LMFDB - Embedded newform 2052.3.m.a.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(881,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 5, 2]))

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

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

chi := DirichletCharacter("2052.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.m (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			881.1
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.40

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.28650i q^{5} +(-2.28067 - 3.95023i) q^{7} +O(q^{10})$	$q-9.28650i q^{5} +(-2.28067 - 3.95023i) q^{7} +(1.10584 - 0.638454i) q^{11} +(-3.30958 - 5.73235i) q^{13} +(11.0966 - 6.40664i) q^{17} +(-18.7585 - 3.01961i) q^{19} +(-16.3480 + 9.43853i) q^{23} -61.2390 q^{25} -7.10485i q^{29} +(7.90302 - 13.6884i) q^{31} +(-36.6838 + 21.1794i) q^{35} -16.9760 q^{37} -17.6906i q^{41} +(20.8849 - 36.1738i) q^{43} -22.7792i q^{47} +(14.0971 - 24.4169i) q^{49} +(-5.86723 - 3.38745i) q^{53} +(-5.92901 - 10.2693i) q^{55} +82.7526i q^{59} +47.4838 q^{61} +(-53.2335 + 30.7344i) q^{65} +(21.3601 + 36.9968i) q^{67} +(21.4678 - 12.3945i) q^{71} +(-39.6934 - 68.7509i) q^{73} +(-5.04409 - 2.91221i) q^{77} +(-54.2353 + 93.9382i) q^{79} +(-34.7029 + 20.0357i) q^{83} +(-59.4952 - 103.049i) q^{85} +(49.2445 + 28.4313i) q^{89} +(-15.0961 + 26.1472i) q^{91} +(-28.0416 + 174.201i) q^{95} +(-86.8612 + 150.448i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	9.28650i		−	1.85730i	−0.370958	−	0.928650i	$-0.620971\pi$
$5$		−	9.28650i		−	1.85730i	0.370958	−	0.928650i	$-0.379029\pi$
$6$		0			0
$7$	−2.28067	−	3.95023i	−0.325810	−	0.564319i	0.655866	−	0.754877i	$-0.272304\pi$
$7$	−2.28067	−	3.95023i	−0.325810	−	0.564319i	−0.981676	+	0.190558i	$0.938970\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.10584	−	0.638454i	0.100531	−	0.0580413i	−0.448892	−	0.893586i	$-0.648181\pi$
$11$	1.10584	−	0.638454i	0.100531	−	0.0580413i	0.549422	+	0.835545i	$0.314848\pi$
$12$		0			0
$13$	−3.30958	−	5.73235i	−0.254583	−	0.440950i	0.710199	−	0.704001i	$-0.248605\pi$
$13$	−3.30958	−	5.73235i	−0.254583	−	0.440950i	−0.964782	+	0.263050i	$0.915271\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	11.0966	−	6.40664i	0.652743	−	0.376861i	−0.136764	−	0.990604i	$-0.543670\pi$
$17$	11.0966	−	6.40664i	0.652743	−	0.376861i	0.789506	+	0.613743i	$0.210337\pi$
$18$		0			0
$19$	−18.7585	−	3.01961i	−0.987290	−	0.158927i
$19$	−18.7585	−	3.01961i	−0.987290	−	0.158927i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−16.3480	+	9.43853i	−0.710783	+	0.410371i	−0.811351	−	0.584559i	$-0.801267\pi$
$23$	−16.3480	+	9.43853i	−0.710783	+	0.410371i	0.100568	+	0.994930i	$0.467934\pi$
$24$		0			0
$25$	−61.2390			−2.44956
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.10485i		−	0.244995i	−0.992469	−	0.122497i	$-0.960910\pi$
$29$		−	7.10485i		−	0.244995i	0.992469	−	0.122497i	$-0.0390903\pi$
$30$		0			0
$31$	7.90302	−	13.6884i	0.254936	−	0.441562i	−0.709942	−	0.704260i	$-0.751279\pi$
$31$	7.90302	−	13.6884i	0.254936	−	0.441562i	0.964878	+	0.262698i	$0.0846123\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−36.6838	+	21.1794i	−1.04811	+	0.605126i
$36$		0			0
$37$	−16.9760			−0.458810			−0.229405	−	0.973331i	$-0.573678\pi$
$37$	−16.9760			−0.458810			−0.229405	+	0.973331i	$0.573678\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	17.6906i		−	0.431477i	−0.976451	−	0.215739i	$-0.930784\pi$
$41$		−	17.6906i		−	0.431477i	0.976451	−	0.215739i	$-0.0692159\pi$
$42$		0			0
$43$	20.8849	−	36.1738i	0.485696	−	0.841251i	−0.514169	−	0.857689i	$-0.671899\pi$
$43$	20.8849	−	36.1738i	0.485696	−	0.841251i	0.999865	+	0.0164385i	$0.00523277\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	22.7792i		−	0.484665i	−0.970193	−	0.242332i	$-0.922088\pi$
$47$		−	22.7792i		−	0.484665i	0.970193	−	0.242332i	$-0.0779124\pi$
$48$		0			0
$49$	14.0971	−	24.4169i	0.287696	−	0.498304i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.86723	−	3.38745i	−0.110702	−	0.0639141i	0.443626	−	0.896212i	$-0.353692\pi$
$53$	−5.86723	−	3.38745i	−0.110702	−	0.0639141i	−0.554329	+	0.832298i	$0.687025\pi$
$54$		0			0
$55$	−5.92901	−	10.2693i	−0.107800	−	0.186715i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			82.7526i			1.40259i	0.712873	+	0.701293i	$0.247394\pi$
$59$			82.7526i			1.40259i	−0.712873	+	0.701293i	$0.752606\pi$
$60$		0			0
$61$	47.4838			0.778423			0.389211	−	0.921148i	$-0.372748\pi$
$61$	47.4838			0.778423			0.389211	+	0.921148i	$0.372748\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−53.2335	+	30.7344i	−0.818977	+	0.472836i
$66$		0			0
$67$	21.3601	+	36.9968i	0.318807	+	0.552190i	0.980239	−	0.197815i	$-0.0633844\pi$
$67$	21.3601	+	36.9968i	0.318807	+	0.552190i	−0.661432	+	0.750005i	$0.730051\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	21.4678	−	12.3945i	0.302364	−	0.174570i	−0.341140	−	0.940012i	$-0.610813\pi$
$71$	21.4678	−	12.3945i	0.302364	−	0.174570i	0.643504	+	0.765442i	$0.277480\pi$
$72$		0			0
$73$	−39.6934	−	68.7509i	−0.543745	−	0.941793i	−0.998685	−	0.0512713i	$-0.983673\pi$
$73$	−39.6934	−	68.7509i	−0.543745	−	0.941793i	0.454940	−	0.890522i	$-0.349661\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.04409	−	2.91221i	−0.0655076	−	0.0378209i
$78$		0			0
$79$	−54.2353	+	93.9382i	−0.686522	+	1.18909i	0.286434	+	0.958100i	$0.407530\pi$
$79$	−54.2353	+	93.9382i	−0.686522	+	1.18909i	−0.972956	+	0.230991i	$0.925803\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−34.7029	+	20.0357i	−0.418107	+	0.241394i	−0.694267	−	0.719717i	$-0.744271\pi$
$83$	−34.7029	+	20.0357i	−0.418107	+	0.241394i	0.276160	+	0.961112i	$0.410938\pi$
$84$		0			0
