518	−	0.613728i	$-0.789669\pi$
$571$	78.0813	−	135.241i	0.136745	−	0.236849i	0.926263	+	0.376879i	$0.123003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−91.4798	−	158.448i	−0.159095	−	0.275561i
$576$		0			0
$577$	−5.72191			−0.00991665			−0.00495832	−	0.999988i	$-0.501578\pi$
$577$	−5.72191			−0.00991665			−0.00495832	+	0.999988i	$0.501578\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−311.950	+	540.313i	−0.536918	+	0.929970i
$582$		0			0
$583$		−	37.7957i		−	0.0648298i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−142.479	+	246.780i	−0.242723	+	0.420409i	−0.961489	−	0.274843i	$-0.911374\pi$
$587$	−142.479	+	246.780i	−0.242723	+	0.420409i	0.718766	+	0.695252i	$0.244707\pi$
$588$		0			0
$589$	−176.330	−	31.9119i	−0.299372	−	0.0541799i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	513.723	−	889.794i	0.866312	−	1.50050i	0.000572553	−	1.00000i	$-0.499818\pi$
$593$	513.723	−	889.794i	0.866312	−	1.50050i	0.865739	−	0.500496i	$-0.166849\pi$
$594$		0			0
$595$	−359.040	+	621.876i	−0.603429	+	1.04517i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	656.418	−	378.983i	1.09586	−	0.632693i	0.160727	−	0.986999i	$-0.448616\pi$
$599$	656.418	−	378.983i	1.09586	−	0.632693i	0.935130	+	0.354306i	$0.115283\pi$
$600$		0			0
$601$	−951.713	+	549.472i	−1.58355	+	0.914262i	−0.589213	+	0.807978i	$0.700562\pi$
$601$	−951.713	+	549.472i	−1.58355	+	0.914262i	−0.994336	+	0.106284i	$0.966105\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−347.868	+	602.525i	−0.574988	+	0.995908i
$606$		0			0
$607$	−751.726	−	434.009i	−1.23843	−	0.715007i	−0.269656	−	0.962957i	$-0.586910\pi$
$607$	−751.726	−	434.009i	−1.23843	−	0.715007i	−0.968773	+	0.247950i	$0.920243\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−210.511	+	121.539i	−0.344536	+	0.198918i
$612$		0			0
$613$	483.508	+	837.461i	0.788757	+	1.36617i	0.926729	+	0.375731i	$0.122608\pi$
$613$	483.508	+	837.461i	0.788757	+	1.36617i	−0.137972	+	0.990436i	$0.544058\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−267.125	−	462.673i	−0.432941	−	0.749876i	0.564184	−	0.825649i	$-0.309191\pi$
$617$	−267.125	−	462.673i	−0.432941	−	0.749876i	−0.997125	+	0.0757733i	$0.975857\pi$
$618$		0			0
$619$	92.9003	+	160.908i	0.150081	+	0.259948i	0.931257	−	0.364363i	$-0.118713\pi$
$619$	92.9003	+	160.908i	0.150081	+	0.259948i	−0.781176	+	0.624311i	$0.785380\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			477.459i			0.766387i
$624$		0			0
$625$	−700.417			−1.12067
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−409.218	+	236.262i	−0.650585	+	0.375615i
$630$		0			0
$631$	29.5776	−	51.2300i	0.0468742	−	0.0811885i	−0.841636	−	0.540045i	$-0.818407\pi$
$631$	29.5776	−	51.2300i	0.0468742	−	0.0811885i	0.888511	+	0.458856i	$0.151741\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−379.755	−	219.252i	−0.598040	−	0.345278i
$636$		0			0
$637$		−	29.3815i		−	0.0461248i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−580.465	−	335.131i	−0.905561	−	0.522826i	−0.0265607	−	0.999647i	$-0.508456\pi$
$641$	−580.465	−	335.131i	−0.905561	−	0.522826i	−0.879000	+	0.476821i	$0.841789\pi$
$642$		0			0
$643$	−530.539			−0.825099			−0.412550	−	0.910935i	$-0.635362\pi$
