LMFDB - Embedded newform 2052.3.be.a.125.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2052.125");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			125.10
Character	$\chi$	$=$	2052.125
Dual form			2052.3.be.a.197.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.96437 + 2.86618i) q^{5} +(5.68826 + 9.85235i) q^{7} +O(q^{10})$	$q+(-4.96437 + 2.86618i) q^{5} +(5.68826 + 9.85235i) q^{7} +(7.26930 - 4.19693i) q^{11} +15.9504 q^{13} +(-15.1054 - 8.72110i) q^{17} +(7.75053 - 17.3473i) q^{19} -29.6650i q^{23} +(3.92997 - 6.80691i) q^{25} +(-13.4267 - 7.75191i) q^{29} +(20.6912 - 35.8383i) q^{31} +(-56.4772 - 32.6071i) q^{35} +43.7381 q^{37} +(45.2606 - 26.1312i) q^{41} -56.3193 q^{43} +(50.4293 + 29.1153i) q^{47} +(-40.2126 + 69.6502i) q^{49} +(8.05606 - 4.65117i) q^{53} +(-24.0583 + 41.6702i) q^{55} +(-58.7003 + 33.8906i) q^{59} +(48.9248 - 84.7403i) q^{61} +(-79.1838 + 45.7168i) q^{65} -77.1899 q^{67} +(-7.14871 - 4.12731i) q^{71} +(54.1268 - 93.7503i) q^{73} +(82.6993 + 47.7465i) q^{77} +14.8260 q^{79} +(73.5512 - 42.4648i) q^{83} +99.9850 q^{85} +(-51.8391 + 29.9293i) q^{89} +(90.7301 + 157.149i) q^{91} +(11.2440 + 108.333i) q^{95} -42.9047 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.96437	+	2.86618i	−0.992874	+	0.573236i	−0.906132	−	0.422995i	$-0.860979\pi$
$5$	−4.96437	+	2.86618i	−0.992874	+	0.573236i	−0.0867417	+	0.996231i	$0.527645\pi$
$6$		0			0
$7$	5.68826	+	9.85235i	0.812608	+	1.40748i	0.911033	+	0.412334i	$0.135286\pi$
$7$	5.68826	+	9.85235i	0.812608	+	1.40748i	−0.0984244	+	0.995145i	$0.531380\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.26930	−	4.19693i	0.660845	−	0.381539i	−0.131754	−	0.991282i	$-0.542061\pi$
$11$	7.26930	−	4.19693i	0.660845	−	0.381539i	0.792599	+	0.609743i	$0.208727\pi$
$12$		0			0
$13$	15.9504			1.22696			0.613478	−	0.789712i	$-0.289770\pi$
$13$	15.9504			1.22696			0.613478	+	0.789712i	$0.289770\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.1054	−	8.72110i	−0.888552	−	0.513006i	−0.0150837	−	0.999886i	$-0.504801\pi$
$17$	−15.1054	−	8.72110i	−0.888552	−	0.513006i	−0.873469	+	0.486880i	$0.838135\pi$
$18$		0			0
$19$	7.75053	−	17.3473i	0.407923	−	0.913016i
$19$	7.75053	−	17.3473i	0.407923	−	0.913016i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	29.6650i		−	1.28978i	−0.764275	−	0.644891i	$-0.776903\pi$
$23$		−	29.6650i		−	1.28978i	0.764275	−	0.644891i	$-0.223097\pi$
$24$		0			0
$25$	3.92997	−	6.80691i	0.157199	−	0.272276i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−13.4267	−	7.75191i	−0.462989	−	0.267307i	0.250311	−	0.968165i	$-0.419467\pi$
$29$	−13.4267	−	7.75191i	−0.462989	−	0.267307i	−0.713300	+	0.700858i	$0.752800\pi$
$30$		0			0
$31$	20.6912	−	35.8383i	0.667460	−	1.15607i	−0.311153	−	0.950360i	$-0.600715\pi$
$31$	20.6912	−	35.8383i	0.667460	−	1.15607i	0.978612	−	0.205714i	$-0.0659516\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−56.4772	−	32.6071i	−1.61363	−	0.931632i
$36$		0			0
$37$	43.7381			1.18211			0.591055	−	0.806631i	$-0.298711\pi$
$37$	43.7381			1.18211			0.591055	+	0.806631i	$0.298711\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	45.2606	−	26.1312i	1.10392	−	0.637347i	0.166670	−	0.986013i	$-0.446698\pi$
$41$	45.2606	−	26.1312i	1.10392	−	0.637347i	0.937247	+	0.348666i	$0.113365\pi$
$42$		0			0
$43$	−56.3193			−1.30975			−0.654875	−	0.755737i	$-0.727279\pi$
$43$	−56.3193			−1.30975			−0.654875	+	0.755737i	$0.727279\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	50.4293	+	29.1153i	1.07296	+	0.619476i	0.928990	−	0.370106i	$-0.120679\pi$
$47$	50.4293	+	29.1153i	1.07296	+	0.619476i	0.143974	+	0.989582i	$0.454012\pi$
$48$		0			0
$49$	−40.2126	+	69.6502i	−0.820664	+	1.42143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.05606	−	4.65117i	0.152001	−	0.0877579i	−0.422070	−	0.906563i	$-0.638696\pi$
$53$	8.05606	−	4.65117i	0.152001	−	0.0877579i	0.574072	+	0.818805i	$0.305363\pi$
$54$		0			0
$55$	−24.0583	+	41.6702i	−0.437424	+	0.757641i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−58.7003	+	33.8906i	−0.994921	+	0.574418i	−0.906742	−	0.421687i	$-0.861438\pi$
$59$	−58.7003	+	33.8906i	−0.994921	+	0.574418i	−0.0881792	+	0.996105i	$0.528105\pi$
$60$		0			0
$61$	48.9248	−	84.7403i	0.802046	−	1.38919i	−0.116221	−	0.993223i	$-0.537078\pi$
$61$	48.9248	−	84.7403i	0.802046	−	1.38919i	0.918267	−	0.395962i	$-0.129589\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−79.1838	+	45.7168i	−1.21821	+	0.703335i
$66$		0			0
$67$	−77.1899			−1.15209			−0.576044	−	0.817419i	$-0.695404\pi$
$67$	−77.1899			−1.15209			−0.576044	+	0.817419i	$0.695404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.14871	−	4.12731i	−0.100686	−	0.0581311i	0.448811	−	0.893626i	$-0.351847\pi$
$71$	−7.14871	−	4.12731i	−0.100686	−	0.0581311i	−0.549497	+	0.835495i	$0.685181\pi$
$72$		0			0
$73$	54.1268	−	93.7503i	0.741463	−	1.28425i	−0.210366	−	0.977623i	$-0.567466\pi$
$73$	54.1268	−	93.7503i	0.741463	−	1.28425i	0.951829	−	0.306629i	$-0.0992010\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	82.6993	+	47.7465i	1.07402	+	0.620084i
