LMFDB - Embedded newform 2052.3.bl.a.145.5

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			145.5
Character	$\chi$	$=$	2052.145
Dual form			2052.3.bl.a.1585.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.99525 q^{5} +(0.133800 + 0.231748i) q^{7} +O(q^{10})$	$q-6.99525 q^{5} +(0.133800 + 0.231748i) q^{7} +(-9.29330 - 16.0965i) q^{11} +(-13.0625 + 7.54165i) q^{13} +(7.41969 + 12.8513i) q^{17} +(-10.6722 + 15.7196i) q^{19} +(-9.95881 - 17.2492i) q^{23} +23.9336 q^{25} +1.84793i q^{29} +(-40.4634 - 23.3615i) q^{31} +(-0.935965 - 1.62114i) q^{35} +19.8245i q^{37} -17.4765i q^{41} +(-37.5697 + 65.0726i) q^{43} -12.5523 q^{47} +(24.4642 - 42.3732i) q^{49} +(-1.06598 - 0.615444i) q^{53} +(65.0090 + 112.599i) q^{55} -88.5490i q^{59} +102.475 q^{61} +(91.3756 - 52.7558i) q^{65} +(-68.2631 + 39.4117i) q^{67} +(85.6563 - 49.4537i) q^{71} +(-10.6721 - 18.4847i) q^{73} +(2.48689 - 4.30742i) q^{77} +(26.0253 + 15.0257i) q^{79} +(27.5237 + 47.6725i) q^{83} +(-51.9026 - 89.8979i) q^{85} +(-122.281 - 70.5992i) q^{89} +(-3.49553 - 2.01815i) q^{91} +(74.6546 - 109.962i) q^{95} +(101.528 + 58.6171i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−6.99525			−1.39905			−0.699525	−	0.714608i	$-0.746605\pi$
$5$	−6.99525			−1.39905			−0.699525	+	0.714608i	$0.746605\pi$
$6$		0			0
$7$	0.133800	+	0.231748i	0.0191143	+	0.0331069i	0.875424	−	0.483355i	$-0.160582\pi$
$7$	0.133800	+	0.231748i	0.0191143	+	0.0331069i	−0.856310	+	0.516462i	$0.827249\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.29330	−	16.0965i	−0.844846	−	1.46332i	−0.885755	−	0.464153i	$-0.846359\pi$
$11$	−9.29330	−	16.0965i	−0.844846	−	1.46332i	0.0409092	−	0.999163i	$-0.486975\pi$
$12$		0			0
$13$	−13.0625	+	7.54165i	−1.00481	+	0.580127i	−0.909668	−	0.415337i	$-0.863664\pi$
$13$	−13.0625	+	7.54165i	−1.00481	+	0.580127i	−0.0951417	+	0.995464i	$0.530330\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.41969	+	12.8513i	0.436452	+	0.755958i	0.997413	−	0.0718849i	$-0.0229014\pi$
$17$	7.41969	+	12.8513i	0.436452	+	0.755958i	−0.560961	+	0.827842i	$0.689568\pi$
$18$		0			0
$19$	−10.6722	+	15.7196i	−0.561694	+	0.827345i
$19$	−10.6722	+	15.7196i	−0.561694	+	0.827345i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.95881	−	17.2492i	−0.432992	−	0.749964i	0.564137	−	0.825681i	$-0.309209\pi$
$23$	−9.95881	−	17.2492i	−0.432992	−	0.749964i	−0.997129	+	0.0757169i	$0.975875\pi$
$24$		0			0
$25$	23.9336			0.957342
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.84793i			0.0637218i	0.999492	+	0.0318609i	$0.0101434\pi$
$29$			1.84793i			0.0637218i	−0.999492	+	0.0318609i	$0.989857\pi$
$30$		0			0
$31$	−40.4634	−	23.3615i	−1.30527	−	0.753598i	−0.323967	−	0.946068i	$-0.605017\pi$
$31$	−40.4634	−	23.3615i	−1.30527	−	0.753598i	−0.981303	+	0.192470i	$0.938350\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.935965	−	1.62114i	−0.0267419	−	0.0463182i
$36$		0			0
$37$			19.8245i			0.535798i	0.963447	+	0.267899i	$0.0863293\pi$
$37$			19.8245i			0.535798i	−0.963447	+	0.267899i	$0.913671\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	17.4765i		−	0.426255i	−0.977024	−	0.213127i	$-0.931635\pi$
$41$		−	17.4765i		−	0.426255i	0.977024	−	0.213127i	$-0.0683650\pi$
$42$		0			0
$43$	−37.5697	+	65.0726i	−0.873713	+	1.51332i	−0.0155865	+	0.999879i	$0.504962\pi$
$43$	−37.5697	+	65.0726i	−0.873713	+	1.51332i	−0.858127	+	0.513438i	$0.828372\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12.5523			−0.267070			−0.133535	−	0.991044i	$-0.542633\pi$
$47$	−12.5523			−0.267070			−0.133535	+	0.991044i	$0.542633\pi$
$48$		0			0
$49$	24.4642	−	42.3732i	0.499269	−	0.864760i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.06598	−	0.615444i	−0.0201128	−	0.0116122i	0.489910	−	0.871773i	$-0.337030\pi$
$53$	−1.06598	−	0.615444i	−0.0201128	−	0.0116122i	−0.510023	+	0.860161i	$0.670363\pi$
$54$		0			0
$55$	65.0090	+	112.599i	1.18198	+	2.04725i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	88.5490i		−	1.50083i	−0.660967	−	0.750415i	$-0.729854\pi$
$59$		−	88.5490i		−	1.50083i	0.660967	−	0.750415i	$-0.270146\pi$
$60$		0			0
$61$	102.475			1.67992			0.839960	−	0.542649i	$-0.182578\pi$
$61$	102.475			1.67992			0.839960	+	0.542649i	$0.182578\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	91.3756	−	52.7558i	1.40578	−	0.811627i
$66$		0			0
$67$	−68.2631	+	39.4117i	−1.01885	+	0.588235i	−0.913771	−	0.406229i	$-0.866844\pi$
$67$	−68.2631	+	39.4117i	−1.01885	+	0.588235i	−0.105081	+	0.994464i	$0.533510\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	85.6563	−	49.4537i	1.20643	−	0.696531i	0.244450	−	0.969662i	$-0.421393\pi$
$71$	85.6563	−	49.4537i	1.20643	−	0.696531i	0.961977	+	0.273131i	$0.0880592\pi$
$72$		0			0
$73$	−10.6721	−	18.4847i	−0.146194	−	0.253215i	0.783624	−	0.621235i	$-0.213369\pi$
$73$	−10.6721	−	18.4847i	−0.146194	−	0.253215i	−0.929818	+	0.368021i	$0.880036\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.48689	−	4.30742i	0.0322973	−	0.0559405i
$78$		0			0
$79$	26.0253	+	15.0257i	0.329434	+	0.190199i	0.655590	−	0.755117i	$-0.272420\pi$
$79$	26.0253	+	15.0257i	0.329434	+	0.190199i	−0.326156	+	0.945316i	$0.605753\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	27.5237	+	47.6725i	0.331611	+	0.574368i	0.982828	−	0.184524i	$-0.0590744\pi$
$83$	27.5237	+	47.6725i	0.331611	+	0.574368i	−0.651217	+	0.758892i	$0.725741\pi$
$84$		0			0
$85$	−51.9026	−	89.8979i	−0.610619	−	1.05762i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−122.281	−	70.5992i	−1.37395	−	0.793249i	−0.382526	−	0.923945i	$-0.624946\pi$
$89$	−122.281	−	70.5992i	−1.37395	−	0.793249i	−0.991423	+	0.130695i	$0.958279\pi$
$90$		0			0