$643$	−530.539			−0.825099			−0.412550	+	0.910935i	$0.635362\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	982.685			1.51883			0.759416	−	0.650605i	$-0.225485\pi$
$647$	982.685			1.51883			0.759416	+	0.650605i	$0.225485\pi$
$648$		0			0
$649$	245.634	−	141.817i	0.378480	−	0.218516i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.3213	+	62.9103i	−0.0556222	+	0.0963404i	−0.892496	−	0.451056i	$-0.851048\pi$
$653$	−36.3213	+	62.9103i	−0.0556222	+	0.0963404i	0.836874	+	0.547396i	$0.184381\pi$
$654$		0			0
$655$	−519.363			−0.792920
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	367.642i		−	0.557878i	−0.960309	−	0.278939i	$-0.910017\pi$
$659$		−	367.642i		−	0.557878i	0.960309	−	0.278939i	$-0.0899828\pi$
$660$		0			0
$661$	225.094	−	129.958i	0.340535	−	0.196608i	−0.319973	−	0.947427i	$-0.603674\pi$
$661$	225.094	−	129.958i	0.340535	−	0.196608i	0.660509	+	0.750818i	$0.270341\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−731.635	−	619.556i	−1.10020	−	0.931663i
$666$		0			0
$667$	−83.4860	+	48.2006i	−0.125166	+	0.0722648i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.0590	−	27.8150i	−0.0239330	−	0.0414531i
$672$		0			0
$673$	−630.689	+	364.128i	−0.937130	+	0.541052i	−0.889060	−	0.457792i	$-0.848641\pi$
$673$	−630.689	+	364.128i	−0.937130	+	0.541052i	−0.0480706	+	0.998844i	$0.515307\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−937.670	+	541.364i	−1.38504	+	0.799651i	−0.992751	−	0.120192i	$-0.961649\pi$
$677$	−937.670	+	541.364i	−1.38504	+	0.799651i	−0.392286	+	0.919843i	$0.628316\pi$
$678$		0			0
$679$		−	763.426i		−	1.12434i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			350.099i			0.512590i	0.966599	+	0.256295i	$0.0825019\pi$
$683$			350.099i			0.512590i	−0.966599	+	0.256295i	$0.917498\pi$
$684$		0			0
$685$	−750.783			−1.09603
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−44.2312			−0.0641962
$690$		0			0
$691$	−145.503	−	252.019i	−0.210569	−	0.364717i	0.741324	−	0.671148i	$-0.234198\pi$
$691$	−145.503	−	252.019i	−0.210569	−	0.364717i	−0.951893	+	0.306431i	$0.900865\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−776.365	−	1344.70i	−1.11707	−	1.93483i
$696$		0			0
$697$	−5.31347	+	3.06773i	−0.00762335	+	0.00440134i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.5413	−	39.0426i	−0.0321559	−	0.0556956i	0.849500	−	0.527589i	$-0.176904\pi$
$701$	−22.5413	−	39.0426i	−0.0321559	−	0.0556956i	−0.881655	+	0.471894i	$0.843571\pi$
$702$		0			0
$703$	−213.096	−	593.792i	−0.303124	−	0.844655i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−355.406	−	615.581i	−0.502695	−	0.870694i
$708$		0			0
$709$	841.192			1.18645			0.593224	−	0.805037i	$-0.297855\pi$
$709$	841.192			1.18645			0.593224	+	0.805037i	$0.297855\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			80.2924i			0.112612i
$714$		0			0
$715$	−131.035	−	75.6534i	−0.183266	−	0.105809i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−470.810	−	815.466i	−0.654812	−	1.13417i	−0.981941	−	0.189188i	$-0.939414\pi$
$719$	−470.810	−	815.466i	−0.654812	−	1.13417i	0.327129	−	0.944980i	$-0.393919\pi$
$720$		0			0
$721$			853.692i			1.18404i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	243.350i		−	0.335655i