$85$	−59.4952	−	103.049i	−0.699944	−	1.21234i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	49.2445	+	28.4313i	0.553309	+	0.319453i	0.750455	−	0.660921i	$-0.229834\pi$
$89$	49.2445	+	28.4313i	0.553309	+	0.319453i	−0.197147	+	0.980374i	$0.563168\pi$
$90$		0			0
$91$	−15.0961	+	26.1472i	−0.165891	+	0.287332i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−28.0416	+	174.201i	−0.295175	+	1.83369i
$96$		0			0
$97$	−86.8612	+	150.448i	−0.895477	+	1.55101i	−0.0622629	+	0.998060i	$0.519832\pi$
$97$	−86.8612	+	150.448i	−0.895477	+	1.55101i	−0.833214	+	0.552951i	$0.813502\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			88.8592i			0.879794i	0.898048	+	0.439897i	$0.144985\pi$
$101$			88.8592i			0.879794i	−0.898048	+	0.439897i	$0.855015\pi$
$102$		0			0
$103$	43.5924	−	75.5043i	0.423227	−	0.733051i	−0.573026	−	0.819537i	$-0.694231\pi$
$103$	43.5924	−	75.5043i	0.423227	−	0.733051i	0.996253	+	0.0864862i	$0.0275639\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			85.1392i			0.795693i	0.917452	+	0.397847i	$0.130242\pi$
$107$			85.1392i			0.795693i	−0.917452	+	0.397847i	$0.869758\pi$
$108$		0			0
$109$	−103.317	−	178.950i	−0.947861	−	1.64174i	−0.749919	−	0.661529i	$-0.769908\pi$
$109$	−103.317	−	178.950i	−0.947861	−	1.64174i	−0.197941	−	0.980214i	$-0.563426\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	180.905	+	104.445i	1.60093	+	0.924296i	0.991302	+	0.131605i	$0.0420132\pi$
$113$	180.905	+	104.445i	1.60093	+	0.924296i	0.609625	+	0.792690i	$0.291320\pi$
$114$		0			0
$115$	87.6509	+	151.816i	0.762181	+	1.32014i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−50.6154	−	29.2228i	−0.425340	−	0.245570i
$120$		0			0
$121$	−59.6848	+	103.377i	−0.493262	+	0.854356i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			336.534i			2.69227i
$126$		0			0
$127$	56.4818	−	97.8294i	0.444739	−	0.770310i	−0.553295	−	0.832985i	$-0.686630\pi$
$127$	56.4818	−	97.8294i	0.444739	−	0.770310i	0.998034	+	0.0626751i	$0.0199632\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.51243i			0.0649803i	0.999472	+	0.0324902i	$0.0103438\pi$
$131$			8.51243i			0.0649803i	−0.999472	+	0.0324902i	$0.989656\pi$
$132$		0			0
$133$	30.8538	+	80.9873i	0.231983	+	0.608927i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	214.194i		−	1.56346i	−0.623618	−	0.781729i	$-0.714338\pi$
$137$		−	214.194i		−	1.56346i	0.623618	−	0.781729i	$-0.285662\pi$
$138$		0			0
$139$	25.7715	+	44.6376i	0.185407	+	0.321134i	0.943713	−	0.330764i	$-0.107306\pi$
$139$	25.7715	+	44.6376i	0.185407	+	0.321134i	−0.758307	+	0.651898i	$0.773973\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.31969	−	4.22603i	−0.0511867	−	0.0295526i
$144$		0			0
$145$	−65.9792			−0.455029
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	143.809i		−	0.965164i	−0.875851	−	0.482582i	$-0.839699\pi$
$149$		−	143.809i		−	0.965164i	0.875851	−	0.482582i	$-0.160301\pi$
$150$		0			0
$151$	109.749	+	190.090i	0.726813	+	1.25888i	0.958223	+	0.286021i	$0.0923326\pi$
$151$	109.749	+	190.090i	0.726813	+	1.25888i	−0.231411	+	0.972856i	$0.574334\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−127.118	−	73.3913i	−0.820113	−	0.473493i
$156$		0			0
$157$	−31.4750			−0.200477			−0.100239	−	0.994963i	$-0.531961\pi$
$157$	−31.4750			−0.200477			−0.100239	+	0.994963i	$0.531961\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	74.5688	+	43.0523i	0.463160	+	0.267406i
$162$		0			0
$163$	−27.3667			−0.167894			−0.0839469	−	0.996470i	$-0.526753\pi$
$163$	−27.3667			−0.167894			−0.0839469	+	0.996470i	$0.526753\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−200.623	+	115.830i	−1.20134	+	0.693592i	−0.960852	−	0.277061i	$-0.910640\pi$
$167$	−200.623	+	115.830i	−1.20134	+	0.693592i	−0.240485	+	0.970653i	$0.577306\pi$
$168$		0			0
$169$	62.5934	−	108.415i	0.370375	−	0.641509i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	103.634	+	59.8331i	0.599041	+	0.345856i	0.768664	−	0.639653i	$-0.220922\pi$
$173$	103.634	+	59.8331i	0.599041	+	0.345856i	−0.169623	+	0.985509i	$0.554255\pi$
$174$		0			0
$175$	139.666	+	241.908i	0.798091	+	1.38233i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			182.887i			1.02171i	0.859666	+	0.510856i	$0.170672\pi$
$179$			182.887i			1.02171i	−0.859666	+	0.510856i	$0.829328\pi$
$180$		0			0
$181$	77.9550	−	135.022i	0.430691	−	0.745978i	−0.566242	−	0.824239i	$-0.691603\pi$
$181$	77.9550	−	135.022i	0.430691	−	0.745978i	0.996933	+	0.0782606i	$0.0249366\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			157.647i			0.852148i
$186$		0			0
$187$	8.18069	−	14.1694i	0.0437470	−	0.0757721i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	292.923	−	169.119i	1.53363	−	0.885440i	0.534436	−	0.845209i	$-0.320524\pi$
$191$	292.923	−	169.119i	1.53363	−	0.885440i	0.999190	−	0.0402304i	$-0.0128092\pi$
$192$		0			0
$193$	−213.007			−1.10366			−0.551832	−	0.833955i	$-0.686071\pi$
$193$	−213.007			−1.10366			−0.551832	+	0.833955i	$0.686071\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.74954i			0.0393378i	0.999807	+	0.0196689i	$0.00626120\pi$
$197$			7.74954i			0.0393378i	−0.999807	+	0.0196689i	$0.993739\pi$
$198$		0			0
$199$	−149.979	+	259.771i	−0.753662	+	1.30538i	0.192375	+	0.981321i	$0.438381\pi$
$199$	−149.979	+	259.771i	−0.753662	+	1.30538i	−0.946037	+	0.324059i	$0.894952\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−28.0658	+	16.2038i	−0.138255	+	0.0798217i
$204$		0			0
$205$	−164.283			−0.801383
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−22.6717	+	8.63726i	−0.108477	+	0.0413266i
$210$		0			0
$211$	−368.255			−1.74528			−0.872641	−	0.488362i	$-0.837595\pi$
$211$	−368.255			−1.74528			−0.872641	+	0.488362i	$0.837595\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−335.928	−	193.948i	−1.56245	−	0.902083i