$91$	−3.49553	−	2.01815i	−0.0384124	−	0.0221774i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	74.6546	−	109.962i	0.785838	−	1.15750i
$96$		0			0
$97$	101.528	+	58.6171i	1.04668	+	0.604300i	0.921717	−	0.387862i	$-0.126786\pi$
$97$	101.528	+	58.6171i	1.04668	+	0.604300i	0.124960	+	0.992162i	$0.460120\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−95.2161			−0.942734			−0.471367	−	0.881937i	$-0.656239\pi$
$101$	−95.2161			−0.942734			−0.471367	+	0.881937i	$0.656239\pi$
$102$		0			0
$103$	103.934	+	60.0065i	1.00907	+	0.582588i	0.910920	−	0.412584i	$-0.135374\pi$
$103$	103.934	+	60.0065i	1.00907	+	0.582588i	0.0981518	+	0.995171i	$0.468707\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			63.7390i			0.595692i	0.954614	+	0.297846i	$0.0962681\pi$
$107$			63.7390i			0.595692i	−0.954614	+	0.297846i	$0.903732\pi$
$108$		0			0
$109$	134.646	−	77.7377i	1.23528	−	0.713190i	0.267155	−	0.963654i	$-0.413916\pi$
$109$	134.646	−	77.7377i	1.23528	−	0.713190i	0.968126	+	0.250464i	$0.0805831\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	178.339	+	102.964i	1.57823	+	0.911189i	0.995107	+	0.0988014i	$0.0315009\pi$
$113$	178.339	+	102.964i	1.57823	+	0.911189i	0.583118	+	0.812387i	$0.301832\pi$
$114$		0			0
$115$	69.6644	+	120.662i	0.605778	+	1.04924i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.98551	+	3.43900i	−0.0166850	+	0.0288992i
$120$		0			0
$121$	−112.231	+	194.390i	−0.927529	+	1.60653i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.46006			0.0596804
$126$		0			0
$127$	−2.83897	−	1.63908i	−0.0223541	−	0.0129061i	0.488781	−	0.872406i	$-0.337442\pi$
$127$	−2.83897	−	1.63908i	−0.0223541	−	0.0129061i	−0.511135	+	0.859500i	$0.670775\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−167.051			−1.27520			−0.637599	−	0.770369i	$-0.720072\pi$
$131$	−167.051			−1.27520			−0.637599	+	0.770369i	$0.720072\pi$
$132$		0			0
$133$	−5.07092	−	0.369985i	−0.0381272	−	0.00278185i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.0716			−0.0881139			−0.0440570	−	0.999029i	$-0.514028\pi$
$137$	−12.0716			−0.0881139			−0.0440570	+	0.999029i	$0.514028\pi$
$138$		0			0
$139$	16.4935	+	28.5675i	0.118658	+	0.205522i	0.919236	−	0.393707i	$-0.128807\pi$
$139$	16.4935	+	28.5675i	0.118658	+	0.205522i	−0.800578	+	0.599228i	$0.795474\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	242.788	+	140.174i	1.69782	+	0.980236i
$144$		0			0
$145$		−	12.9268i		−	0.0891500i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	188.156			1.26279			0.631395	−	0.775462i	$-0.282483\pi$
$149$	188.156			1.26279			0.631395	+	0.775462i	$0.282483\pi$
$150$		0			0
$151$	146.502	−	84.5828i	0.970210	−	0.560151i	0.0709097	−	0.997483i	$-0.477410\pi$
$151$	146.502	−	84.5828i	0.970210	−	0.560151i	0.899300	+	0.437332i	$0.144076\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	283.051	+	163.420i	1.82614	+	1.05432i
$156$		0			0
$157$	−76.3501			−0.486306			−0.243153	−	0.969988i	$-0.578182\pi$
$157$	−76.3501			−0.486306			−0.243153	+	0.969988i	$0.578182\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.66498	−	4.61588i	0.0165527	−	0.0286701i
$162$		0			0
$163$	119.907			0.735625			0.367813	−	0.929900i	$-0.380107\pi$
$163$	119.907			0.735625			0.367813	+	0.929900i	$0.380107\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−154.456	+	89.1754i	−0.924888	+	0.533985i	−0.885191	−	0.465227i	$-0.845973\pi$
$167$	−154.456	+	89.1754i	−0.924888	+	0.533985i	−0.0396970	+	0.999212i	$0.512639\pi$
$168$		0			0
$169$	29.2530	−	50.6677i	0.173095	−	0.299809i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	40.4348	+	23.3451i	0.233727	+	0.134943i	0.612290	−	0.790633i	$-0.290248\pi$
$173$	40.4348	+	23.3451i	0.233727	+	0.134943i	−0.378563	+	0.925575i	$0.623582\pi$
$174$		0			0
$175$	3.20231	+	5.54656i	0.0182989	+	0.0316946i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			181.221i			1.01241i	0.862413	+	0.506205i	$0.168952\pi$
$179$			181.221i			1.01241i	−0.862413	+	0.506205i	$0.831048\pi$
$180$		0			0
$181$	58.3219	+	33.6721i	0.322220	+	0.186034i	0.652382	−	0.757891i	$-0.273770\pi$
$181$	58.3219	+	33.6721i	0.322220	+	0.186034i	−0.330162	+	0.943924i	$0.607103\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	138.678i		−	0.749608i
$186$		0			0
$187$	137.907	−	238.862i	0.737470	−	1.27734i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.69790	+	16.7973i	0.0507744	+	0.0879438i	0.890296	−	0.455383i	$-0.150498\pi$
$191$	9.69790	+	16.7973i	0.0507744	+	0.0879438i	−0.839521	+	0.543327i	$0.817164\pi$
$192$		0			0
$193$		−	92.0948i		−	0.477175i	−0.971121	−	0.238588i	$-0.923316\pi$
$193$		−	92.0948i		−	0.477175i	0.971121	−	0.238588i	$-0.0766844\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	274.390			1.39284			0.696421	−	0.717633i	$-0.254775\pi$
$197$	274.390			1.39284			0.696421	+	0.717633i	$0.254775\pi$
$198$		0			0
$199$	163.455	−	283.112i	0.821379	−	1.42267i	−0.0832757	−	0.996527i	$-0.526538\pi$
$199$	163.455	−	283.112i	0.821379	−	1.42267i	0.904655	−	0.426144i	$-0.140128\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.428255	+	0.247253i	−0.00210963	+	0.00121800i
$204$		0			0
$205$			122.252i			0.596352i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	352.209	+	25.6980i	1.68521	+	0.122957i
$210$		0			0
$211$			374.799i			1.77630i	0.459557	+	0.888148i	$0.348008\pi$
$211$			374.799i			1.77630i	−0.459557	+	0.888148i	$0.651992\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	262.809	−	455.199i	1.22237	−	2.11721i
$216$		0			0
$217$		−	12.5031i		−	0.0576180i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−193.840	−	111.913i	−0.877103	−	0.506396i
$222$		0			0
$223$	−355.548	−	205.276i	−1.59438	−	0.920518i	−0.992542	−	0.121903i	$-0.961100\pi$