$726$		0			0
$727$	−451.105	+	781.336i	−0.620502	+	1.07474i	0.368891	+	0.929473i	$0.379738\pi$
$727$	−451.105	+	781.336i	−0.620502	+	1.07474i	−0.989392	+	0.145268i	$0.953596\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1086.93			1.48691
$732$		0			0
$733$	177.764	−	307.897i	0.242516	−	0.420051i	−0.718914	−	0.695099i	$-0.755361\pi$
$733$	177.764	−	307.897i	0.242516	−	0.420051i	0.961430	+	0.275048i	$0.0886939\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−208.454	−	120.351i	−0.282841	−	0.163298i
$738$		0			0
$739$	−620.346	−	1074.47i	−0.839440	−	1.45395i	−0.890363	−	0.455251i	$-0.849550\pi$
$739$	−620.346	−	1074.47i	−0.839440	−	1.45395i	0.0509230	−	0.998703i	$-0.483784\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1250.68i			1.68328i	0.540035	+	0.841642i	$0.318411\pi$
$743$			1250.68i			1.68328i	−0.540035	+	0.841642i	$0.681589\pi$
$744$		0			0
$745$	−777.546			−1.04369
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−693.596	+	400.448i	−0.926030	+	0.534644i
$750$		0			0
$751$	1150.62	−	664.313i	1.53212	−	0.884572i	0.532859	−	0.846204i	$-0.321117\pi$
$751$	1150.62	−	664.313i	1.53212	−	0.884572i	0.999264	−	0.0383679i	$-0.0122159\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1186.74	+	685.165i	−1.57184	+	0.907503i
$756$		0			0
$757$	−708.415	−	1227.01i	−0.935819	−	1.62089i	−0.773167	−	0.634203i	$-0.781328\pi$
$757$	−708.415	−	1227.01i	−0.935819	−	1.62089i	−0.162652	−	0.986683i	$-0.552005\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	172.332	−	298.487i	0.226454	−	0.392231i	−0.730300	−	0.683126i	$-0.760620\pi$
$761$	172.332	−	298.487i	0.226454	−	0.392231i	0.956755	+	0.290896i	$0.0939532\pi$
$762$		0			0
$763$	−672.099	−	388.036i	−0.880863	−	0.508567i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−165.963	−	287.457i	−0.216380	−	0.374781i
$768$		0			0
$769$	−377.710	−	654.213i	−0.491170	−	0.850732i	0.508778	−	0.860898i	$-0.330097\pi$
$769$	−377.710	−	654.213i	−0.491170	−	0.850732i	−0.999948	+	0.0101656i	$0.996764\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	328.092	+	189.424i	0.424440	+	0.245050i	0.696975	−	0.717095i	$-0.254529\pi$
$773$	328.092	+	189.424i	0.424440	+	0.245050i	−0.272535	+	0.962146i	$0.587862\pi$
$774$		0			0
$775$	−175.531	−	101.343i	−0.226491	−	0.130765i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.76694	−	7.71008i	−0.00355192	−	0.00989740i
$780$		0			0
$781$	183.744	+	106.085i	0.235268	+	0.135832i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−375.721			−0.478625
$786$		0			0
$787$	−148.831	−	85.9278i	−0.189112	−	0.109184i	0.402455	−	0.915440i	$-0.368157\pi$
$787$	−148.831	−	85.9278i	−0.189112	−	0.109184i	−0.591567	+	0.806256i	$0.701490\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			552.558i			0.698556i
$792$		0			0
$793$	−32.5510	+	18.7934i	−0.0410480	+	0.0236991i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−715.645	−	413.178i	−0.897924	−	0.518417i	−0.0213978	−	0.999771i	$-0.506812\pi$
$797$	−715.645	−	413.178i	−0.897924	−	0.518417i	−0.876526	+	0.481355i	$0.840145\pi$
$798$		0			0
$799$	−339.407	−	587.870i	−0.424789	−	0.735757i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−181.114	+	313.698i	−0.225547	+	0.390658i
$804$		0			0