$216$		0			0
$217$	−72.0966			−0.332243
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−73.4502	−	42.4065i	−0.332354	−	0.191885i
$222$		0			0
$223$	76.6100	−	132.692i	0.343543	−	0.595033i	−0.641545	−	0.767085i	$-0.721706\pi$
$223$	76.6100	−	132.692i	0.343543	−	0.595033i	0.985088	+	0.172052i	$0.0550397\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−119.451	+	68.9649i	−0.526214	+	0.303810i	−0.739474	−	0.673186i	$-0.764925\pi$
$227$	−119.451	+	68.9649i	−0.526214	+	0.303810i	0.213259	+	0.976996i	$0.431592\pi$
$228$		0			0
$229$	−84.2944	+	146.002i	−0.368098	+	0.637564i	−0.989268	−	0.146112i	$-0.953324\pi$
$229$	−84.2944	+	146.002i	−0.368098	+	0.637564i	0.621170	+	0.783676i	$0.286657\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	257.617	−	148.735i	1.10565	−	0.638348i	0.167952	−	0.985795i	$-0.446285\pi$
$233$	257.617	−	148.735i	1.10565	−	0.638348i	0.937700	+	0.347447i	$0.112951\pi$
$234$		0			0
$235$	−211.539			−0.900167
$236$		0			0
$237$		0			0
$238$		0			0
$239$	277.929	+	160.462i	1.16288	+	0.671391i	0.951993	−	0.306119i	$-0.0990307\pi$
$239$	277.929	+	160.462i	1.16288	+	0.671391i	0.210889	+	0.977510i	$0.432364\pi$
$240$		0			0
$241$	−393.874			−1.63433			−0.817167	−	0.576402i	$-0.804456\pi$
$241$	−393.874			−1.63433			−0.817167	+	0.576402i	$0.804456\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−226.747	−	130.913i	−0.925500	−	0.534338i
$246$		0			0
$247$	44.7732	+	117.524i	0.181268	+	0.475806i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−320.034	−	184.772i	−1.27504	−	0.736142i	−0.299104	−	0.954220i	$-0.596688\pi$
$251$	−320.034	−	184.772i	−1.27504	−	0.736142i	−0.975931	+	0.218078i	$0.930021\pi$
$252$		0			0
$253$	−12.0521	+	20.8749i	−0.0476369	+	0.0825096i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−159.326	+	91.9868i	−0.619945	+	0.357925i	−0.776847	−	0.629689i	$-0.783182\pi$
$257$	−159.326	+	91.9868i	−0.619945	+	0.357925i	0.156903	+	0.987614i	$0.449849\pi$
$258$		0			0
$259$	38.7166	+	67.0591i	0.149485	+	0.258915i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	146.456	+	84.5564i	0.556867	+	0.321507i	0.751887	−	0.659292i	$-0.229144\pi$
$263$	146.456	+	84.5564i	0.556867	+	0.321507i	−0.195020	+	0.980799i	$0.562477\pi$
$264$		0			0
$265$	−31.4575	+	54.4860i	−0.118708	+	0.205608i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.505277	+	0.291722i	−0.00187835	+	0.00108447i	−0.500939	−	0.865483i	$-0.667012\pi$
$269$	−0.505277	+	0.291722i	−0.00187835	+	0.00108447i	0.499061	+	0.866567i	$0.333679\pi$
$270$		0			0
$271$	−173.903	−	301.208i	−0.641707	−	1.11147i	−0.985052	−	0.172260i	$-0.944893\pi$
$271$	−173.903	−	301.208i	−0.641707	−	1.11147i	0.343344	−	0.939210i	$-0.388440\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−67.7203	+	39.0983i	−0.246256	+	0.142176i
$276$		0			0
$277$	−115.851	−	200.660i	−0.418235	−	0.724404i	0.577527	−	0.816371i	$-0.304018\pi$
$277$	−115.851	−	200.660i	−0.418235	−	0.724404i	−0.995762	+	0.0919678i	$0.970684\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	501.455i		−	1.78454i	−0.451505	−	0.892268i	$-0.649113\pi$
$281$		−	501.455i		−	1.78454i	0.451505	−	0.892268i	$-0.350887\pi$
$282$		0			0
$283$	−46.8874			−0.165680			−0.0828399	−	0.996563i	$-0.526399\pi$
$283$	−46.8874			−0.165680			−0.0828399	+	0.996563i	$0.526399\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−69.8819	+	40.3463i	−0.243491	+	0.140580i
$288$		0			0
$289$	−62.4099	+	108.097i	−0.215951	+	0.374039i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	67.5883	+	39.0221i	0.230677	+	0.133181i	0.610884	−	0.791720i	$-0.290814\pi$
$293$	67.5883	+	39.0221i	0.230677	+	0.133181i	−0.380207	+	0.924901i	$0.624147\pi$
$294$		0			0
$295$	768.482			2.60502
$296$		0			0
$297$		0			0
$298$		0			0
$299$	108.210	+	62.4750i	0.361906	+	0.208947i
$300$		0			0
$301$	−190.526			−0.632978
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	440.958i		−	1.44576i
$306$		0			0
$307$	−13.1550	−	22.7852i	−0.0428503	−	0.0742188i	0.843805	−	0.536650i	$-0.180311\pi$
$307$	−13.1550	−	22.7852i	−0.0428503	−	0.0742188i	−0.886655	+	0.462431i	$0.846977\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	339.125	+	195.794i	1.09043	+	0.629562i	0.933692	−	0.358078i	$-0.116568\pi$
$311$	339.125	+	195.794i	1.09043	+	0.629562i	0.156741	+	0.987640i	$0.449901\pi$
$312$		0			0
$313$	−93.3871			−0.298361			−0.149181	−	0.988810i	$-0.547664\pi$
$313$	−93.3871			−0.298361			−0.149181	+	0.988810i	$0.547664\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			57.0992i			0.180124i	0.995936	+	0.0900618i	$0.0287064\pi$
$317$			57.0992i			0.180124i	−0.995936	+	0.0900618i	$0.971294\pi$
$318$		0			0
$319$	−4.53612	−	7.85679i	−0.0142198	−	0.0246294i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−227.502	+	86.6715i	−0.704340	+	0.268333i
$324$		0			0
$325$	202.675	+	351.044i	0.623616	+	1.08013i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−89.9833	+	51.9519i	−0.273505	+	0.157908i
$330$		0			0
$331$	−5.97582	−	10.3504i	−0.0180538	−	0.0312702i	0.856857	−	0.515554i	$-0.172414\pi$
$331$	−5.97582	−	10.3504i	−0.0180538	−	0.0312702i	−0.874911	+	0.484283i	$0.839080\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	343.570	−	198.360i	1.02558	−	0.592121i
$336$		0			0
$337$	−100.450			−0.298071			−0.149035	−	0.988832i	$-0.547617\pi$
$337$	−100.450			−0.298071			−0.149035	+	0.988832i	$0.547617\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	20.1829i		−	0.0591873i
$342$		0			0
$343$	−352.109			−1.02656
$344$		0			0
$345$		0			0
$346$		0			0
$347$			151.374i			0.436236i	0.975922	+	0.218118i	$0.0699917\pi$
$347$			151.374i			0.436236i	−0.975922	+	0.218118i	$0.930008\pi$
$348$		0			0