$223$	−355.548	−	205.276i	−1.59438	−	0.920518i	−0.601842	−	0.798615i	$-0.705566\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−253.428	+	146.316i	−1.11642	+	0.644566i	−0.940485	−	0.339836i	$-0.889629\pi$
$227$	−253.428	+	146.316i	−1.11642	+	0.644566i	−0.175936	+	0.984402i	$0.556295\pi$
$228$		0			0
$229$	63.7894	−	110.486i	0.278556	−	0.482474i	−0.692470	−	0.721447i	$-0.743477\pi$
$229$	63.7894	−	110.486i	0.278556	−	0.482474i	0.971026	+	0.238973i	$0.0768108\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	32.8387	+	56.8782i	0.140938	+	0.244113i	0.927850	−	0.372953i	$-0.121655\pi$
$233$	32.8387	+	56.8782i	0.140938	+	0.244113i	−0.786912	+	0.617065i	$0.788321\pi$
$234$		0			0
$235$	87.8064			0.373644
$236$		0			0
$237$		0			0
$238$		0			0
$239$	218.108	−	377.773i	0.912584	−	1.58064i	0.102183	−	0.994766i	$-0.467417\pi$
$239$	218.108	−	377.773i	0.912584	−	1.58064i	0.810401	−	0.585876i	$-0.199249\pi$
$240$		0			0
$241$			239.143i			0.992293i	0.868239	+	0.496146i	$0.165252\pi$
$241$			239.143i			0.992293i	−0.868239	+	0.496146i	$0.834748\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−171.133	+	296.411i	−0.698503	+	1.20984i
$246$		0			0
$247$	20.8543	−	285.823i	0.0844302	−	1.15718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−62.2217	+	107.771i	−0.247895	+	0.429367i	−0.962942	−	0.269710i	$-0.913072\pi$
$251$	−62.2217	+	107.771i	−0.247895	+	0.429367i	0.715046	+	0.699077i	$0.246406\pi$
$252$		0			0
$253$	−185.101	+	320.604i	−0.731623	+	1.26721i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	112.554	−	64.9829i	0.437952	−	0.252852i	−0.264777	−	0.964310i	$-0.585298\pi$
$257$	112.554	−	64.9829i	0.437952	−	0.252852i	0.702729	+	0.711458i	$0.251965\pi$
$258$		0			0
$259$	−4.59430	+	2.65252i	−0.0177386	+	0.0102414i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−35.3328	+	61.1982i	−0.134345	+	0.232693i	−0.925347	−	0.379121i	$-0.876226\pi$
$263$	−35.3328	+	61.1982i	−0.134345	+	0.232693i	0.791002	+	0.611814i	$0.209560\pi$
$264$		0			0
$265$	7.45681	+	4.30519i	0.0281389	+	0.0162460i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	108.899	−	62.8726i	0.404827	−	0.233727i	−0.283737	−	0.958902i	$-0.591574\pi$
$269$	108.899	−	62.8726i	0.404827	−	0.233727i	0.688565	+	0.725175i	$0.258241\pi$
$270$		0			0
$271$	98.9843	+	171.446i	0.365256	+	0.632642i	0.988817	−	0.149133i	$-0.0476482\pi$
$271$	98.9843	+	171.446i	0.365256	+	0.632642i	−0.623561	+	0.781774i	$0.714315\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−222.422	−	385.246i	−0.808807	−	1.40089i
$276$		0			0
$277$	−20.6263	−	35.7258i	−0.0744632	−	0.128974i	0.826390	−	0.563099i	$-0.190391\pi$
$277$	−20.6263	−	35.7258i	−0.0744632	−	0.128974i	−0.900853	+	0.434125i	$0.857058\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	126.286i		−	0.449417i	−0.974426	−	0.224708i	$-0.927857\pi$
$281$		−	126.286i		−	0.449417i	0.974426	−	0.224708i	$-0.0721430\pi$
$282$		0			0
$283$	190.119			0.671800			0.335900	−	0.941898i	$-0.390960\pi$
$283$	190.119			0.671800			0.335900	+	0.941898i	$0.390960\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.05014	−	2.33835i	0.0141120	−	0.00814756i
$288$		0			0
$289$	34.3964	−	59.5763i	0.119019	−	0.206147i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−197.636	−	114.105i	−0.674527	−	0.389438i	0.123263	−	0.992374i	$-0.460664\pi$
$293$	−197.636	−	114.105i	−0.674527	−	0.389438i	−0.797790	+	0.602936i	$0.793997\pi$
$294$		0			0
$295$			619.422i			2.09974i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	260.174	+	150.212i	0.870149	+	0.502381i
$300$		0			0
$301$	−20.1073			−0.0668016
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−716.839			−2.35029
$306$		0			0
$307$	−27.2921	+	15.7571i	−0.0888994	+	0.0513261i	−0.543791	−	0.839221i	$-0.683011\pi$
$307$	−27.2921	+	15.7571i	−0.0888994	+	0.0513261i	0.454891	+	0.890547i	$0.349678\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	86.5119	−	149.843i	0.278173	−	0.481810i	−0.692757	−	0.721171i	$-0.743604\pi$
$311$	86.5119	−	149.843i	0.278173	−	0.481810i	0.970931	+	0.239360i	$0.0769377\pi$
$312$		0			0
$313$	−121.542			−0.388315			−0.194157	−	0.980970i	$-0.562197\pi$
$313$	−121.542			−0.388315			−0.194157	+	0.980970i	$0.562197\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			242.092i			0.763696i	0.924225	+	0.381848i	$0.124712\pi$
$317$			242.092i			0.763696i	−0.924225	+	0.381848i	$0.875288\pi$
$318$		0			0
$319$	29.7452	−	17.1734i	0.0932451	−	0.0538351i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−281.201	−	20.5170i	−0.870590	−	0.0635202i
$324$		0			0
$325$	−312.633	+	180.499i	−0.961947	+	0.555380i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.67950	−	2.90897i	−0.00510485	−	0.00884186i
$330$		0			0
$331$	113.130	−	65.3154i	0.341781	−	0.197327i	−0.319278	−	0.947661i	$-0.603440\pi$
$331$	113.130	−	65.3154i	0.341781	−	0.197327i	0.661059	+	0.750334i	$0.270107\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	477.518	−	275.695i	1.42543	−	0.822971i
$336$		0			0
$337$		−	254.281i		−	0.754542i	−0.926103	−	0.377271i	$-0.876863\pi$
$337$		−	254.281i		−	0.754542i	0.926103	−	0.377271i	$-0.123137\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			868.424i			2.54670i
$342$		0			0
$343$	26.2056			0.0764013
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−665.971			−1.91922			−0.959612	−	0.281327i	$-0.909225\pi$
$347$	−665.971			−1.91922			−0.959612	+	0.281327i	$0.909225\pi$
$348$		0			0
$349$	96.7248	+	167.532i	0.277149	+	0.480035i	0.970675	−	0.240396i	$-0.0772772\pi$
$349$	96.7248	+	167.532i	0.277149	+	0.480035i	−0.693526	+	0.720431i	$0.743944\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−167.622	−	290.329i	−0.474849	−	0.822462i	0.524736	−	0.851265i	$-0.324164\pi$