$805$	−214.788	+	372.024i	−0.266818	+	0.462142i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	432.334			0.534405			0.267203	−	0.963640i	$-0.413901\pi$
$809$	432.334			0.534405			0.267203	+	0.963640i	$0.413901\pi$
$810$		0			0
$811$	600.564	+	346.736i	0.740523	+	0.427541i	0.822259	−	0.569113i	$-0.192713\pi$
$811$	600.564	+	346.736i	0.740523	+	0.427541i	−0.0817364	+	0.996654i	$0.526047\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	849.938			1.04287
$816$		0			0
$817$	−258.433	+	1427.98i	−0.316320	+	1.74783i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.32284			−0.00404731			−0.00202366	−	0.999998i	$-0.500644\pi$
$821$	−3.32284			−0.00404731			−0.00202366	+	0.999998i	$0.500644\pi$
$822$		0			0
$823$	−293.664	−	508.641i	−0.356822	−	0.618033i	0.630606	−	0.776103i	$-0.282806\pi$
$823$	−293.664	−	508.641i	−0.356822	−	0.618033i	−0.987428	+	0.158070i	$0.949473\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−165.584	−	95.5997i	−0.200222	−	0.115598i	0.396537	−	0.918019i	$-0.370212\pi$
$827$	−165.584	−	95.5997i	−0.200222	−	0.115598i	−0.596759	+	0.802421i	$0.703545\pi$
$828$		0			0
$829$			94.1061i			0.113518i	0.998388	+	0.0567588i	$0.0180766\pi$
$829$			94.1061i			0.113518i	−0.998388	+	0.0567588i	$0.981923\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	82.0502			0.0984996
$834$		0			0
$835$	−481.394	+	277.933i	−0.576520	+	0.332854i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	964.660	+	556.947i	1.14977	+	0.663822i	0.948832	−	0.315781i	$-0.102266\pi$
$839$	964.660	+	556.947i	1.14977	+	0.663822i	0.200942	+	0.979603i	$0.435600\pi$
$840$		0			0
$841$	712.779			0.847538
$842$		0			0
$843$		0			0
$844$		0			0
$845$	487.621	−	844.584i	0.577066	−	0.999508i
$846$		0			0
$847$	755.118			0.891521
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−244.806	+	141.339i	−0.287669	+	0.166086i
$852$		0			0
$853$	231.576	−	401.102i	0.271484	−	0.470225i	−0.697758	−	0.716334i	$-0.745819\pi$
$853$	231.576	−	401.102i	0.271484	−	0.470225i	0.969242	+	0.246109i	$0.0791521\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1038.43	−	599.539i	−1.21171	−	0.699579i	−0.248576	−	0.968612i	$-0.579962\pi$
$857$	−1038.43	−	599.539i	−1.21171	−	0.699579i	−0.963131	+	0.269033i	$0.913296\pi$
$858$		0			0
$859$	107.708	+	186.555i	0.125387	+	0.217177i	0.921884	−	0.387465i	$-0.126649\pi$
$859$	107.708	+	186.555i	0.125387	+	0.217177i	−0.796497	+	0.604642i	$0.793316\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1437.80i		−	1.66605i	−0.553235	−	0.833025i	$-0.686607\pi$
$863$		−	1437.80i		−	1.66605i	0.553235	−	0.833025i	$-0.313393\pi$
$864$		0			0
$865$	−1452.44	−	838.566i	−1.67912	−	0.969440i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	459.134i		−	0.528347i
$870$		0			0
$871$	−140.843	+	243.947i	−0.161702	+	0.280077i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	88.5366	+	153.350i	0.101185	+	0.175257i
$876$		0			0
$877$		−	860.554i		−	0.981248i	−0.871371	−	0.490624i	$-0.836769\pi$
$877$		−	860.554i		−	0.981248i	0.871371	−	0.490624i	$-0.163231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	532.775			0.604739			0.302369	−	0.953191i	$-0.402222\pi$
$881$	532.775			0.604739			0.302369	+	0.953191i	$0.402222\pi$