$78$		0			0
$79$	14.8260			0.187671			0.0938355	−	0.995588i	$-0.470087\pi$
$79$	14.8260			0.187671			0.0938355	+	0.995588i	$0.470087\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	73.5512	−	42.4648i	0.886158	−	0.511624i	0.0134745	−	0.999909i	$-0.495711\pi$
$83$	73.5512	−	42.4648i	0.886158	−	0.511624i	0.872684	+	0.488285i	$0.162377\pi$
$84$		0			0
$85$	99.9850			1.17629
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−51.8391	+	29.9293i	−0.582462	+	0.336285i	−0.762111	−	0.647446i	$-0.775837\pi$
$89$	−51.8391	+	29.9293i	−0.582462	+	0.336285i	0.179649	+	0.983731i	$0.442504\pi$
$90$		0			0
$91$	90.7301	+	157.149i	0.997034	+	1.72691i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.2440	+	108.333i	0.118358	+	1.14035i
$96$		0			0
$97$	−42.9047			−0.442316			−0.221158	−	0.975238i	$-0.570984\pi$
$97$	−42.9047			−0.442316			−0.221158	+	0.975238i	$0.570984\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	80.9893	+	46.7592i	0.801874	+	0.462962i	0.844126	−	0.536145i	$-0.180120\pi$
$101$	80.9893	+	46.7592i	0.801874	+	0.462962i	−0.0422519	+	0.999107i	$0.513453\pi$
$102$		0			0
$103$	−43.4426	+	75.2447i	−0.421772	+	0.730531i	−0.996113	−	0.0880853i	$-0.971925\pi$
$103$	−43.4426	+	75.2447i	−0.421772	+	0.730531i	0.574341	+	0.818616i	$0.305259\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.388947i		−	0.00363502i	−0.999998	−	0.00181751i	$-0.999421\pi$
$107$		−	0.388947i		−	0.00363502i	0.999998	−	0.00181751i	$-0.000578531\pi$
$108$		0			0
$109$	51.7429	−	89.6213i	0.474705	−	0.822213i	−0.524875	−	0.851179i	$-0.675888\pi$
$109$	51.7429	−	89.6213i	0.474705	−	0.822213i	0.999580	+	0.0289658i	$0.00922138\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	42.0647	+	24.2861i	0.372254	+	0.214921i	0.674443	−	0.738327i	$-0.264384\pi$
$113$	42.0647	+	24.2861i	0.372254	+	0.214921i	−0.302189	+	0.953248i	$0.597717\pi$
$114$		0			0
$115$	85.0251	+	147.268i	0.739349	+	1.28059i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	198.431i		−	1.66749i
$120$		0			0
$121$	−25.2715	+	43.7716i	−0.208856	+	0.361749i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	98.2530i		−	0.786024i
$126$		0			0
$127$	56.3382	+	97.5806i	0.443608	+	0.768351i	0.997954	−	0.0639349i	$-0.0203650\pi$
$127$	56.3382	+	97.5806i	0.443608	+	0.768351i	−0.554346	+	0.832286i	$0.687032\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	152.184	−	87.8636i	1.16171	−	0.670715i	0.209998	−	0.977702i	$-0.432654\pi$
$131$	152.184	−	87.8636i	1.16171	−	0.670715i	0.951714	+	0.306987i	$0.0993209\pi$
$132$		0			0
$133$	214.999	−	22.3150i	1.61653	−	0.167782i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	167.426	+	96.6632i	1.22208	+	0.705571i	0.965362	−	0.260915i	$-0.0840241\pi$
$137$	167.426	+	96.6632i	1.22208	+	0.705571i	0.256722	+	0.966485i	$0.417357\pi$
$138$		0			0
$139$	146.074			1.05089			0.525446	−	0.850827i	$-0.323898\pi$
$139$	146.074			1.05089			0.525446	+	0.850827i	$0.323898\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	115.948	−	66.9428i	0.810828	−	0.468132i
$144$		0			0
$145$	88.8734			0.612920
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−212.965	+	122.955i	−1.42929	+	0.825202i	−0.997065	−	0.0765620i	$-0.975606\pi$
$149$	−212.965	+	122.955i	−1.42929	+	0.825202i	−0.432228	+	0.901764i	$0.642272\pi$
$150$		0			0
$151$	−79.6886	−	138.025i	−0.527739	−	0.914070i	−0.999477	−	0.0323319i	$-0.989707\pi$
$151$	−79.6886	−	138.025i	−0.527739	−	0.914070i	0.471738	−	0.881739i	$-0.343627\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			237.219i			1.53045i
$156$		0			0
$157$	−64.8220	−	112.275i	−0.412879	−	0.715128i	0.582324	−	0.812957i	$-0.302143\pi$
$157$	−64.8220	−	112.275i	−0.412879	−	0.715128i	−0.995203	+	0.0978291i	$0.968810\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	292.270	−	168.742i	1.81534	−	1.04809i
$162$		0			0
$163$	74.7496			0.458587			0.229293	−	0.973357i	$-0.426358\pi$
$163$	74.7496			0.458587			0.229293	+	0.973357i	$0.426358\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			60.4753i			0.362128i	0.983471	+	0.181064i	$0.0579541\pi$
$167$			60.4753i			0.362128i	−0.983471	+	0.181064i	$0.942046\pi$
$168$		0			0
$169$	85.4158			0.505419
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	147.057i		−	0.850042i	−0.905184	−	0.425021i	$-0.860267\pi$
$173$		−	147.057i		−	0.850042i	0.905184	−	0.425021i	$-0.139733\pi$
$174$		0			0
$175$	89.4188			0.510964
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	105.180i		−	0.587596i	−0.955868	−	0.293798i	$-0.905081\pi$
$179$		−	105.180i		−	0.587596i	0.955868	−	0.293798i	$-0.0949193\pi$
$180$		0			0
$181$	133.711	+	231.594i	0.738734	+	1.27952i	0.953066	+	0.302763i	$0.0979091\pi$
$181$	133.711	+	231.594i	0.738734	+	1.27952i	−0.214332	+	0.976761i	$0.568758\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−217.132	+	125.361i	−1.17369	+	0.677628i
$186$		0			0
$187$	−146.407			−0.782928
$188$		0			0
$189$		0			0