$353$	−167.622	−	290.329i	−0.474849	−	0.822462i	−0.999585	+	0.0288025i	$0.990831\pi$
$354$		0			0
$355$	−599.188	+	345.941i	−1.68785	+	0.974482i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	143.086	+	247.832i	0.398568	+	0.690341i	0.993550	−	0.113399i	$-0.0361738\pi$
$359$	143.086	+	247.832i	0.398568	+	0.690341i	−0.594981	+	0.803740i	$0.702840\pi$
$360$		0			0
$361$	−133.209	−	335.524i	−0.369000	−	0.929430i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	74.6543	+	129.305i	0.204532	+	0.354260i
$366$		0			0
$367$	−551.097			−1.50163			−0.750813	−	0.660515i	$-0.770338\pi$
$367$	−551.097			−1.50163			−0.750813	+	0.660515i	$0.770338\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	0.329386i		−	0.000887833i
$372$		0			0
$373$	444.407	+	256.579i	1.19144	+	0.687878i	0.958633	−	0.284645i	$-0.0918756\pi$
$373$	444.407	+	256.579i	1.19144	+	0.687878i	0.232807	+	0.972523i	$0.425209\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13.9365	−	24.1387i	−0.0369667	−	0.0640283i
$378$		0			0
$379$		−	647.818i		−	1.70928i	−0.519220	−	0.854641i	$-0.673777\pi$
$379$		−	647.818i		−	1.70928i	0.519220	−	0.854641i	$-0.326223\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			183.882i			0.480108i	0.970759	+	0.240054i	$0.0771652\pi$
$383$			183.882i			0.480108i	−0.970759	+	0.240054i	$0.922835\pi$
$384$		0			0
$385$	−17.3964	+	30.1315i	−0.0451855	+	0.0782636i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	602.686			1.54932			0.774661	−	0.632377i	$-0.217921\pi$
$389$	602.686			1.54932			0.774661	+	0.632377i	$0.217921\pi$
$390$		0			0
$391$	147.783	−	255.967i	0.377961	−	0.654647i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−182.053	−	105.109i	−0.460895	−	0.266098i
$396$		0			0
$397$	60.0424	+	103.996i	0.151240	+	0.261956i	0.931684	−	0.363271i	$-0.118340\pi$
$397$	60.0424	+	103.996i	0.151240	+	0.261956i	−0.780443	+	0.625226i	$0.785007\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	466.323i		−	1.16290i	−0.813582	−	0.581450i	$-0.802486\pi$
$401$		−	466.323i		−	1.16290i	0.813582	−	0.581450i	$-0.197514\pi$
$402$		0			0
$403$	704.738			1.74873
$404$		0			0
$405$		0			0
$406$		0			0
$407$	319.105	−	184.235i	0.784042	−	0.452667i
$408$		0			0
$409$	24.5546	−	14.1766i	0.0600356	−	0.0346616i	−0.469682	−	0.882836i	$-0.655631\pi$
$409$	24.5546	−	14.1766i	0.0600356	−	0.0346616i	0.529717	+	0.848174i	$0.322298\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	20.5211	−	11.8479i	0.0496878	−	0.0286873i
$414$		0			0
$415$	−192.536	−	333.481i	−0.463941	−	0.803570i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	151.556	−	262.502i	0.361708	−	0.626497i	−0.626534	−	0.779394i	$-0.715527\pi$
$419$	151.556	−	262.502i	0.361708	−	0.626497i	0.988242	+	0.152897i	$0.0488604\pi$
$420$		0			0
$421$	526.909	+	304.211i	1.25156	+	0.722591i	0.971420	−	0.237367i	$-0.0762844\pi$
$421$	526.909	+	304.211i	1.25156	+	0.722591i	0.280144	+	0.959958i	$0.409618\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	177.580	+	307.577i	0.417834	+	0.723710i
$426$		0			0
$427$	13.7112	+	23.7484i	0.0321105	+	0.0556170i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.5148	−	10.6895i	−0.0429577	−	0.0248016i	0.478367	−	0.878160i	$-0.341229\pi$
$431$	−18.5148	−	10.6895i	−0.0429577	−	0.0248016i	−0.521325	+	0.853358i	$0.674562\pi$
$432$		0			0
$433$	−107.300	−	61.9498i	−0.247807	−	0.143071i	0.370953	−	0.928652i	$-0.379031\pi$
$433$	−107.300	−	61.9498i	−0.247807	−	0.143071i	−0.618760	+	0.785580i	$0.712364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	377.432	+	27.5382i	0.863688	+	0.0630166i
$438$		0			0
$439$	−478.498	−	276.261i	−1.08997	−	0.629296i	−0.156404	−	0.987693i	$-0.549990\pi$
$439$	−478.498	−	276.261i	−1.08997	−	0.629296i	−0.933569	+	0.358397i	$0.883323\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−248.445			−0.560823			−0.280411	−	0.959880i	$-0.590471\pi$
$443$	−248.445			−0.560823			−0.280411	+	0.959880i	$0.590471\pi$
$444$		0			0
$445$	855.389	+	493.859i	1.92222	+	1.10980i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	105.045i		−	0.233954i	−0.993135	−	0.116977i	$-0.962680\pi$
$449$		−	105.045i		−	0.233954i	0.993135	−	0.116977i	$-0.0373204\pi$
$450$		0			0
$451$	−281.309	+	162.414i	−0.623746	+	0.360120i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	24.4521	+	14.1174i	0.0537409	+	0.0310273i
$456$		0			0
$457$	128.074	+	221.830i	0.280249	+	0.485406i	0.971446	−	0.237261i	$-0.0762496\pi$
$457$	128.074	+	221.830i	0.280249	+	0.485406i	−0.691197	+	0.722667i	$0.742916\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.1628	−	26.2628i	0.0328912	−	0.0569692i	−0.849111	−	0.528214i	$-0.822862\pi$
$461$	15.1628	−	26.2628i	0.0328912	−	0.0569692i	0.882002	+	0.471245i	$0.156195\pi$
$462$		0			0
$463$	227.150	−	393.435i	0.490604	−	0.849752i	−0.509337	−	0.860567i	$-0.670109\pi$
$463$	227.150	−	393.435i	0.490604	−	0.849752i	0.999942	+	0.0108153i	$0.00344267\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	53.0719			0.113644			0.0568222	−	0.998384i	$-0.481903\pi$
$467$	53.0719			0.113644			0.0568222	+	0.998384i	$0.481903\pi$
$468$		0			0
$469$	−18.2672	−	10.5466i	−0.0389493	−	0.0224874i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1396.59			2.95261
$474$		0			0
$475$	−255.423	+	376.225i	−0.537733	+	0.792052i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.5493			0.0282866			0.0141433	−	0.999900i	$-0.495498\pi$
$479$	13.5493			0.0282866			0.0141433	+	0.999900i	$0.495498\pi$
$480$		0			0
$481$	−149.510	−	258.958i	−0.310831	−	0.538375i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−710.212	−	410.041i	−1.46435	−	0.845446i