$349$	−219.737	−	380.596i	−0.629619	−	1.09053i	−0.987628	−	0.156814i	$-0.949878\pi$
$349$	−219.737	−	380.596i	−0.629619	−	1.09053i	0.358009	−	0.933718i	$-0.383456\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−524.353	+	302.736i	−1.48542	+	0.857608i	−0.999862	−	0.0165964i	$-0.994717\pi$
$353$	−524.353	+	302.736i	−1.48542	+	0.857608i	−0.485558	+	0.874204i	$0.661384\pi$
$354$		0			0
$355$	−115.101	−	199.361i	−0.324229	−	0.561580i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	84.9553	−	49.0490i	0.236644	−	0.136627i	−0.376989	−	0.926218i	$-0.623041\pi$
$359$	84.9553	−	49.0490i	0.236644	−	0.136627i	0.613633	+	0.789591i	$0.289707\pi$
$360$		0			0
$361$	342.764	+	113.287i	0.949484	+	0.313814i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−638.455	+	368.612i	−1.74919	+	1.00990i
$366$		0			0
$367$	−324.259			−0.883539			−0.441769	−	0.897129i	$-0.645649\pi$
$367$	−324.259			−0.883539			−0.441769	+	0.897129i	$0.645649\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			30.9026i			0.0832954i
$372$		0			0
$373$	−79.8812	+	138.358i	−0.214159	+	0.370934i	−0.953012	−	0.302933i	$-0.902034\pi$
$373$	−79.8812	+	138.358i	−0.214159	+	0.370934i	0.738853	+	0.673866i	$0.235368\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−40.7275	+	23.5140i	−0.108030	+	0.0623714i
$378$		0			0
$379$	366.244			0.966343			0.483171	−	0.875526i	$-0.339485\pi$
$379$	366.244			0.966343			0.483171	+	0.875526i	$0.339485\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			124.052i			0.323895i	0.986799	+	0.161948i	$0.0517775\pi$
$383$			124.052i			0.323895i	−0.986799	+	0.161948i	$0.948222\pi$
$384$		0			0
$385$	−27.0442	+	46.8419i	−0.0702446	+	0.121667i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	415.200i		−	1.06735i	−0.845689	−	0.533676i	$-0.820810\pi$
$389$		−	415.200i		−	1.06735i	0.845689	−	0.533676i	$-0.179190\pi$
$390$		0			0
$391$	−120.938	+	209.472i	−0.309306	+	0.535733i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	872.357	+	503.656i	2.20850	+	1.27508i
$396$		0			0
$397$	247.236	+	428.225i	0.622760	+	1.07865i	0.988969	+	0.148120i	$0.0473220\pi$
$397$	247.236	+	428.225i	0.622760	+	1.07865i	−0.366209	+	0.930532i	$0.619345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	299.577i		−	0.747075i	−0.927615	−	0.373538i	$-0.878145\pi$
$401$		−	299.577i		−	0.747075i	0.927615	−	0.373538i	$-0.121855\pi$
$402$		0			0
$403$	−104.623			−0.259609
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−18.7726	+	10.8384i	−0.0461244	+	0.0266299i
$408$		0			0
$409$	80.2674	+	139.027i	0.196253	+	0.339920i	0.947310	−	0.320317i	$-0.103789\pi$
$409$	80.2674	+	139.027i	0.196253	+	0.339920i	−0.751058	+	0.660237i	$0.770456\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	326.892	−	188.731i	0.791506	−	0.456976i
$414$		0			0
$415$	186.062	+	322.268i	0.448342	+	0.776550i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−442.295	−	255.359i	−1.05560	−	0.609449i	−0.131386	−	0.991331i	$-0.541943\pi$
$419$	−442.295	−	255.359i	−1.05560	−	0.609449i	−0.924211	+	0.381882i	$0.875276\pi$
$420$		0			0
$421$	−176.079	+	304.978i	−0.418241	+	0.724414i	−0.995763	−	0.0919610i	$-0.970686\pi$
$421$	−176.079	+	304.978i	−0.418241	+	0.724414i	0.577522	+	0.816375i	$0.304020\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−679.547	+	392.336i	−1.59893	+	0.923144i
$426$		0			0
$427$	−108.295	−	187.572i	−0.253618	−	0.439279i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	561.436	+	324.145i	1.30264	+	0.752077i	0.980855	−	0.194738i	$-0.0623856\pi$
$431$	561.436	+	324.145i	1.30264	+	0.752077i	0.321780	+	0.946815i	$0.395719\pi$
$432$		0			0
$433$	270.752	−	468.956i	0.625293	−	1.08304i	−0.363191	−	0.931715i	$-0.618313\pi$
$433$	270.752	−	468.956i	0.625293	−	1.08304i	0.988484	−	0.151325i	$-0.0483539\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	335.165	−	127.688i	0.766968	−	0.292192i
$438$		0			0
$439$	230.430	−	399.117i	0.524898	−	0.909151i	−0.474681	−	0.880158i	$-0.657437\pi$
$439$	230.430	−	399.117i	0.524898	−	0.909151i	0.999580	−	0.0289930i	$-0.00923004\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	798.745i		−	1.80304i	−0.432740	−	0.901519i	$-0.642453\pi$
$443$		−	798.745i		−	1.80304i	0.432740	−	0.901519i	$-0.357547\pi$
$444$		0			0
$445$	264.027	−	457.309i	0.593320	−	1.02766i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			751.768i			1.67432i	0.546962	+	0.837158i	$0.315784\pi$
$449$			751.768i			1.67432i	−0.546962	+	0.837158i	$0.684216\pi$
$450$		0			0
$451$	−11.2946	−	19.5629i	−0.0250435	−	0.0433766i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	242.816	+	140.190i	0.533661	+	0.308109i
$456$		0			0
$457$	−341.536	−	591.557i	−0.747343	−	1.29444i	−0.949092	−	0.314999i	$-0.897996\pi$
$457$	−341.536	−	591.557i	−0.747343	−	1.29444i	0.201749	−	0.979437i	$-0.435338\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	495.618	+	286.145i	1.07509	+	0.620706i	0.929569	−	0.368649i	$-0.120180\pi$
$461$	495.618	+	286.145i	1.07509	+	0.620706i	0.145525	+	0.989355i	$0.453513\pi$
$462$		0			0
$463$	221.398	−	383.473i	0.478182	−	0.828236i	−0.521505	−	0.853248i	$-0.674629\pi$
$463$	221.398	−	383.473i	0.478182	−	0.828236i	0.999687	+	0.0250123i	$0.00796250\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			246.829i			0.528542i	0.964449	+	0.264271i	$0.0851313\pi$
$467$			246.829i			0.528542i	−0.964449	+	0.264271i	$0.914869\pi$
$468$		0			0
$469$	97.4306	−	168.755i	0.207741	−	0.359818i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	53.3363i		−	0.112762i
$474$		0			0
$475$	1148.75	+	184.918i	2.41843	+	0.389301i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			208.455i			0.435188i	0.976039	+	0.217594i	$0.0698209\pi$