$190$		0			0
$191$	191.871	−	110.777i	1.00456	−	0.579983i	0.0949654	−	0.995481i	$-0.469726\pi$
$191$	191.871	−	110.777i	1.00456	−	0.579983i	0.909594	+	0.415498i	$0.136393\pi$
$192$		0			0
$193$	−33.1705	−	57.4529i	−0.171868	−	0.297684i	0.767205	−	0.641402i	$-0.221647\pi$
$193$	−33.1705	−	57.4529i	−0.171868	−	0.297684i	−0.939073	+	0.343718i	$0.888314\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	7.31542i		−	0.0371341i	−0.999828	−	0.0185670i	$-0.994090\pi$
$197$		−	7.31542i		−	0.0371341i	0.999828	−	0.0185670i	$-0.00591041\pi$
$198$		0			0
$199$	60.3316	+	104.497i	0.303174	+	0.525112i	0.976853	−	0.213912i	$-0.0686204\pi$
$199$	60.3316	+	104.497i	0.303174	+	0.525112i	−0.673679	+	0.739024i	$0.735287\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	176.379i		−	0.868864i
$204$		0			0
$205$	−149.794	+	259.450i	−0.730700	+	1.26561i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.4646	−	158.631i	−0.0787778	−	0.759001i
$210$		0			0
$211$	−88.7486	−	153.717i	−0.420609	−	0.728517i	0.575390	−	0.817879i	$-0.304850\pi$
$211$	−88.7486	−	153.717i	−0.420609	−	0.728517i	−0.995999	+	0.0893624i	$0.971517\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	279.590	−	161.421i	1.30042	−	0.750796i
$216$		0			0
$217$	470.789			2.16953
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−240.937	−	139.105i	−1.09021	−	0.629435i
$222$		0			0
$223$	−338.152			−1.51638			−0.758189	−	0.652035i	$-0.773916\pi$
$223$	−338.152			−1.51638			−0.758189	+	0.652035i	$0.773916\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	322.258	−	186.056i	1.41964	−	0.819630i	0.423373	−	0.905955i	$-0.360846\pi$
$227$	322.258	−	186.056i	1.41964	−	0.819630i	0.996267	+	0.0863255i	$0.0275125\pi$
$228$		0			0
$229$	−9.51699	+	16.4839i	−0.0415589	+	0.0719821i	−0.886057	−	0.463577i	$-0.846566\pi$
$229$	−9.51699	+	16.4839i	−0.0415589	+	0.0719821i	0.844498	+	0.535559i	$0.179899\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	380.517	+	219.692i	1.63312	+	0.942882i	0.983123	+	0.182946i	$0.0585633\pi$
$233$	380.517	+	219.692i	1.63312	+	0.942882i	0.649997	+	0.759937i	$0.274770\pi$
$234$		0			0
$235$	−333.799			−1.42042
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−241.431	−	139.390i	−1.01017	−	0.583222i	−0.0989287	−	0.995095i	$-0.531542\pi$
$239$	−241.431	−	139.390i	−1.01017	−	0.583222i	−0.911241	+	0.411872i	$0.864875\pi$
$240$		0			0
$241$	−131.420	+	227.627i	−0.545313	+	0.944510i	0.453274	+	0.891371i	$0.350256\pi$
$241$	−131.420	+	227.627i	−0.545313	+	0.944510i	−0.998587	+	0.0531389i	$0.983077\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	461.026i		−	1.88174i
$246$		0			0
$247$	123.624	−	276.697i	0.500503	−	1.12023i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−259.537	+	149.843i	−1.03401	+	0.596986i	−0.918131	−	0.396277i	$-0.870302\pi$
$251$	−259.537	+	149.843i	−1.03401	+	0.596986i	−0.115879	+	0.993263i	$0.536969\pi$
$252$		0			0
$253$	−124.502	−	215.644i	−0.492102	−	0.852346i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	445.962i		−	1.73526i	−0.497211	−	0.867630i	$-0.665642\pi$
$257$		−	445.962i		−	1.73526i	0.497211	−	0.867630i	$-0.334358\pi$
$258$		0			0
$259$	248.794	+	430.923i	0.960593	+	1.66380i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.66326i			0.0215333i	0.999942	+	0.0107667i	$0.00342720\pi$
$263$			5.66326i			0.0215333i	−0.999942	+	0.0107667i	$0.996573\pi$
$264$		0			0
$265$	−26.6622	+	46.1802i	−0.100612	+	0.174265i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−157.523	−	90.9458i	−0.585586	−	0.338088i	0.177764	−	0.984073i	$-0.443114\pi$
$269$	−157.523	−	90.9458i	−0.585586	−	0.338088i	−0.763350	+	0.645985i	$0.776447\pi$
$270$		0			0
$271$	49.1844	−	85.1899i	0.181492	−	0.314354i	−0.760897	−	0.648873i	$-0.775241\pi$
$271$	49.1844	−	85.1899i	0.181492	−	0.314354i	0.942389	+	0.334519i	$0.108574\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	65.9753i		−	0.239910i
$276$		0			0
$277$	170.275	+	294.926i	0.614712	+	1.06471i	0.990435	+	0.137981i	$0.0440613\pi$
$277$	170.275	+	294.926i	0.614712	+	1.06471i	−0.375722	+	0.926732i	$0.622605\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	287.533	+	166.007i	1.02325	+	0.590772i	0.915043	−	0.403356i	$-0.132156\pi$
$281$	287.533	+	166.007i	1.02325	+	0.590772i	0.108205	+	0.994129i	$0.465490\pi$
$282$		0			0
$283$	180.330	+	312.341i	0.637209	+	1.10368i	0.986042	+	0.166494i	$0.0532447\pi$
$283$	180.330	+	312.341i	0.637209	+	1.10368i	−0.348833	+	0.937185i	$0.613422\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	514.908	+	297.282i	1.79410	+	1.03583i
$288$		0			0
$289$	7.61522	+	13.1900i	0.0263502	+	0.0456400i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	280.323	+	161.844i	0.956732	+	0.552370i	0.895166	−	0.445733i	$-0.147057\pi$
$293$	280.323	+	161.844i	0.956732	+	0.552370i	0.0615664	+	0.998103i	$0.480390\pi$
$294$		0			0
$295$	194.273	−	336.491i	0.658554	−	1.14065i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	473.169i		−	1.58250i
$300$		0			0