$486$		0			0
$487$			318.476i			0.653956i	0.945032	+	0.326978i	$0.106030\pi$
$487$			318.476i			0.653956i	−0.945032	+	0.326978i	$0.893970\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	444.969			0.906251			0.453126	−	0.891447i	$-0.350309\pi$
$491$	444.969			0.906251			0.453126	+	0.891447i	$0.350309\pi$
$492$		0			0
$493$	−23.7483	+	13.7111i	−0.0481710	+	0.0278115i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	22.9216	+	13.2338i	0.0461200	+	0.0266274i
$498$		0			0
$499$	−815.470			−1.63421			−0.817104	−	0.576490i	$-0.804422\pi$
$499$	−815.470			−1.63421			−0.817104	+	0.576490i	$0.804422\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−201.885	+	349.675i	−0.401361	+	0.695178i	−0.993890	−	0.110371i	$-0.964796\pi$
$503$	−201.885	+	349.675i	−0.401361	+	0.695178i	0.592529	+	0.805549i	$0.298129\pi$
$504$		0			0
$505$	666.061			1.31893
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−696.774	+	402.282i	−1.36891	+	0.790339i	−0.990789	−	0.135418i	$-0.956762\pi$
$509$	−696.774	+	402.282i	−1.36891	+	0.790339i	−0.378119	+	0.925757i	$0.623429\pi$
$510$		0			0
$511$	2.85586	−	4.94650i	0.00558878	−	0.00968004i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−727.047	−	419.761i	−1.41174	−	0.815070i
$516$		0			0
$517$	116.652	+	202.047i	0.225633	+	0.390807i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			191.360i			0.367294i	0.982992	+	0.183647i	$0.0587903\pi$
$521$			191.360i			0.367294i	−0.982992	+	0.183647i	$0.941210\pi$
$522$		0			0
$523$	353.793	+	204.262i	0.676468	+	0.390559i	0.798523	−	0.601965i	$-0.205615\pi$
$523$	353.793	+	204.262i	0.676468	+	0.390559i	−0.122055	+	0.992523i	$0.538949\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	693.341i		−	1.31564i
$528$		0			0
$529$	66.1441	−	114.565i	0.125036	−	0.216569i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	131.801	+	228.287i	0.247282	+	0.428305i
$534$		0			0
$535$		−	445.870i		−	0.833403i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−909.413			−1.68722
$540$		0			0
$541$	28.1693	−	48.7906i	0.0520689	−	0.0901860i	−0.838816	−	0.544415i	$-0.816752\pi$
$541$	28.1693	−	48.7906i	0.0520689	−	0.0901860i	0.890885	+	0.454229i	$0.150085\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−941.880	+	543.795i	−1.72822	+	0.997789i
$546$		0			0
$547$		−	463.249i		−	0.846891i	−0.905922	−	0.423445i	$-0.860821\pi$
$547$		−	463.249i		−	0.846891i	0.905922	−	0.423445i	$-0.139179\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−29.0487	−	19.7215i	−0.0527199	−	0.0357922i
$552$		0			0
$553$			8.04176i			0.0145421i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−190.449	+	329.868i	−0.341920	+	0.592223i	−0.984789	−	0.173753i	$-0.944411\pi$
$557$	−190.449	+	329.868i	−0.341920	+	0.592223i	0.642869	+	0.765976i	$0.277744\pi$
$558$		0			0
$559$		−	1133.35i		−	2.02746i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−145.097	−	83.7716i	−0.257720	−	0.148795i	0.365574	−	0.930782i	$-0.380873\pi$
$563$	−145.097	−	83.7716i	−0.257720	−	0.148795i	−0.623294	+	0.781987i	$0.714206\pi$
$564$		0			0
$565$	−1247.53	−	720.261i	−2.20802	−	1.27480i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−165.234	+	95.3978i	−0.290394	+	0.167659i	−0.638119	−	0.769938i	$-0.720287\pi$
$569$	−165.234	+	95.3978i	−0.290394	+	0.167659i	0.347726	+	0.937596i	$0.386954\pi$
$570$		0			0
$571$	−537.327	+	930.678i	−0.941028	+	1.62991i	−0.177513	+	0.984118i	$0.556805\pi$
$571$	−537.327	+	930.678i	−0.941028	+	1.62991i	−0.763515	+	0.645790i	$0.776528\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−238.350	−	412.834i	−0.414521	−	0.717972i
$576$		0			0
$577$	−335.073			−0.580716			−0.290358	−	0.956918i	$-0.593774\pi$
$577$	−335.073			−0.580716			−0.290358	+	0.956918i	$0.593774\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.36536	+	12.7572i	−0.0126770	+	0.0219573i
$582$		0			0
$583$			22.8780i			0.0392419i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	353.588	−	612.433i	0.602365	−	1.04333i	−0.390097	−	0.920774i	$-0.627558\pi$
$587$	353.588	−	612.433i	0.602365	−	1.04333i	0.992462	−	0.122553i	$-0.0391082\pi$
$588$		0			0
$589$	799.066	−	386.748i	1.35665	−	0.656617i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−517.354	+	896.083i	−0.872434	+	1.51110i	−0.0129636	+	0.999916i	$0.504127\pi$
$593$	−517.354	+	896.083i	−0.872434	+	1.51110i	−0.859471	+	0.511185i	$0.829207\pi$
$594$		0			0
$595$	13.8891	−	24.0567i	0.0233431	−	0.0404314i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−506.558	+	292.462i	−0.845674	+	0.488250i	−0.859189	−	0.511659i	$-0.829031\pi$
$599$	−506.558	+	292.462i	−0.845674	+	0.488250i	0.0135152	+	0.999909i	$0.495698\pi$
$600$		0			0
$601$	−71.6749	+	41.3815i	−0.119259	+	0.0688544i	−0.558443	−	0.829543i	$-0.688601\pi$
$601$	−71.6749	+	41.3815i	−0.119259	+	0.0688544i	0.439184	+	0.898397i	$0.355268\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	785.084	−	1359.81i	1.29766	−	2.24761i
$606$		0			0
$607$	459.170	+	265.102i	0.756458	+	0.436741i	0.828023	−	0.560695i	$-0.189466\pi$
$607$	459.170	+	265.102i	0.756458	+	0.436741i	−0.0715644	+	0.997436i	$0.522799\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	163.964	−	94.6649i	0.268354	−	0.154934i
$612$		0			0
$613$	328.438	+	568.872i	0.535788	+	0.928013i	0.999125	+	0.0418302i	$0.0133189\pi$
$613$	328.438	+	568.872i	0.535788	+	0.928013i	−0.463336	+	0.886183i	$0.653348\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	552.704	+	957.311i	0.895792	+	1.55156i	0.832821	+	0.553542i	$0.186724\pi$
$617$	552.704	+	957.311i	0.895792	+	1.55156i	0.0629714	+	0.998015i	$0.479942\pi$
$618$		0			0
$619$	−579.962	−	1004.52i	−0.936933	−	1.62282i	−0.771150	−	0.636654i	$-0.780318\pi$