$479$			208.455i			0.435188i	−0.976039	+	0.217594i	$0.930179\pi$
$480$		0			0
$481$	56.1833	+	97.3123i	0.116805	+	0.202312i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1397.14	+	806.637i	2.88069	+	1.66317i
$486$		0			0
$487$	467.236			0.959416			0.479708	−	0.877428i	$-0.340743\pi$
$487$	467.236			0.959416			0.479708	+	0.877428i	$0.340743\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	263.317i		−	0.536287i	−0.963379	−	0.268143i	$-0.913590\pi$
$491$		−	263.317i		−	0.536287i	0.963379	−	0.268143i	$-0.0864101\pi$
$492$		0			0
$493$	−45.5182	−	78.8398i	−0.0923290	−	0.159919i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−97.9220	−	56.5353i	−0.197026	−	0.113753i
$498$		0			0
$499$	−891.558			−1.78669			−0.893345	−	0.449372i	$-0.851648\pi$
$499$	−891.558			−1.78669			−0.893345	+	0.449372i	$0.851648\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	383.876	+	221.631i	0.763174	+	0.440619i	0.830434	−	0.557117i	$-0.188092\pi$
$503$	383.876	+	221.631i	0.763174	+	0.440619i	−0.0672604	+	0.997735i	$0.521426\pi$
$504$		0			0
$505$	825.191			1.63404
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−306.042	+	176.693i	−0.601261	+	0.347138i	−0.769538	−	0.638602i	$-0.779513\pi$
$509$	−306.042	+	176.693i	−0.601261	+	0.347138i	0.168276	+	0.985740i	$0.446180\pi$
$510$		0			0
$511$	−181.055	+	313.596i	−0.354315	+	0.613691i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−701.170	−	404.821i	−1.36150	−	0.786060i
$516$		0			0
$517$	−14.5435	−	25.1901i	−0.0281306	−	0.0487236i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	530.816i		−	1.01884i	−0.860518	−	0.509420i	$-0.829860\pi$
$521$		−	530.816i		−	1.01884i	0.860518	−	0.509420i	$-0.170140\pi$
$522$		0			0
$523$	518.783	−	898.559i	0.991938	−	1.71809i	0.386221	−	0.922406i	$-0.373780\pi$
$523$	518.783	−	898.559i	0.991938	−	1.71809i	0.605717	−	0.795680i	$-0.292887\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	202.527i		−	0.384302i
$528$		0			0
$529$	−86.3284	+	149.525i	−0.163192	+	0.282656i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−101.409	+	58.5483i	−0.190260	+	0.109847i
$534$		0			0
$535$	790.645			1.47784
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	36.0014i		−	0.0667930i
$540$		0			0
$541$	−427.224	+	739.974i	−0.789693	+	1.36779i	0.136462	+	0.990645i	$0.456427\pi$
$541$	−427.224	+	739.974i	−0.789693	+	1.36779i	−0.926155	+	0.377143i	$0.876906\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1661.82	+	959.451i	−3.04921	+	1.76046i
$546$		0			0
$547$	−412.287			−0.753724			−0.376862	−	0.926269i	$-0.622997\pi$
$547$	−412.287			−0.753724			−0.376862	+	0.926269i	$0.622997\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21.4539	+	133.276i	−0.0389363	+	0.241881i
$552$		0			0
$553$	494.770			0.894703
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−863.794	−	498.712i	−1.55080	−	0.895353i	−0.998077	−	0.0619859i	$-0.980257\pi$
$557$	−863.794	−	498.712i	−1.55080	−	0.895353i	−0.552720	−	0.833367i	$-0.686410\pi$
$558$		0			0
$559$	−276.481			−0.494600
$560$		0			0
$561$		0			0
$562$		0			0
$563$	305.679	+	176.484i	0.542947	+	0.313470i	0.746272	−	0.665641i	$-0.231842\pi$
$563$	305.679	+	176.484i	0.542947	+	0.313470i	−0.203326	+	0.979111i	$0.565175\pi$
$564$		0			0
$565$	969.932	−	1679.97i	1.71669	−	2.97340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−276.835	+	159.831i	−0.486529	+	0.280897i	−0.723133	−	0.690709i	$-0.757299\pi$
$569$	−276.835	+	159.831i	−0.486529	+	0.280897i	0.236605	+	0.971606i	$0.423965\pi$
$570$		0			0
$571$	352.313	−	610.224i	0.617010	−	1.06869i	−0.373018	−	0.927824i	$-0.621677\pi$
$571$	352.313	−	610.224i	0.617010	−	1.06869i	0.990028	−	0.140869i	$-0.0449897\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1001.14	−	578.006i	1.74111	−	1.00523i
$576$		0			0
$577$	322.338			0.558644			0.279322	−	0.960197i	$-0.409890\pi$
$577$	322.338			0.558644			0.279322	+	0.960197i	$0.409890\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	158.292	+	91.3897i	0.272447	+	0.157297i
$582$		0			0
$583$	−8.65093			−0.0148386
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−974.121	−	562.409i	−1.65949	−	0.958107i	−0.972950	−	0.231016i	$-0.925795\pi$
$587$	−974.121	−	562.409i	−1.65949	−	0.958107i	−0.686541	−	0.727091i	$-0.740872\pi$
$588$		0			0
$589$	−189.583	+	232.911i	−0.321872	+	0.395434i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−686.343	−	396.261i	−1.15741	−	0.668230i	−0.206727	−	0.978399i	$-0.566281\pi$
$593$	−686.343	−	396.261i	−1.15741	−	0.668230i	−0.950682	+	0.310168i	$0.899615\pi$
$594$		0			0
$595$	−271.378	+	470.040i	−0.456097	+	0.789983i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	695.903	−	401.780i	1.16177	−	0.670750i	0.210045	−	0.977692i	$-0.432639\pi$
$599$	695.903	−	401.780i	1.16177	−	0.670750i	0.951728	+	0.306941i	$0.0993055\pi$
$600$		0			0
$601$	3.87039	+	6.70371i	0.00643992	+	0.0111543i	0.869227	−	0.494412i	$-0.164617\pi$
$601$	3.87039	+	6.70371i	0.00643992	+	0.0111543i	−0.862787	+	0.505567i	$0.831283\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	960.010	+	554.262i	1.58679	+	0.916136i
$606$		0			0
$607$	111.655	−	193.392i	0.183945	−	0.318602i	−0.759275	−	0.650769i	$-0.774446\pi$
$607$	111.655	−	193.392i	0.183945	−	0.318602i	0.943221	+	0.332167i	$0.107780\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−130.579	+	75.3896i	−0.213713	+	0.123387i
$612$		0			0
$613$	188.469	+	326.438i	0.307454	+	0.532525i	0.977805	−	0.209519i	$-0.0671898\pi$
$613$	188.469	+	326.438i	0.307454	+	0.532525i	−0.670351	+	0.742044i	$0.733856\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	404.809	−	233.717i	0.656093	−	0.378795i	−0.134694	−	0.990887i	$-0.543005\pi$
$617$	404.809	−	233.717i	0.656093	−	0.378795i	0.790787	+	0.612092i	$0.209672\pi$