$301$	−320.359	−	554.877i	−1.06431	−	1.84345i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			560.909i			1.83905i
$306$		0			0
$307$	−186.149	+	322.419i	−0.606347	+	1.05022i	0.385490	+	0.922712i	$0.374033\pi$
$307$	−186.149	+	322.419i	−0.606347	+	1.05022i	−0.991837	+	0.127512i	$0.959301\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−244.080	−	140.919i	−0.784822	−	0.453117i	0.0533148	−	0.998578i	$-0.483021\pi$
$311$	−244.080	−	140.919i	−0.784822	−	0.453117i	−0.838136	+	0.545461i	$0.816355\pi$
$312$		0			0
$313$	−105.566	+	182.846i	−0.337273	+	0.584173i	−0.983919	−	0.178617i	$-0.942838\pi$
$313$	−105.566	+	182.846i	−0.337273	+	0.584173i	0.646646	+	0.762790i	$0.276171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−39.6520	−	22.8931i	−0.125085	−	0.0722180i	0.436152	−	0.899873i	$-0.356341\pi$
$317$	−39.6520	−	22.8931i	−0.125085	−	0.0722180i	−0.561237	+	0.827655i	$0.689674\pi$
$318$		0			0
$319$	−130.137			−0.407953
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−268.362	+	194.445i	−0.830844	+	0.601996i
$324$		0			0
$325$	62.6847	−	108.573i	0.192876	−	0.334071i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			662.462i			2.01356i
$330$		0			0
$331$	−34.6411	−	60.0002i	−0.104656	−	0.181269i	0.808942	−	0.587889i	$-0.200041\pi$
$331$	−34.6411	−	60.0002i	−0.104656	−	0.181269i	−0.913598	+	0.406620i	$0.866707\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	383.199	−	221.240i	1.14388	−	0.660418i
$336$		0			0
$337$	−36.4436	−	63.1222i	−0.108141	−	0.187306i	0.806876	−	0.590721i	$-0.201157\pi$
$337$	−36.4436	−	63.1222i	−0.108141	−	0.187306i	−0.915017	+	0.403415i	$0.867823\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	347.359i		−	1.01865i
$342$		0			0
$343$	−357.508			−1.04230
$344$		0			0
$345$		0			0
$346$		0			0
$347$	260.564	−	150.436i	0.750904	−	0.433535i	−0.0751166	−	0.997175i	$-0.523933\pi$
$347$	260.564	−	150.436i	0.750904	−	0.433535i	0.826020	+	0.563640i	$0.190600\pi$
$348$		0			0
$349$	39.0504	+	67.6373i	0.111892	+	0.193803i	0.916533	−	0.399959i	$-0.130975\pi$
$349$	39.0504	+	67.6373i	0.111892	+	0.193803i	−0.804641	+	0.593762i	$0.797642\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−564.032	+	325.644i	−1.59782	+	0.922504i	−0.605918	+	0.795527i	$0.707194\pi$
$353$	−564.032	+	325.644i	−1.59782	+	0.922504i	−0.991906	+	0.126977i	$0.959473\pi$
$354$		0			0
$355$	47.3184			0.133291
$356$		0			0
$357$		0			0
$358$		0			0
$359$	523.756	+	302.390i	1.45893	+	0.842313i	0.998959	−	0.0456210i	$-0.0145267\pi$
$359$	523.756	+	302.390i	1.45893	+	0.842313i	0.459970	+	0.887934i	$0.347860\pi$
$360$		0			0
$361$	−240.859	−	268.902i	−0.667198	−	0.744880i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			620.548i			1.70013i
$366$		0			0
$367$	−64.8792	+	112.374i	−0.176783	+	0.306196i	−0.940777	−	0.339027i	$-0.889902\pi$
$367$	−64.8792	+	112.374i	−0.176783	+	0.306196i	0.763994	+	0.645223i	$0.223236\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	91.6499	+	52.9141i	0.247035	+	0.142626i
$372$		0			0
$373$	−246.184	+	426.404i	−0.660012	+	1.14317i	0.320600	+	0.947215i	$0.396116\pi$
$373$	−246.184	+	426.404i	−0.660012	+	1.14317i	−0.980612	+	0.195960i	$0.937218\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−214.161	−	123.646i	−0.568067	−	0.327974i
$378$		0			0
$379$	702.141			1.85262			0.926308	−	0.376768i	$-0.122964\pi$
$379$	702.141			1.85262			0.926308	+	0.376768i	$0.122964\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	127.315	−	73.5051i	0.332414	−	0.191919i	−0.324498	−	0.945886i	$-0.605196\pi$
$383$	127.315	−	73.5051i	0.332414	−	0.191919i	0.656912	+	0.753967i	$0.271862\pi$
$384$		0			0
$385$	−547.400			−1.42182
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−317.836	−	183.503i	−0.817060	−	0.471730i	0.0323416	−	0.999477i	$-0.489704\pi$
$389$	−317.836	−	183.503i	−0.817060	−	0.471730i	−0.849402	+	0.527747i	$0.823037\pi$
$390$		0			0
$391$	−258.711	+	448.101i	−0.661666	+	1.14604i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−73.6018	+	42.4940i	−0.186334	+	0.107580i
$396$		0			0
$397$	−33.6521	+	58.2871i	−0.0847660	+	0.146819i	−0.905291	−	0.424791i	$-0.860348\pi$
$397$	−33.6521	+	58.2871i	−0.0847660	+	0.146819i	0.820525	+	0.571610i	$0.193681\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−610.132	+	352.260i	−1.52153	+	0.878454i	−0.521849	+	0.853038i	$0.674758\pi$
$401$	−610.132	+	352.260i	−1.52153	+	0.878454i	−0.999677	+	0.0254160i	$0.991909\pi$
$402$		0			0
$403$	330.034	−	571.636i	0.818943	−	1.41845i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	317.945	−	183.566i	0.781192	−	0.451022i
$408$		0			0
$409$	161.612			0.395140			0.197570	−	0.980289i	$-0.436695\pi$
$409$	161.612			0.395140			0.197570	+	0.980289i	$0.436695\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−667.805	−	385.557i	−1.61696	−	0.933553i
$414$		0			0
$415$	−243.423	+	421.622i	−0.586562	+	1.01596i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	101.045	+	58.3385i	0.241158	+	0.139233i	0.615709	−	0.787974i	$-0.288870\pi$