$619$	−579.962	−	1004.52i	−0.936933	−	1.62282i	−0.165783	−	0.986162i	$-0.553015\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	37.7847i		−	0.0606496i
$624$		0			0
$625$	−650.524			−1.04084
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−254.770	+	147.092i	−0.405040	+	0.233850i
$630$		0			0
$631$	−134.726	+	233.352i	−0.213511	+	0.369812i	−0.952811	−	0.303564i	$-0.901823\pi$
$631$	−134.726	+	233.352i	−0.213511	+	0.369812i	0.739300	+	0.673376i	$0.235157\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.8593	+	11.4658i	0.0312745	+	0.0180563i
$636$		0			0
$637$			738.002i			1.15856i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	787.130	+	454.450i	1.22797	+	0.708970i	0.966605	−	0.256269i	$-0.0824934\pi$
$641$	787.130	+	454.450i	1.22797	+	0.708970i	0.261367	+	0.965240i	$0.415827\pi$
$642$		0			0
$643$	115.202			0.179163			0.0895813	−	0.995980i	$-0.471447\pi$
$643$	115.202			0.179163			0.0895813	+	0.995980i	$0.471447\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1207.11			−1.86570			−0.932850	−	0.360264i	$-0.882687\pi$
$647$	−1207.11			−1.86570			−0.932850	+	0.360264i	$0.882687\pi$
$648$		0			0
$649$	−1425.33	+	822.912i	−2.19619	+	1.26797i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−293.553	+	508.449i	−0.449545	+	0.778635i	−0.998356	−	0.0573114i	$-0.981747\pi$
$653$	−293.553	+	508.449i	−0.449545	+	0.778635i	0.548811	+	0.835946i	$0.315081\pi$
$654$		0			0
$655$	1168.56			1.78407
$656$		0			0
$657$		0			0
$658$		0			0
$659$			683.322i			1.03691i	0.855106	+	0.518453i	$0.173492\pi$
$659$			683.322i			1.03691i	−0.855106	+	0.518453i	$0.826508\pi$
$660$		0			0
$661$	382.613	−	220.902i	0.578839	−	0.334193i	−0.181833	−	0.983329i	$-0.558203\pi$
$661$	382.613	−	220.902i	0.578839	−	0.334193i	0.760672	+	0.649136i	$0.224870\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	35.4724	+	2.58814i	0.0533419	+	0.00389194i
$666$		0			0
$667$	31.8753	−	18.4032i	0.0477891	−	0.0275910i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−952.332	−	1649.49i	−1.41927	−	2.45825i
$672$		0			0
$673$	−592.886	+	342.303i	−0.880960	+	0.508623i	−0.870975	−	0.491328i	$-0.836512\pi$
$673$	−592.886	+	342.303i	−0.880960	+	0.508623i	−0.00998520	+	0.999950i	$0.503178\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−496.365	+	286.576i	−0.733183	+	0.423303i	−0.819585	−	0.572957i	$-0.805796\pi$
$677$	−496.365	+	286.576i	−0.733183	+	0.423303i	0.0864027	+	0.996260i	$0.472463\pi$
$678$		0			0
$679$			31.3719i			0.0462030i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	313.554i		−	0.459083i	−0.973299	−	0.229542i	$-0.926277\pi$
$683$		−	313.554i		−	0.459083i	0.973299	−	0.229542i	$-0.0737227\pi$
$684$		0			0
$685$	84.4440			0.123276
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18.5659			0.0269461
$690$		0			0
$691$	−326.726	−	565.905i	−0.472830	−	0.818966i	0.526686	−	0.850060i	$-0.323434\pi$
$691$	−326.726	−	565.905i	−0.472830	−	0.818966i	−0.999516	+	0.0310940i	$0.990101\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−115.376	−	199.837i	−0.166009	−	0.287535i
$696$		0			0
$697$	224.595	−	129.670i	0.322231	−	0.186040i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−581.857	−	1007.81i	−0.830039	−	1.43767i	−0.898007	−	0.439982i	$-0.854985\pi$
$701$	−581.857	−	1007.81i	−0.830039	−	1.43767i	0.0679678	−	0.997688i	$-0.478348\pi$
$702$		0			0
$703$	−311.633	−	211.571i	−0.443290	−	0.300954i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.7399	−	22.0662i	−0.0180197	−	0.0312110i
$708$		0			0
$709$	75.4560			0.106426			0.0532129	−	0.998583i	$-0.483054\pi$
$709$	75.4560			0.106426			0.0532129	+	0.998583i	$0.483054\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			930.613i			1.30521i
$714$		0			0
$715$	−1698.36	−	980.551i	−2.37533	−	1.37140i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	495.695	+	858.570i	0.689423	+	1.19412i	0.972025	+	0.234879i	$0.0754694\pi$
$719$	495.695	+	858.570i	0.689423	+	1.19412i	−0.282601	+	0.959237i	$0.591197\pi$
$720$		0			0
$721$			32.1155i			0.0445430i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			44.2276i			0.0610036i
$726$		0			0
$727$	553.074	−	957.953i	0.760763	−	1.31768i	−0.181696	−	0.983355i	$-0.558159\pi$
$727$	553.074	−	957.953i	0.760763	−	1.31768i	0.942458	−	0.334324i	$-0.108508\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1115.02			−1.52534
$732$		0			0
$733$	−283.607	+	491.222i	−0.386913	+	0.670153i	−0.992033	−	0.125982i	$-0.959792\pi$
$733$	−283.607	+	491.222i	−0.386913	+	0.670153i	0.605120	+	0.796134i	$0.293125\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1268.78	+	732.531i	1.72155	+	0.993936i
$738$		0			0
$739$	229.797	+	398.021i	0.310957	+	0.538594i	0.978570	−	0.205915i	$-0.0660171\pi$
$739$	229.797	+	398.021i	0.310957	+	0.538594i	−0.667613	+	0.744509i	$0.732684\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	318.126i		−	0.428164i	−0.976816	−	0.214082i	$-0.931324\pi$
$743$		−	318.126i		−	0.428164i	0.976816	−	0.214082i	$-0.0686760\pi$
$744$		0			0
$745$	−1316.20			−1.76671
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14.7714	+	8.52828i	−0.0197215	+	0.0113862i
$750$		0			0
$751$	432.098	−	249.472i	0.575364	−	0.332186i	−0.183925	−	0.982940i	$-0.558880\pi$
$751$	432.098	−	249.472i	0.575364	−	0.332186i	0.759289	+	0.650754i	$0.225547\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1024.82	+	591.678i	−1.35737	+	0.783679i
$756$		0			0
$757$	63.9290	+	110.728i	0.0844505	+	0.146273i	0.905157	−	0.425078i	$-0.139753\pi$
$757$	63.9290	+	110.728i	0.0844505	+	0.146273i	−0.820706	+	0.571350i	$0.806420\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	243.975	−	422.577i	0.320598	−	0.555292i	−0.660014	−	0.751254i	$-0.729450\pi$