$618$		0			0
$619$	481.347	+	833.718i	0.777621	+	1.34688i	0.933310	+	0.359073i	$0.116907\pi$
$619$	481.347	+	833.718i	0.777621	+	1.34688i	−0.155689	+	0.987806i	$0.549760\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	259.370i		−	0.416324i
$624$		0			0
$625$	1594.24			2.55079
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−188.376	+	108.759i	−0.299485	+	0.172908i
$630$		0			0
$631$	132.578	−	229.632i	0.210108	−	0.363917i	−0.741640	−	0.670798i	$-0.765952\pi$
$631$	132.578	−	229.632i	0.210108	−	0.363917i	0.951748	+	0.306880i	$0.0992852\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−908.492	−	524.518i	−1.43070	−	0.826013i
$636$		0			0
$637$	−186.622			−0.292970
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.8673	+	12.6251i	0.0341143	+	0.0196959i	0.516960	−	0.856009i	$-0.327064\pi$
$641$	21.8673	+	12.6251i	0.0341143	+	0.0196959i	−0.482846	+	0.875705i	$0.660397\pi$
$642$		0			0
$643$	−335.777			−0.522204			−0.261102	−	0.965311i	$-0.584086\pi$
$643$	−335.777			−0.522204			−0.261102	+	0.965311i	$0.584086\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	905.889i		−	1.40014i	−0.714075	−	0.700069i	$-0.753152\pi$
$647$		−	905.889i		−	1.40014i	0.714075	−	0.700069i	$-0.246848\pi$
$648$		0			0
$649$	52.8338	+	91.5108i	0.0814080	+	0.141003i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−929.143	−	536.441i	−1.42288	−	0.821502i	−0.426339	−	0.904564i	$-0.640197\pi$
$653$	−929.143	−	536.441i	−1.42288	−	0.821502i	−0.996544	+	0.0830615i	$0.973530\pi$
$654$		0			0
$655$	79.0506			0.120688
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1036.04i		−	1.57214i	−0.618135	−	0.786072i	$-0.712111\pi$
$659$		−	1036.04i		−	1.57214i	0.618135	−	0.786072i	$-0.287889\pi$
$660$		0			0
$661$	47.1772	+	81.7133i	0.0713725	+	0.123621i	0.899503	−	0.436914i	$-0.143929\pi$
$661$	47.1772	+	81.7133i	0.0713725	+	0.123621i	−0.828131	+	0.560535i	$0.810595\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	752.088	−	286.524i	1.13096	−	0.430862i
$666$		0			0
$667$	67.0593	+	116.150i	0.100539	+	0.174138i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	52.5093	−	30.3162i	0.0782553	−	0.0451807i
$672$		0			0
$673$	−156.533	−	271.123i	−0.232590	−	0.402858i	0.725979	−	0.687716i	$-0.241387\pi$
$673$	−156.533	−	271.123i	−0.232590	−	0.402858i	−0.958570	+	0.284858i	$0.908053\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	523.760	−	302.393i	0.773649	−	0.446666i	−0.0605259	−	0.998167i	$-0.519278\pi$
$677$	523.760	−	302.393i	0.773649	−	0.446666i	0.834175	+	0.551500i	$0.185944\pi$
$678$		0			0
$679$	792.407			1.16702
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	27.4860i		−	0.0402430i	−0.999798	−	0.0201215i	$-0.993595\pi$
$683$		−	27.4860i		−	0.0402430i	0.999798	−	0.0201215i	$-0.00640531\pi$
$684$		0			0
$685$	−1989.11			−2.90381
$686$		0			0
$687$		0			0
$688$		0			0
$689$			44.8441i			0.0650857i
$690$		0			0
$691$	2.35922	+	4.08630i	0.00341422	+	0.00591360i	0.867727	−	0.497040i	$-0.165580\pi$
$691$	2.35922	+	4.08630i	0.00341422	+	0.00591360i	−0.864313	+	0.502954i	$0.832247\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	414.527	−	239.327i	0.596442	−	0.344356i
$696$		0			0
$697$	−113.337	−	196.306i	−0.162607	−	0.281644i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	81.6577	−	47.1451i	0.116487	−	0.0672541i	−0.440624	−	0.897692i	$-0.645243\pi$
$701$	81.6577	−	47.1451i	0.116487	−	0.0672541i	0.557112	+	0.830438i	$0.311910\pi$
$702$		0			0
$703$	318.444	+	51.2609i	0.452979	+	0.0729173i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	351.014	−	202.658i	0.496484	−	0.286645i
$708$		0			0
$709$	−866.770			−1.22252			−0.611262	−	0.791428i	$-0.709338\pi$
$709$	−866.770			−1.22252			−0.611262	+	0.791428i	$0.709338\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			298.371i			0.418473i
$714$		0			0
$715$	−39.2450	+	67.9743i	−0.0548881	+	0.0950689i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	960.272	−	554.413i	1.33557	−	0.771089i	0.349419	−	0.936966i	$-0.386379\pi$
$719$	960.272	−	554.413i	1.33557	−	0.771089i	0.986146	+	0.165877i	$0.0530456\pi$
$720$		0			0
$721$	−397.679			−0.551566
$722$		0			0
$723$		0			0
$724$		0			0
$725$			435.094i			0.600130i
$726$		0			0
$727$	202.057	−	349.973i	0.277933	−	0.481393i	−0.692938	−	0.720997i	$-0.743684\pi$
$727$	202.057	−	349.973i	0.277933	−	0.481393i	0.970871	+	0.239604i	$0.0770175\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	535.209i		−	0.732160i
$732$		0			0
$733$	−564.680	+	978.054i	−0.770368	+	1.33432i	0.166994	+	0.985958i	$0.446594\pi$
$733$	−564.680	+	978.054i	−0.770368	+	1.33432i	−0.937362	+	0.348358i	$0.886739\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	47.2415	+	27.2749i	0.0640997	+	0.0370080i
$738$		0			0
$739$	321.557	+	556.953i	0.435124	+	0.753658i	0.997306	−	0.0733563i	$-0.0233710\pi$
$739$	321.557	+	556.953i	0.435124	+	0.753658i	−0.562181	+	0.827014i	$0.690038\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1205.28i			1.62219i	0.584917	+	0.811093i	$0.301127\pi$
$743$			1205.28i			1.62219i	−0.584917	+	0.811093i	$0.698873\pi$
$744$		0			0
$745$	−1335.49			−1.79260
$746$		0			0
$747$		0			0
$748$		0			0
$749$	336.320	−	194.174i	0.449025	−	0.259245i
$750$		0			0
$751$	−123.979	−	214.738i	−0.165086	−	0.285937i	0.771600	−	0.636108i	$-0.219457\pi$
$751$	−123.979	−	214.738i	−0.165086	−	0.285937i	−0.936686	+	0.350171i	$0.886123\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1765.27	−	1019.18i	2.33811	−	1.34991i
$756$		0			0
$757$	−313.951	−	543.779i	−0.414731	−	0.718335i	0.580669	−	0.814139i	$-0.302791\pi$