$419$	101.045	+	58.3385i	0.241158	+	0.139233i	−0.374551	+	0.927206i	$0.622203\pi$
$420$		0			0
$421$	273.938			0.650684			0.325342	−	0.945596i	$-0.394521\pi$
$421$	273.938			0.650684			0.325342	+	0.945596i	$0.394521\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−118.728	+	68.5474i	−0.279359	+	0.161288i
$426$		0			0
$427$	1113.19			2.60700
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−82.9390	+	47.8849i	−0.192434	+	0.111102i	−0.593122	−	0.805113i	$-0.702105\pi$
$431$	−82.9390	+	47.8849i	−0.192434	+	0.111102i	0.400688	+	0.916215i	$0.368771\pi$
$432$		0			0
$433$	−45.1476	−	78.1980i	−0.104267	−	0.180596i	0.809171	−	0.587573i	$-0.199916\pi$
$433$	−45.1476	−	78.1980i	−0.104267	−	0.180596i	−0.913439	+	0.406977i	$0.866583\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−514.608	−	229.919i	−1.17759	−	0.526131i
$438$		0			0
$439$	324.843			0.739960			0.369980	−	0.929040i	$-0.379365\pi$
$439$	324.843			0.739960			0.369980	+	0.929040i	$0.379365\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−39.9943	−	23.0907i	−0.0902806	−	0.0521235i	0.454180	−	0.890910i	$-0.349932\pi$
$443$	−39.9943	−	23.0907i	−0.0902806	−	0.0521235i	−0.544461	+	0.838786i	$0.683266\pi$
$444$		0			0
$445$	171.566	−	297.160i	0.385541	−	0.667776i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			434.800i			0.968373i	0.874965	+	0.484187i	$0.160884\pi$
$449$			434.800i			0.968373i	−0.874965	+	0.484187i	$0.839116\pi$
$450$		0			0
$451$	219.342	−	379.911i	0.486346	−	0.842375i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−900.835	−	520.097i	−1.97986	−	1.14307i
$456$		0			0
$457$	−243.869	−	422.393i	−0.533630	−	0.924274i	−0.999228	−	0.0392779i	$-0.987494\pi$
$457$	−243.869	−	422.393i	−0.533630	−	0.924274i	0.465598	−	0.884996i	$-0.345839\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			223.322i			0.484428i	0.970223	+	0.242214i	$0.0778737\pi$
$461$			223.322i			0.484428i	−0.970223	+	0.242214i	$0.922126\pi$
$462$		0			0
$463$	−293.054	+	507.585i	−0.632946	+	1.09630i	0.354000	+	0.935245i	$0.384821\pi$
$463$	−293.054	+	507.585i	−0.632946	+	1.09630i	−0.986946	+	0.161050i	$0.948512\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			527.930i			1.13047i	0.824930	+	0.565235i	$0.191215\pi$
$467$			527.930i			1.13047i	−0.824930	+	0.565235i	$0.808785\pi$
$468$		0			0
$469$	−439.076	−	760.502i	−0.936196	−	1.62154i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−409.402	+	236.368i	−0.865543	+	0.499721i
$474$		0			0
$475$	−87.6222	−	120.932i	−0.184468	−	0.254593i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−132.723	−	76.6278i	−0.277084	−	0.159975i	0.355019	−	0.934859i	$-0.384475\pi$
$479$	−132.723	−	76.6278i	−0.277084	−	0.159975i	−0.632103	+	0.774885i	$0.717808\pi$
$480$		0			0
$481$	697.641			1.45040
$482$		0			0
$483$		0			0
$484$		0			0
$485$	212.995	−	122.973i	0.439164	−	0.253552i
$486$		0			0
$487$	255.475			0.524590			0.262295	−	0.964988i	$-0.415521\pi$
$487$	255.475			0.524590			0.262295	+	0.964988i	$0.415521\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	309.990	−	178.973i	0.631344	−	0.364506i	−0.149929	−	0.988697i	$-0.547904\pi$
$491$	309.990	−	178.973i	0.631344	−	0.364506i	0.781272	+	0.624190i	$0.214571\pi$
$492$		0			0
$493$	135.210	+	234.191i	0.274260	+	0.475033i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	93.9087i		−	0.188951i
$498$		0			0
$499$	−109.971	−	190.476i	−0.220383	−	0.381715i	0.734541	−	0.678564i	$-0.237398\pi$
$499$	−109.971	−	190.476i	−0.220383	−	0.381715i	−0.954924	+	0.296849i	$0.904064\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	482.957	−	278.835i	0.960154	−	0.554345i	0.0639332	−	0.997954i	$-0.479636\pi$
$503$	482.957	−	278.835i	0.960154	−	0.554345i	0.896220	+	0.443609i	$0.146302\pi$
$504$		0			0
$505$	−536.081			−1.06155
$506$		0			0
$507$		0			0
$508$		0			0
$509$			354.509i			0.696481i	0.937405	+	0.348240i	$0.113221\pi$
$509$			354.509i			0.696481i	−0.937405	+	0.348240i	$0.886779\pi$
$510$		0			0
$511$	1231.55			2.41008
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	498.057i		−	0.967100i
$516$		0			0
$517$	488.781			0.945417
$518$		0			0
$519$		0			0
$520$		0			0
$521$			249.110i			0.478139i	0.971003	+	0.239069i	$0.0768423\pi$
$521$			249.110i			0.478139i	−0.971003	+	0.239069i	$0.923158\pi$
$522$		0			0
$523$	−78.8721	−	136.611i	−0.150807	−	0.261206i	0.780717	−	0.624884i	$-0.214854\pi$
$523$	−78.8721	−	136.611i	−0.150807	−	0.261206i	−0.931524	+	0.363679i	$0.881521\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−625.099	+	360.901i	−1.18615	+	0.684821i
$528$		0			0
$529$	−351.011			−0.663536
$530$		0			0
$531$		0			0
$532$		0			0
$533$	721.926	−	416.804i	1.35446	−	0.781996i
$534$		0			0
$535$	1.11479	+	1.93088i	0.00208372	+	0.00360911i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			675.077i			1.25246i
$540$		0			0
$541$	−313.961	−	543.797i	−0.580335	−	1.00517i	−0.995439	−	0.0953961i	$-0.969588\pi$