$761$	243.975	−	422.577i	0.320598	−	0.555292i	0.980612	+	0.195962i	$0.0627828\pi$
$762$		0			0
$763$	36.0312	+	20.8026i	0.0472230	+	0.0272642i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	667.805	+	1156.67i	0.870672	+	1.50805i
$768$		0			0
$769$	381.726	+	661.169i	0.496393	+	0.859778i	0.999991	−	0.00416002i	$-0.00132418\pi$
$769$	381.726	+	661.169i	0.496393	+	0.859778i	−0.503598	+	0.863938i	$0.667991\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−181.681	−	104.893i	−0.235033	−	0.135696i	0.377859	−	0.925863i	$-0.376661\pi$
$773$	−181.681	−	104.893i	−0.235033	−	0.135696i	−0.612892	+	0.790167i	$0.709994\pi$
$774$		0			0
$775$	−968.432	−	559.125i	−1.24959	−	0.721451i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	274.722	+	186.512i	0.352660	+	0.239425i
$780$		0			0
$781$	−1592.06	−	919.176i	−2.03849	−	1.17692i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	534.088			0.680367
$786$		0			0
$787$	−44.6872	−	25.8001i	−0.0567816	−	0.0327829i	0.471340	−	0.881951i	$-0.343770\pi$
$787$	−44.6872	−	25.8001i	−0.0567816	−	0.0327829i	−0.528122	+	0.849168i	$0.677104\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			55.1065i			0.0696669i
$792$		0			0
$793$	−1338.58	+	772.831i	−1.68800	+	0.974567i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1186.04	−	684.759i	−1.48813	−	0.859170i	−0.488218	−	0.872721i	$-0.662353\pi$
$797$	−1186.04	−	684.759i	−1.48813	−	0.859170i	−0.999908	+	0.0135512i	$0.995686\pi$
$798$		0			0
$799$	−93.1340	−	161.313i	−0.116563	−	0.201893i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−198.359	+	343.568i	−0.247022	+	0.427855i
$804$		0			0
$805$	−18.6422	+	32.2892i	−0.0231580	+	0.0401109i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	656.453			0.811438			0.405719	−	0.913998i	$-0.367021\pi$
$809$	656.453			0.811438			0.405719	+	0.913998i	$0.367021\pi$
$810$		0			0
$811$	1122.54	+	648.101i	1.38415	+	0.799138i	0.992648	−	0.121039i	$-0.0386228\pi$
$811$	1122.54	+	648.101i	1.38415	+	0.799138i	0.391501	+	0.920178i	$0.371956\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−838.779			−1.02918
$816$		0			0
$817$	−621.962	−	1285.05i	−0.761275	−	1.57288i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−993.523			−1.21014			−0.605069	−	0.796173i	$-0.706854\pi$
$821$	−993.523			−1.21014			−0.605069	+	0.796173i	$0.706854\pi$
$822$		0			0
$823$	−563.254	−	975.584i	−0.684391	−	1.18540i	−0.973628	−	0.228142i	$-0.926735\pi$
$823$	−563.254	−	975.584i	−0.684391	−	1.18540i	0.289237	−	0.957257i	$-0.406598\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	135.268	+	78.0970i	0.163565	+	0.0944341i	0.579548	−	0.814938i	$-0.303229\pi$
$827$	135.268	+	78.0970i	0.163565	+	0.0944341i	−0.415983	+	0.909372i	$0.636563\pi$
$828$		0			0
$829$		−	3.38334i		−	0.00408123i	−0.999998	−	0.00204062i	$-0.999350\pi$
$829$		−	3.38334i		−	0.00408123i	0.999998	−	0.00204062i	$-0.000649549\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	726.067			0.871629
$834$		0			0
$835$	1080.46	−	623.805i	1.29397	−	0.747071i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	546.285	+	315.398i	0.651115	+	0.375921i	0.788883	−	0.614543i	$-0.210660\pi$
$839$	546.285	+	315.398i	0.651115	+	0.375921i	−0.137768	+	0.990464i	$0.543993\pi$
$840$		0			0
$841$	837.585			0.995940
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−204.632	+	354.433i	−0.242168	+	0.419448i
$846$		0			0
$847$	−60.0660			−0.0709162
$848$		0			0
$849$		0			0
$850$		0			0
$851$	341.957	−	197.429i	0.401829	−	0.231996i
$852$		0			0
$853$	507.394	−	878.832i	0.594834	−	1.03028i	−0.398736	−	0.917066i	$-0.630551\pi$
$853$	507.394	−	878.832i	0.594834	−	1.03028i	0.993570	−	0.113217i	$-0.0361156\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	427.275	+	246.687i	0.498570	+	0.287850i	0.728123	−	0.685447i	$-0.240393\pi$
$857$	427.275	+	246.687i	0.498570	+	0.287850i	−0.229553	+	0.973296i	$0.573726\pi$
$858$		0			0
$859$	−400.876	−	694.338i	−0.466678	−	0.808310i	0.532598	−	0.846369i	$-0.321216\pi$
$859$	−400.876	−	694.338i	−0.466678	−	0.808310i	−0.999276	+	0.0380587i	$0.987883\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	563.699i		−	0.653185i	−0.945165	−	0.326593i	$-0.894099\pi$
$863$		−	563.699i		−	0.653185i	0.945165	−	0.326593i	$-0.105901\pi$
$864$		0			0
$865$	−282.852	−	163.305i	−0.326996	−	0.188791i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	558.554i		−	0.642754i
$870$		0			0
$871$	594.459	−	1029.63i	0.682502	−	1.18213i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.998155	+	1.72886i	0.00114075	+	0.00197584i
$876$		0			0
$877$		−	157.606i		−	0.179710i	−0.995955	−	0.0898552i	$-0.971360\pi$
$877$		−	157.606i		−	0.179710i	0.995955	−	0.0898552i	$-0.0286404\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1304.72			−1.48095			−0.740476	−	0.672083i	$-0.765400\pi$
$881$	−1304.72			−1.48095			−0.740476	+	0.672083i	$0.765400\pi$
$882$		0			0
$883$	388.325	−	672.599i	0.439779	−	0.761720i	−0.557893	−	0.829913i	$-0.688390\pi$
$883$	388.325	−	672.599i	0.439779	−	0.761720i	0.997672	+	0.0681929i	$0.0217233\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−59.4386	+	34.3169i	−0.0670108	+	0.0386887i	−0.533131	−	0.846033i	$-0.678985\pi$
$887$	−59.4386	+	34.3169i	−0.0670108	+	0.0386887i	0.466120	+	0.884721i	$0.345651\pi$
$888$		0			0
$889$		−	0.877235i		−	0.000986765i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	133.960	−	197.316i	0.150011	−	0.220959i
$894$		0			0
$895$		−	1267.69i		−	1.41641i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	43.1706	−	74.7736i	0.0480206	−	0.0831742i
$900$		0			0
$901$		−	18.2656i		−	0.0202726i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−407.976	−	235.545i	−0.450802	−	0.260271i