$757$	−313.951	−	543.779i	−0.414731	−	0.718335i	−0.995400	+	0.0958047i	$0.969458\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−856.714	−	494.624i	−1.12577	−	0.649966i	−0.182905	−	0.983131i	$-0.558550\pi$
$761$	−856.714	−	494.624i	−1.12577	−	0.649966i	−0.942869	+	0.333165i	$0.891883\pi$
$762$		0			0
$763$	−471.263	+	816.251i	−0.617645	+	1.06979i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	474.367	−	273.876i	0.618471	−	0.357074i
$768$		0			0
$769$	−256.815	−	444.816i	−0.333959	−	0.578434i	0.649325	−	0.760511i	$-0.275051\pi$
$769$	−256.815	−	444.816i	−0.333959	−	0.578434i	−0.983284	+	0.182077i	$0.941718\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−464.596	−	268.235i	−0.601030	−	0.347005i	0.168417	−	0.985716i	$-0.446135\pi$
$773$	−464.596	−	268.235i	−0.601030	−	0.347005i	−0.769447	+	0.638711i	$0.779468\pi$
$774$		0			0
$775$	−483.973	+	838.266i	−0.624481	+	1.08163i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−53.4187	+	331.849i	−0.0685734	+	0.425993i
$780$		0			0
$781$	15.8266	−	27.4125i	0.0202645	−	0.0350992i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			292.292i			0.372347i
$786$		0			0
$787$	60.3896	−	104.598i	0.0767339	−	0.132907i	−0.825105	−	0.564979i	$-0.808884\pi$
$787$	60.3896	−	104.598i	0.0767339	−	0.132907i	0.901839	+	0.432072i	$0.142217\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	952.821i		−	1.20458i
$792$		0			0
$793$	−157.151	−	272.194i	−0.198173	−	0.343246i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−525.724	−	303.527i	−0.659629	−	0.380837i	0.132507	−	0.991182i	$-0.457697\pi$
$797$	−525.724	−	303.527i	−0.659629	−	0.380837i	−0.792136	+	0.610345i	$0.791031\pi$
$798$		0			0
$799$	−145.938	−	252.773i	−0.182651	−	0.316361i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−87.7886	−	50.6848i	−0.109326	−	0.0631193i
$804$		0			0
$805$	399.805	−	692.483i	0.496652	−	0.860227i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	144.203i		−	0.178248i	−0.996021	−	0.0891241i	$-0.971593\pi$
$809$		−	144.203i		−	0.178248i	0.996021	−	0.0891241i	$-0.0284068\pi$
$810$		0			0
$811$	−487.189	+	843.837i	−0.600727	+	1.04049i	0.391984	+	0.919972i	$0.371789\pi$
$811$	−487.189	+	843.837i	−0.600727	+	1.04049i	−0.992711	+	0.120518i	$0.961545\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			254.141i			0.311829i
$816$		0			0
$817$	−501.001	+	615.502i	−0.613221	+	0.753368i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1392.69i		−	1.69634i	−0.529726	−	0.848169i	$-0.677705\pi$
$821$		−	1392.69i		−	1.69634i	0.529726	−	0.848169i	$-0.322295\pi$
$822$		0			0
$823$	−327.037	−	566.445i	−0.397372	−	0.688268i	0.596029	−	0.802963i	$-0.296744\pi$
$823$	−327.037	−	566.445i	−0.397372	−	0.688268i	−0.993401	+	0.114695i	$0.963411\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	966.498	+	558.008i	1.16868	+	0.674737i	0.953369	−	0.301808i	$-0.0975903\pi$
$827$	966.498	+	558.008i	1.16868	+	0.674737i	0.215310	+	0.976546i	$0.430924\pi$
$828$		0			0
$829$	−42.5592			−0.0513380			−0.0256690	−	0.999670i	$-0.508172\pi$
$829$	−42.5592			−0.0513380			−0.0256690	+	0.999670i	$0.508172\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	361.260i		−	0.433686i
$834$		0			0
$835$	1075.65	+	1863.09i	1.28821	+	2.23124i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−413.860	−	238.942i	−0.493278	−	0.284794i	0.232656	−	0.972559i	$-0.425259\pi$
$839$	−413.860	−	238.942i	−0.493278	−	0.284794i	−0.725933	+	0.687765i	$0.758592\pi$
$840$		0			0
$841$	790.521			0.939978
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1006.80	−	581.274i	−1.19147	−	0.687898i
$846$		0			0
$847$	544.484			0.642839
$848$		0			0
$849$		0			0
$850$		0			0
$851$	277.523	−	160.228i	0.326114	−	0.188282i
$852$		0			0
$853$	440.866	−	763.602i	0.516841	−	0.895195i	−0.482968	−	0.875638i	$-0.660441\pi$
$853$	440.866	−	763.602i	0.516841	−	0.895195i	0.999809	−	0.0195570i	$-0.00622558\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−727.761	−	420.173i	−0.849197	−	0.490284i	0.0111831	−	0.999937i	$-0.496440\pi$
$857$	−727.761	−	420.173i	−0.849197	−	0.490284i	−0.860380	+	0.509654i	$0.829774\pi$
$858$		0			0
$859$	578.204	+	1001.48i	0.673113	+	1.16587i	0.977017	+	0.213163i	$0.0683765\pi$
$859$	578.204	+	1001.48i	0.673113	+	1.16587i	−0.303904	+	0.952703i	$0.598290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	267.381i		−	0.309828i	−0.987928	−	0.154914i	$-0.950490\pi$
$863$		−	267.381i		−	0.309828i	0.987928	−	0.154914i	$-0.0495100\pi$
$864$		0			0
$865$	555.640	−	962.397i	0.642359	−	1.11260i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			138.507i			0.159387i
$870$		0			0
$871$	141.386	−	244.887i	0.162326	−	0.281156i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1329.39	−	767.522i	1.51930	−	0.877168i
$876$		0			0
$877$	1435.34			1.63665			0.818326	−	0.574754i	$-0.194902\pi$
$877$	1435.34			1.63665			0.818326	+	0.574754i	$0.194902\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			920.034i			1.04431i	0.852852	+	0.522153i	$0.174871\pi$
$881$			920.034i			1.04431i	−0.852852	+	0.522153i	$0.825129\pi$
$882$		0			0
$883$	261.135	−	452.299i	0.295736	−	0.512230i	−0.679420	−	0.733750i	$-0.737768\pi$
$883$	261.135	−	452.299i	0.295736	−	0.512230i	0.975156	+	0.221520i	$0.0711018\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	239.539	−	138.298i	0.270055	−	0.155917i	−0.358857	−	0.933392i	$-0.616834\pi$
$887$	239.539	−	138.298i	0.270055	−	0.155917i	0.628913	+	0.777476i	$0.283500\pi$
$888$		0			0
$889$	−515.265			−0.579601
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−68.7845	+	427.305i	−0.0770263	+	0.478505i
$894$		0			0
$895$	1698.38			1.89763
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−97.2542	−	56.1497i	−0.108180	−	0.0624580i
$900$		0			0
$901$	−86.8086			−0.0963470