$541$	−313.961	−	543.797i	−0.580335	−	1.00517i	0.415104	−	0.909774i	$-0.363745\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			593.217i			1.08847i
$546$		0			0
$547$	427.963	−	741.254i	0.782383	−	1.35513i	−0.148167	−	0.988962i	$-0.547337\pi$
$547$	427.963	−	741.254i	0.782383	−	1.35513i	0.930550	−	0.366164i	$-0.119329\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−238.539	+	172.836i	−0.432920	+	0.313676i
$552$		0			0
$553$	84.3341	+	146.071i	0.152503	+	0.264143i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	868.882	−	501.649i	1.55993	−	0.900627i	0.562670	−	0.826682i	$-0.309774\pi$
$557$	868.882	−	501.649i	1.55993	−	0.900627i	0.997262	−	0.0739453i	$-0.0235590\pi$
$558$		0			0
$559$	−898.316			−1.60701
$560$		0			0
$561$		0			0
$562$		0			0
$563$	357.546	+	206.430i	0.635074	+	0.366660i	0.782714	−	0.622381i	$-0.213835\pi$
$563$	357.546	+	206.430i	0.635074	+	0.366660i	−0.147641	+	0.989041i	$0.547168\pi$
$564$		0			0
$565$	−278.433			−0.492802
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−849.629	+	490.534i	−1.49320	+	0.862098i	−0.999970	−	0.00780246i	$-0.997516\pi$
$569$	−849.629	+	490.534i	−1.49320	+	0.862098i	−0.493228	+	0.869900i	$0.664183\pi$
$570$		0			0
$571$	316.350	−	547.933i	0.554027	−	0.959603i	−0.443951	−	0.896051i	$-0.646424\pi$
$571$	316.350	−	547.933i	0.554027	−	0.959603i	0.997979	−	0.0635523i	$-0.0202430\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−201.927	−	116.582i	−0.351177	−	0.202752i
$576$		0			0
$577$	−67.1923			−0.116451			−0.0582256	−	0.998303i	$-0.518544\pi$
$577$	−67.1923			−0.116451			−0.0582256	+	0.998303i	$0.518544\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	836.756	+	483.101i	1.44020	+	0.831500i
$582$		0			0
$583$	39.0413	−	67.6215i	0.0669662	−	0.115989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			843.872i			1.43760i	0.695216	+	0.718800i	$0.255309\pi$
$587$			843.872i			1.43760i	−0.695216	+	0.718800i	$0.744691\pi$
$588$		0			0
$589$	−461.330	−	636.703i	−0.783243	−	1.08099i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−479.399	+	276.781i	−0.808429	+	0.466747i	−0.846410	−	0.532532i	$-0.821241\pi$
$593$	−479.399	+	276.781i	−0.808429	+	0.466747i	0.0379808	+	0.999278i	$0.487907\pi$
$594$		0			0
$595$	568.740	+	985.087i	0.955866	+	1.65561i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1051.62i		−	1.75563i	−0.479000	−	0.877815i	$-0.659001\pi$
$599$		−	1051.62i		−	1.75563i	0.479000	−	0.877815i	$-0.340999\pi$
$600$		0			0
$601$	−510.289	−	883.846i	−0.849066	−	1.47063i	−0.882042	−	0.471170i	$-0.843832\pi$
$601$	−510.289	−	883.846i	−0.849066	−	1.47063i	0.0329759	−	0.999456i	$-0.489502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	289.731i		−	0.478894i
$606$		0			0
$607$	73.8601	−	127.929i	0.121681	−	0.210757i	−0.798750	−	0.601663i	$-0.794505\pi$
$607$	73.8601	−	127.929i	0.121681	−	0.210757i	0.920431	+	0.390906i	$0.127838\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	804.368	+	464.402i	1.31648	+	0.760069i
$612$		0			0
$613$	−36.8003	+	63.7400i	−0.0600331	+	0.103980i	−0.894480	−	0.447108i	$-0.852454\pi$
$613$	−36.8003	+	63.7400i	−0.0600331	+	0.103980i	0.834447	+	0.551088i	$0.185787\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	855.348i		−	1.38630i	−0.720792	−	0.693151i	$-0.756222\pi$
$617$		−	855.348i		−	1.38630i	0.720792	−	0.693151i	$-0.243778\pi$
$618$		0			0
$619$	−403.417	−	698.738i	−0.651723	−	1.12882i	−0.982705	−	0.185180i	$-0.940713\pi$
$619$	−403.417	−	698.738i	−0.651723	−	1.12882i	0.330981	−	0.943637i	$-0.392620\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−589.749	−	340.492i	−0.946627	−	0.546535i
$624$		0			0
$625$	379.860	+	657.937i	0.607776	+	1.05270i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−660.681	−	381.444i	−1.05037	−	0.606430i
$630$		0			0
$631$	46.4117	+	80.3874i	0.0735526	+	0.127397i	0.900456	−	0.434947i	$-0.143233\pi$
$631$	46.4117	+	80.3874i	0.0735526	+	0.127397i	−0.826903	+	0.562344i	$0.809900\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−559.367	−	322.951i	−0.880893	−	0.508584i
$636$		0			0
$637$	−641.407	+	1110.95i	−1.00692	+	1.74403i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	866.084i		−	1.35115i	−0.737293	−	0.675573i	$-0.763896\pi$
$641$		−	866.084i		−	1.35115i	0.737293	−	0.675573i	$-0.236104\pi$
$642$		0			0
$643$	−201.250	−	348.576i	−0.312986	−	0.542108i	0.666021	−	0.745933i	$-0.267996\pi$
$643$	−201.250	−	348.576i	−0.312986	−	0.542108i	−0.979007	+	0.203825i	$0.934663\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			688.511i			1.06416i	0.846695	+	0.532079i	$0.178589\pi$
$647$			688.511i			1.06416i	−0.846695	+	0.532079i	$0.821411\pi$
$648$		0			0
$649$	−284.473	+	492.722i	−0.438326	+	0.759203i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	757.524	+	437.357i	1.16007	+	0.669765i	0.951320	−	0.308205i	$-0.0997282\pi$
$653$	757.524	+	437.357i	1.16007	+	0.669765i	0.208747	+	0.977970i	$0.433062\pi$
$654$		0			0
$655$	−503.666	+	872.375i	−0.768955	+	1.33187i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0793	+	6.97398i	0.0183297	+	0.0105827i	0.509137	−	0.860686i	$-0.329965\pi$