$906$		0			0
$907$	−247.248	−	142.749i	−0.272600	−	0.157386i	0.357469	−	0.933925i	$-0.383640\pi$
$907$	−247.248	−	142.749i	−0.272600	−	0.157386i	−0.630069	+	0.776540i	$0.716973\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	327.463	−	189.061i	0.359454	−	0.207531i	−0.309387	−	0.950936i	$-0.600124\pi$
$911$	327.463	−	189.061i	0.359454	−	0.207531i	0.668841	+	0.743405i	$0.266791\pi$
$912$		0			0
$913$	511.573	−	886.071i	0.560321	−	0.970505i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.3514	−	38.7138i	−0.0243745	−	0.0422178i
$918$		0			0
$919$	956.436			1.04074			0.520368	−	0.853942i	$-0.325795\pi$
$919$	956.436			1.04074			0.520368	+	0.853942i	$0.325795\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−745.925	+	1291.98i	−0.808153	+	1.39976i
$924$		0			0
$925$			474.471i			0.512942i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	493.442	−	854.667i	0.531154	−	0.919986i	−0.468185	−	0.883631i	$-0.655092\pi$
$929$	493.442	−	854.667i	0.531154	−	0.919986i	0.999339	−	0.0363555i	$-0.0115749\pi$
$930$		0			0
$931$	405.002	+	836.781i	0.435018	+	0.898798i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−964.693	+	1670.90i	−1.03176	+	1.78706i
$936$		0			0
$937$	−858.403	+	1486.80i	−0.916118	+	1.58676i	−0.110862	+	0.993836i	$0.535361\pi$
$937$	−858.403	+	1486.80i	−0.916118	+	1.58676i	−0.805256	+	0.592927i	$0.797972\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1004.00	−	579.658i	1.06695	−	0.616002i	0.139601	−	0.990208i	$-0.455418\pi$
$941$	1004.00	−	579.658i	1.06695	−	0.616002i	0.927346	+	0.374206i	$0.122085\pi$
$942$		0			0
$943$	−301.454	+	174.045i	−0.319676	+	0.184565i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	848.667	−	1469.93i	0.896163	−	1.55220i	0.0638053	−	0.997962i	$-0.479676\pi$
$947$	848.667	−	1469.93i	0.896163	−	1.55220i	0.832358	−	0.554238i	$-0.186990\pi$
$948$		0			0
$949$	278.810	+	160.971i	0.293794	+	0.169622i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−895.156	+	516.819i	−0.939303	+	0.542307i	−0.889742	−	0.456464i	$-0.849116\pi$
$953$	−895.156	+	516.819i	−0.939303	+	0.542307i	−0.0495615	+	0.998771i	$0.515782\pi$
$954$		0			0
$955$	−67.8393	−	117.501i	−0.0710359	−	0.123038i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.61518	−	2.79758i	−0.00168424	−	0.00291718i
$960$		0			0
$961$	611.023	+	1058.32i	0.635820	+	1.10127i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			644.226i			0.667592i
$966$		0			0
$967$	−1534.08			−1.58644			−0.793219	−	0.608937i	$-0.791596\pi$
$967$	−1534.08			−1.58644			−0.793219	+	0.608937i	$0.791596\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−145.053	+	83.7464i	−0.149385	+	0.0862475i	−0.572829	−	0.819675i	$-0.694154\pi$
$971$	−145.053	+	83.7464i	−0.149385	+	0.0862475i	0.423444	+	0.905922i	$0.360821\pi$
$972$		0			0
$973$	−4.41365	+	7.64467i	−0.00453613	+	0.00785680i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	265.644	+	153.369i	0.271897	+	0.156980i	0.629750	−	0.776798i	$-0.283158\pi$
$977$	265.644	+	153.369i	0.271897	+	0.156980i	−0.357852	+	0.933778i	$0.616491\pi$
$978$		0			0
$979$			2624.40i			2.68069i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1417.32	+	818.292i	1.44183	+	0.832444i	0.997972	−	0.0636515i	$-0.0202746\pi$
$983$	1417.32	+	818.292i	1.44183	+	0.832444i	0.443862	+	0.896095i	$0.353608\pi$
$984$		0			0
$985$	−1919.43			−1.94866
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1496.60			1.51324
$990$		0			0
$991$	468.337	−	270.394i	0.472590	−	0.272850i	−0.244733	−	0.969590i	$-0.578700\pi$
$991$	468.337	−	270.394i	0.472590	−	0.272850i	0.717323	+	0.696740i	$0.245367\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1143.41	+	1980.44i	−1.14915	+	1.99039i
$996$		0			0
$997$	1897.28			1.90299			0.951493	−	0.307669i	$-0.0995490\pi$
$997$	1897.28			1.90299			0.951493	+	0.307669i	$0.0995490\pi$
$998$		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.145.5		80
3.2	odd	2		684.3.bl.a.373.5	yes	80
9.2	odd	6		684.3.s.a.601.17	yes	80
9.7	even	3		2052.3.s.a.829.36		80
19.8	odd	6		2052.3.s.a.901.36		80
57.8	even	6		684.3.s.a.445.17	✓	80
171.65	even	6		684.3.bl.a.673.5	yes	80
171.160	odd	6	inner	2052.3.bl.a.1585.5		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.17	✓	80	57.8	even	6
684.3.s.a.601.17	yes	80	9.2	odd	6
684.3.bl.a.373.5	yes	80	3.2	odd	2
684.3.bl.a.673.5	yes	80	171.65	even	6
2052.3.s.a.829.36		80	9.7	even	3
2052.3.s.a.901.36		80	19.8	odd	6
2052.3.bl.a.145.5		80	1.1	even	1	trivial
2052.3.bl.a.1585.5		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)

\(f(q)\)	\(=\)	\(q-6.99525 q^{5} +(0.133800 + 0.231748i) q^{7} +O(q^{10})\)	\(q-6.99525 q^{5} +(0.133800 + 0.231748i) q^{7} +(-9.29330 - 16.0965i) q^{11} +(-13.0625 + 7.54165i) q^{13} +(7.41969 + 12.8513i) q^{17} +(-10.6722 + 15.7196i) q^{19} +(-9.95881 - 17.2492i) q^{23} +23.9336 q^{25} +1.84793i q^{29} +(-40.4634 - 23.3615i) q^{31} +(-0.935965 - 1.62114i) q^{35} +19.8245i q^{37} -17.4765i q^{41} +(-37.5697 + 65.0726i) q^{43} -12.5523 q^{47} +(24.4642 - 42.3732i) q^{49} +(-1.06598 - 0.615444i) q^{53} +(65.0090 + 112.599i) q^{55} -88.5490i q^{59} +102.475 q^{61} +(91.3756 - 52.7558i) q^{65} +(-68.2631 + 39.4117i) q^{67} +(85.6563 - 49.4537i) q^{71} +(-10.6721 - 18.4847i) q^{73} +(2.48689 - 4.30742i) q^{77} +(26.0253 + 15.0257i) q^{79} +(27.5237 + 47.6725i) q^{83} +(-51.9026 - 89.8979i) q^{85} +(-122.281 - 70.5992i) q^{89} +(-3.49553 - 2.01815i) q^{91} +(74.6546 - 109.962i) q^{95} +(101.528 + 58.6171i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 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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