$882$		0			0
$883$	−555.585	+	962.302i	−0.629202	+	1.08981i	0.358510	+	0.933526i	$0.383285\pi$
$883$	−555.585	+	962.302i	−0.629202	+	1.08981i	−0.987712	+	0.156284i	$0.950049\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	79.2921	−	45.7793i	0.0893936	−	0.0516114i	−0.454637	−	0.890677i	$-0.650231\pi$
$887$	79.2921	−	45.7793i	0.0893936	−	0.0516114i	0.544030	+	0.839066i	$0.316898\pi$
$888$		0			0
$889$			475.931i			0.535355i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	853.024	−	306.128i	0.955234	−	0.342808i
$894$		0			0
$895$			2121.01i			2.36984i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−53.3974	+	92.4870i	−0.0593964	+	0.102878i
$900$		0			0
$901$		−	123.519i		−	0.137091i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	342.208	+	197.574i	0.378130	+	0.218313i
$906$		0			0
$907$	811.285	+	468.396i	0.894471	+	0.516423i	0.875402	−	0.483395i	$-0.160597\pi$
$907$	811.285	+	468.396i	0.894471	+	0.516423i	0.0190688	+	0.999818i	$0.493930\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	606.515	−	350.171i	0.665768	−	0.384381i	−0.128703	−	0.991683i	$-0.541081\pi$
$911$	606.515	−	350.171i	0.665768	−	0.384381i	0.794471	+	0.607302i	$0.207748\pi$
$912$		0			0
$913$	183.559	−	317.933i	0.201050	−	0.348229i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	281.846	+	488.171i	0.307356	+	0.532357i
$918$		0			0
$919$	905.446			0.985251			0.492625	−	0.870241i	$-0.336037\pi$
$919$	905.446			0.985251			0.492625	+	0.870241i	$0.336037\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	124.148	−	215.030i	0.134505	−	0.232969i
$924$		0			0
$925$		−	713.574i		−	0.771431i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−676.594	+	1171.89i	−0.728303	+	1.26146i	0.229297	+	0.973357i	$0.426357\pi$
$929$	−676.594	+	1171.89i	−0.728303	+	1.26146i	−0.957600	+	0.288102i	$0.906976\pi$
$930$		0			0
$931$	−19.5086	+	107.795i	−0.0209545	+	0.115784i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	211.268	−	365.927i	0.225955	−	0.391366i
$936$		0			0
$937$	124.786	−	216.136i	0.133176	−	0.230668i	−0.791723	−	0.610880i	$-0.790816\pi$
$937$	124.786	−	216.136i	0.133176	−	0.230668i	0.924899	+	0.380212i	$0.124149\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	814.206	−	470.082i	0.865256	−	0.499556i	−0.000512642	−	1.00000i	$-0.500163\pi$
$941$	814.206	−	470.082i	0.865256	−	0.499556i	0.865769	+	0.500444i	$0.166830\pi$
$942$		0			0
$943$	−3.17867	+	1.83521i	−0.00337081	+	0.00194614i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−805.457	+	1395.09i	−0.850535	+	1.47317i	0.0301907	+	0.999544i	$0.490389\pi$
$947$	−805.457	+	1395.09i	−0.850535	+	1.47317i	−0.880726	+	0.473626i	$0.842945\pi$
$948$		0			0
$949$	367.111	+	211.952i	0.386840	+	0.223342i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	171.006	−	98.7303i	0.179440	−	0.103599i	−0.407590	−	0.913165i	$-0.633631\pi$
$953$	171.006	−	98.7303i	0.179440	−	0.103599i	0.587029	+	0.809566i	$0.300297\pi$
$954$		0			0
$955$	−1004.75	−	1740.29i	−1.05210	−	1.82229i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	407.432	+	705.693i	0.424851	+	0.735863i
$960$		0			0
$961$	−436.025	−	755.218i	−0.453721	−	0.785867i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	246.144i		−	0.255072i