$659$	12.0793	+	6.97398i	0.0183297	+	0.0105827i	−0.490807	+	0.871268i	$0.663298\pi$
$660$		0			0
$661$	−1261.07			−1.90782			−0.953908	−	0.300099i	$-0.902980\pi$
$661$	−1261.07			−1.90782			−0.953908	+	0.300099i	$0.902980\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1003.37	+	727.005i	−1.50883	+	1.09324i
$666$		0			0
$667$	−229.960	+	398.303i	−0.344768	+	0.597155i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	821.337i		−	1.22405i
$672$		0			0
$673$	11.2224	+	19.4377i	0.0166751	+	0.0288822i	0.874243	−	0.485489i	$-0.161359\pi$
$673$	11.2224	+	19.4377i	0.0166751	+	0.0288822i	−0.857567	+	0.514372i	$0.828025\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−492.818	+	284.528i	−0.727943	+	0.420278i	−0.817669	−	0.575688i	$-0.804734\pi$
$677$	−492.818	+	284.528i	−0.727943	+	0.420278i	0.0897259	+	0.995966i	$0.471401\pi$
$678$		0			0
$679$	−244.053	−	422.712i	−0.359430	−	0.622551i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			307.792i			0.450647i	0.974284	+	0.225323i	$0.0723439\pi$
$683$			307.792i			0.450647i	−0.974284	+	0.225323i	$0.927656\pi$
$684$		0			0
$685$	−1108.22			−1.61783
$686$		0			0
$687$		0			0
$688$		0			0
$689$	128.498	−	74.1881i	0.186499	−	0.107675i
$690$		0			0
$691$	488.356	+	845.858i	0.706739	+	1.22411i	0.966060	+	0.258316i	$0.0831677\pi$
$691$	488.356	+	845.858i	0.706739	+	1.22411i	−0.259322	+	0.965791i	$0.583499\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−725.165	+	418.674i	−1.04340	+	0.602409i
$696$		0			0
$697$	−911.572			−1.30785
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−66.7694	−	38.5494i	−0.0952489	−	0.0549920i	0.451619	−	0.892211i	$-0.350847\pi$
$701$	−66.7694	−	38.5494i	−0.0952489	−	0.0549920i	−0.546868	+	0.837219i	$0.684180\pi$
$702$		0			0
$703$	338.993	−	758.738i	0.482210	−	1.07929i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1063.91i			1.50483i
$708$		0			0
$709$	364.741	−	631.750i	0.514445	−	0.891044i	−0.485415	−	0.874284i	$-0.661331\pi$
$709$	364.741	−	631.750i	0.514445	−	0.891044i	0.999860	−	0.0167602i	$-0.00533520\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1063.14	−	613.805i	−1.49108	−	0.860877i
$714$		0			0
$715$	−383.740	+	664.658i	−0.536700	+	0.929591i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−53.4606	−	30.8655i	−0.0743541	−	0.0429284i	0.462362	−	0.886691i	$-0.347002\pi$
$719$	−53.4606	−	30.8655i	−0.0743541	−	0.0429284i	−0.536716	+	0.843763i	$0.680335\pi$
$720$		0			0
$721$	−988.450			−1.37094
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−105.533	+	60.9295i	−0.145563	+	0.0840407i
$726$		0			0
$727$	−964.195			−1.32627			−0.663133	−	0.748502i	$-0.730774\pi$
$727$	−964.195			−1.32627			−0.663133	+	0.748502i	$0.730774\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	850.725	+	491.166i	1.16378	+	0.671910i
$732$		0			0
$733$	−348.862	+	604.246i	−0.475937	+	0.824347i	−0.999620	−	0.0275663i	$-0.991224\pi$
$733$	−348.862	+	604.246i	−0.475937	+	0.824347i	0.523683	+	0.851913i	$0.324558\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−561.116	+	323.961i	−0.761352	+	0.439567i
$738$		0			0
$739$	15.5321	−	26.9024i	0.0210177	−	0.0364038i	−0.855325	−	0.518091i	$-0.826643\pi$
$739$	15.5321	−	26.9024i	0.0210177	−	0.0364038i	0.876343	+	0.481688i	$0.159976\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−971.370	+	560.821i	−1.30736	+	0.754806i	−0.981655	−	0.190665i	$-0.938936\pi$
$743$	−971.370	+	560.821i	−1.30736	+	0.754806i	−0.325707	+	0.945471i	$0.605602\pi$
$744$		0			0
$745$	704.823	−	1220.79i	0.946071	−	1.63864i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.83204	−	2.21243i	0.00511621	−	0.00295385i
$750$		0			0
$751$	−1115.08			−1.48480			−0.742398	−	0.669959i	$-0.766312\pi$
$751$	−1115.08			−1.48480			−0.742398	+	0.669959i	$0.766312\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	791.207	+	456.803i	1.04796	+	0.605038i
$756$		0			0
$757$	223.536	−	387.176i	0.295292	−	0.511461i	−0.679761	−	0.733434i	$-0.737916\pi$
$757$	223.536	−	387.176i	0.295292	−	0.511461i	0.975053	+	0.221973i	$0.0712497\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−617.890	−	356.739i	−0.811945	−	0.468777i	0.0356859	−	0.999363i	$-0.488638\pi$
$761$	−617.890	−	356.739i	−0.811945	−	0.468777i	−0.847631	+	0.530586i	$0.821972\pi$
$762$		0			0
$763$	1177.31			1.54300
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−936.295	+	540.570i	−1.22072	+	0.704785i
$768$		0			0
$769$	−87.9835			−0.114413			−0.0572065	−	0.998362i	$-0.518219\pi$
$769$	−87.9835			−0.114413			−0.0572065	+	0.998362i	$0.518219\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	316.894	−	182.959i	0.409953	−	0.236686i	−0.280817	−	0.959761i	$-0.590605\pi$
$773$	316.894	−	182.959i	0.409953	−	0.236686i	0.690769	+	0.723075i	$0.257272\pi$
$774$		0			0
$775$	−162.632	−	281.687i	−0.209848	−	0.363467i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−102.513	−	987.681i	−0.131595	−	1.26788i
$780$		0			0
$781$	−69.2881			−0.0887172