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1253.88	−	723.929i	−1.38551	−	0.799922i
$906$		0			0
$907$	171.465	−	296.986i	0.189046	−	0.327438i	−0.755886	−	0.654703i	$-0.772794\pi$
$907$	171.465	−	296.986i	0.189046	−	0.327438i	0.944933	+	0.327265i	$0.106127\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	629.426	−	363.399i	0.690918	−	0.398901i	−0.113038	−	0.993591i	$-0.536058\pi$
$911$	629.426	−	363.399i	0.690918	−	0.398901i	0.803956	+	0.594689i	$0.202725\pi$
$912$		0			0
$913$	−25.5838	+	44.3124i	−0.0280217	+	0.0485350i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	33.6261	−	19.4140i	0.0366696	−	0.0211712i
$918$		0			0
$919$	−851.166			−0.926188			−0.463094	−	0.886309i	$-0.653261\pi$
$919$	−851.166			−0.926188			−0.463094	+	0.886309i	$0.653261\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−142.099	−	82.0408i	−0.153953	−	0.0888850i
$924$		0			0
$925$	1039.59			1.12388
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−786.926	−	454.332i	−0.847067	−	0.489055i	0.0125930	−	0.999921i	$-0.495991\pi$
$929$	−786.926	−	454.332i	−0.847067	−	0.489055i	−0.859660	+	0.510866i	$0.829325\pi$
$930$		0			0
$931$	−338.170	+	415.457i	−0.363233	+	0.446248i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−131.584	−	75.9700i	−0.140731	−	0.0812513i
$936$		0			0
$937$	−38.1604	+	66.0958i	−0.0407262	+	0.0705398i	−0.885670	−	0.464315i	$-0.846300\pi$
$937$	−38.1604	+	66.0958i	−0.0407262	+	0.0705398i	0.844944	+	0.534855i	$0.179634\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	755.091	−	435.952i	0.802434	−	0.463286i	−0.0418872	−	0.999122i	$-0.513337\pi$
$941$	755.091	−	435.952i	0.802434	−	0.463286i	0.844322	+	0.535837i	$0.180004\pi$
$942$		0			0
$943$	166.973	+	289.206i	0.177066	+	0.306687i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−283.460	−	163.656i	−0.299324	−	0.172815i	0.342815	−	0.939403i	$-0.388620\pi$
$947$	−283.460	−	163.656i	−0.299324	−	0.172815i	−0.642139	+	0.766588i	$0.721953\pi$
$948$		0			0
$949$	−262.736	+	455.073i	−0.276856	+	0.479529i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−87.9770	+	50.7935i	−0.0923158	+	0.0532986i	−0.545447	−	0.838145i	$-0.683640\pi$
$953$	−87.9770	+	50.7935i	−0.0923158	+	0.0532986i	0.453131	+	0.891444i	$0.350307\pi$
$954$		0			0
$955$	−1570.52	−	2720.22i	−1.64453	−	2.84840i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−846.116	+	488.505i	−0.882290	+	0.509390i
$960$		0			0
$961$	355.585	+	615.891i	0.370015	+	0.640885i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1978.09i			2.04984i
$966$		0			0
$967$	645.605			0.667637			0.333819	−	0.942637i	$-0.391663\pi$
$967$	645.605			0.667637			0.333819	+	0.942637i	$0.391663\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1455.94	+	840.588i	−1.49943	+	0.865693i	−1.00000		0.000663529i	$-0.999789\pi$
$971$	−1455.94	+	840.588i	−1.49943	+	0.865693i	−0.499425	+	0.866357i	$0.666455\pi$
$972$		0			0
$973$	117.553	−	203.607i	0.120815	−	0.209257i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−417.952	−	241.305i	−0.427792	−	0.246986i	0.270614	−	0.962688i	$-0.412773\pi$
$977$	−417.952	−	241.305i	−0.427792	−	0.246986i	−0.698405	+	0.715702i	$0.746107\pi$
$978$		0			0
$979$	72.6084			0.0741659
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−285.347	−	164.745i	−0.290282	−	0.167594i	0.347787	−	0.937574i	$-0.386933\pi$
$983$	−285.347	−	164.745i	−0.290282	−	0.167594i	−0.638069	+	0.769979i	$0.720267\pi$
$984$		0			0
$985$	71.9661			0.0730620
$986$		0			0
$987$		0			0
$988$		0			0
$989$			788.492i			0.797262i
$990$		0			0
$991$	130.189	+	225.494i	0.131371	+	0.227542i	0.924205	−	0.381896i	$-0.124729\pi$
$991$	130.189	+	225.494i	0.131371	+	0.227542i	−0.792834	+	0.609438i	$0.791395\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2412.36	+	1392.78i	2.42448	+	1.39978i
$996$		0			0
$997$	1452.05			1.45642			0.728210	−	0.685354i	$-0.240353\pi$
$997$	1452.05			1.45642			0.728210	+	0.685354i	$0.240353\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.m.a.881.1		80
3.2	odd	2		684.3.m.a.653.12	yes	80
9.2	odd	6		2052.3.be.a.197.1		80
9.7	even	3		684.3.be.a.425.38	yes	80
19.11	even	3		2052.3.be.a.125.1		80
57.11	odd	6		684.3.be.a.581.38	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.40		80
171.106	even	3		684.3.m.a.353.12	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.12	✓	80	171.106	even	3
684.3.m.a.653.12	yes	80	3.2	odd	2
684.3.be.a.425.38	yes	80	9.7	even	3
684.3.be.a.581.38	yes	80	57.11	odd	6
2052.3.m.a.881.1		80	1.1	even	1	trivial
2052.3.m.a.1493.40		80	171.11	odd	6	inner
2052.3.be.a.125.1		80	19.11	even	3
2052.3.be.a.197.1		80	9.2	odd	6

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.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-9.28650i q^{5} +(-2.28067 - 3.95023i) q^{7} +O(q^{10})\)	\(q-9.28650i q^{5} +(-2.28067 - 3.95023i) q^{7} +(1.10584 - 0.638454i) q^{11} +(-3.30958 - 5.73235i) q^{13} +(11.0966 - 6.40664i) q^{17} +(-18.7585 - 3.01961i) q^{19} +(-16.3480 + 9.43853i) q^{23} -61.2390 q^{25} -7.10485i q^{29} +(7.90302 - 13.6884i) q^{31} +(-36.6838 + 21.1794i) q^{35} -16.9760 q^{37} -17.6906i q^{41} +(20.8849 - 36.1738i) q^{43} -22.7792i q^{47} +(14.0971 - 24.4169i) q^{49} +(-5.86723 - 3.38745i) q^{53} +(-5.92901 - 10.2693i) q^{55} +82.7526i q^{59} +47.4838 q^{61} +(-53.2335 + 30.7344i) q^{65} +(21.3601 + 36.9968i) q^{67} +(21.4678 - 12.3945i) q^{71} +(-39.6934 - 68.7509i) q^{73} +(-5.04409 - 2.91221i) q^{77} +(-54.2353 + 93.9382i) q^{79} +(-34.7029 + 20.0357i) q^{83} +(-59.4952 - 103.049i) q^{85} +(49.2445 + 28.4313i) q^{89} +(-15.0961 + 26.1472i) q^{91} +(-28.0416 + 174.201i) q^{95} +(-86.8612 + 150.448i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

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