$966$		0			0
$967$	1205.21			1.24634			0.623169	−	0.782087i	$-0.285845\pi$
$967$	1205.21			1.24634			0.623169	+	0.782087i	$0.285845\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	971.217	−	560.732i	1.00022	−	0.577479i	0.0919089	−	0.995767i	$-0.470703\pi$
$971$	971.217	−	560.732i	1.00022	−	0.577479i	0.908314	+	0.418288i	$0.137370\pi$
$972$		0			0
$973$	−842.630	+	1459.48i	−0.866012	+	1.49998i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	724.515	+	418.299i	0.741571	+	0.428146i	0.822640	−	0.568562i	$-0.192500\pi$
$977$	724.515	+	418.299i	0.741571	+	0.428146i	−0.0810693	+	0.996708i	$0.525834\pi$
$978$		0			0
$979$		−	280.949i		−	0.286975i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−410.062	−	236.749i	−0.417154	−	0.240844i	0.276705	−	0.960955i	$-0.410757\pi$
$983$	−410.062	−	236.749i	−0.417154	−	0.240844i	−0.693859	+	0.720111i	$0.744091\pi$
$984$		0			0
$985$	−1571.16			−1.59508
$986$		0			0
$987$		0			0
$988$		0			0
$989$	650.234			0.657466
$990$		0			0
$991$	579.617	−	334.642i	0.584881	−	0.337681i	−0.178190	−	0.983996i	$-0.557024\pi$
$991$	579.617	−	334.642i	0.584881	−	0.337681i	0.763071	+	0.646315i	$0.223691\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	264.803	−	458.652i	0.266134	−	0.460957i
$996$		0			0
$997$	−446.508			−0.447851			−0.223926	−	0.974606i	$-0.571887\pi$
$997$	−446.508			−0.447851			−0.223926	+	0.974606i	$0.571887\pi$
$998$		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.145.6		80
3.2	odd	2		684.3.bl.a.373.32	yes	80
9.2	odd	6		684.3.s.a.601.35	yes	80
9.7	even	3		2052.3.s.a.829.35		80
19.8	odd	6		2052.3.s.a.901.35		80
57.8	even	6		684.3.s.a.445.35	✓	80
171.65	even	6		684.3.bl.a.673.32	yes	80
171.160	odd	6	inner	2052.3.bl.a.1585.6		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.35	✓	80	57.8	even	6
684.3.s.a.601.35	yes	80	9.2	odd	6
684.3.bl.a.373.32	yes	80	3.2	odd	2
684.3.bl.a.673.32	yes	80	171.65	even	6
2052.3.s.a.829.35		80	9.7	even	3
2052.3.s.a.901.35		80	19.8	odd	6
2052.3.bl.a.145.6		80	1.1	even	1	trivial
2052.3.bl.a.1585.6		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)

\(f(q)\)	\(=\)	\(q-6.81841 q^{5} +(3.70019 + 6.40891i) q^{7} +O(q^{10})\)	\(q-6.81841 q^{5} +(3.70019 + 6.40891i) q^{7} +(-2.17728 - 3.77116i) q^{11} +(-4.41327 + 2.54800i) q^{13} +(-7.11551 - 12.3244i) q^{17} +(17.8833 - 6.41783i) q^{19} +(-4.25671 - 7.37283i) q^{23} +21.4907 q^{25} -11.3235i q^{29} +(-8.16773 - 4.71564i) q^{31} +(-25.2294 - 43.6986i) q^{35} -33.2038i q^{37} -0.431133i q^{41} +(-38.1888 + 66.1449i) q^{43} +47.6996 q^{47} +(-2.88279 + 4.99314i) q^{49} +(7.51673 + 4.33978i) q^{53} +(14.8456 + 25.7133i) q^{55} +65.1347i q^{59} +7.37572 q^{61} +(30.0915 - 17.3733i) q^{65} +(47.8702 - 27.6378i) q^{67} +(-42.1958 + 24.3618i) q^{71} +(-41.5917 - 72.0390i) q^{73} +(16.1127 - 27.9080i) q^{77} +(91.3114 + 52.7187i) q^{79} +(42.1532 + 73.0115i) q^{83} +(48.5165 + 84.0330i) q^{85} +(55.8744 + 32.2591i) q^{89} +(-32.6599 - 18.8562i) q^{91} +(-121.936 + 43.7594i) q^{95} +(-89.3395 - 51.5802i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 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

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