$782$		0			0
$783$		0			0
$784$		0			0
$785$	643.601	+	371.583i	0.819874	+	0.473354i
$786$		0			0
$787$	−44.8335	+	77.6540i	−0.0569676	+	0.0986709i	−0.893103	−	0.449853i	$-0.851477\pi$
$787$	−44.8335	+	77.6540i	−0.0569676	+	0.0986709i	0.836135	+	0.548523i	$0.184810\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			552.582i			0.698586i
$792$		0			0
$793$	780.371	−	1351.64i	0.984075	−	1.70447i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	354.295	+	204.552i	0.444536	+	0.256653i	0.705520	−	0.708690i	$-0.250714\pi$
$797$	354.295	+	204.552i	0.444536	+	0.256653i	−0.260984	+	0.965343i	$0.584047\pi$
$798$		0			0
$799$	−507.836	−	879.597i	−0.635589	−	1.10087i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	908.666i		−	1.13159i
$804$		0			0
$805$	−967.290	+	1675.40i	−1.20160	+	2.08124i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	127.641i		−	0.157777i	−0.996883	−	0.0788884i	$-0.974863\pi$
$809$		−	127.641i		−	0.157777i	0.996883	−	0.0788884i	$-0.0251371\pi$
$810$		0			0
$811$	665.457	+	1152.61i	0.820539	+	1.42122i	0.905281	+	0.424812i	$0.139660\pi$
$811$	665.457	+	1152.61i	0.820539	+	1.42122i	−0.0847425	+	0.996403i	$0.527007\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−371.085	+	214.246i	−0.455319	+	0.262878i
$816$		0			0
$817$	−436.504	+	976.988i	−0.534277	+	1.19582i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−566.733	−	327.203i	−0.690296	−	0.398542i	0.113427	−	0.993546i	$-0.463817\pi$
$821$	−566.733	−	327.203i	−0.690296	−	0.398542i	−0.803723	+	0.595004i	$0.797151\pi$
$822$		0			0
$823$	1124.95			1.36689			0.683447	−	0.730001i	$-0.260480\pi$
$823$	1124.95			1.36689			0.683447	+	0.730001i	$0.260480\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	827.815	−	477.939i	1.00099	−	0.577920i	0.0924461	−	0.995718i	$-0.470531\pi$
$827$	827.815	−	477.939i	1.00099	−	0.577920i	0.908540	+	0.417798i	$0.137198\pi$
$828$		0			0
$829$	−259.409			−0.312918			−0.156459	−	0.987684i	$-0.550008\pi$
$829$	−259.409			−0.312918			−0.156459	+	0.987684i	$0.550008\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1214.85	−	701.395i	1.45841	−	0.842011i
$834$		0			0
$835$	−173.333	−	300.222i	−0.207585	−	0.359547i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1.59278i		−	0.00189843i	−1.00000		0.000949215i	$-0.999698\pi$
$839$		−	1.59278i		−	0.00189843i	1.00000		0.000949215i	$-0.000302145\pi$
$840$		0			0
$841$	−300.316	−	520.162i	−0.357094	−	0.618505i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−424.036	+	244.817i	−0.501817	+	0.289724i
$846$		0			0
$847$	−575.004			−0.678871
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1297.49i		−	1.52466i
$852$		0			0
$853$	−75.4108			−0.0884065			−0.0442033	−	0.999023i	$-0.514075\pi$
$853$	−75.4108			−0.0884065			−0.0442033	+	0.999023i	$0.514075\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	481.236i		−	0.561536i	−0.959776	−	0.280768i	$-0.909411\pi$
$857$		−	481.236i		−	0.561536i	0.959776	−	0.280768i	$-0.0905891\pi$
$858$		0			0
$859$	−1449.57			−1.68750			−0.843752	−	0.536733i	$-0.819658\pi$
$859$	−1449.57			−1.68750			−0.843752	+	0.536733i	$0.819658\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	635.739i		−	0.736662i	−0.929695	−	0.368331i	$-0.879929\pi$
$863$		−	635.739i		−	0.736662i	0.929695	−	0.368331i	$-0.120071\pi$
$864$		0			0
$865$	421.492	+	730.046i	0.487275	+	0.843984i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	107.775	−	62.2237i	0.124021	−	0.0716038i
$870$		0			0
$871$	−1231.21			−1.41356
$872$		0			0
$873$		0			0
$874$		0			0
$875$	968.023	−	558.888i	1.10631	−	0.638729i
$876$		0			0
$877$	7.64421	+	13.2402i	0.00871632	+	0.0150971i	0.870351	−	0.492432i	$-0.163892\pi$
$877$	7.64421	+	13.2402i	0.00871632	+	0.0150971i	−0.861634	+	0.507530i	$0.830559\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	104.595i		−	0.118724i	−0.998237	−	0.0593618i	$-0.981093\pi$
$881$		−	104.595i		−	0.118724i	0.998237	−	0.0593618i	$-0.0189066\pi$
$882$		0			0
$883$	−408.633	−	707.773i	−0.462778	−	0.801555i	0.536320	−	0.844014i	$-0.319814\pi$
$883$	−408.633	−	707.773i	−0.462778	−	0.801555i	−0.999098	+	0.0424598i	$0.986481\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1105.00i			1.24578i	0.782311	+	0.622888i	$0.214041\pi$
$887$			1105.00i			1.24578i	−0.782311	+	0.622888i	$0.785959\pi$
$888$		0			0
$889$	−640.932	+	1110.13i	−0.720959	+	1.24874i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	895.927	−	649.153i	1.00328	−	0.726935i
$894$		0			0
$895$	301.464	+	522.150i	0.336831	+	0.583408i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−555.630	+	320.793i	−0.618053	+	0.356833i
$900$		0			0
$901$	−162.253			−0.180081
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1327.58	−	766.478i	−1.46694	−	0.846937i
$906$		0			0
$907$	423.505			0.466929			0.233465	−	0.972365i	$-0.424994\pi$
$907$	423.505			0.466929			0.233465	+	0.972365i	$0.424994\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−950.485	+	548.763i	−1.04334	+	0.602374i	−0.920778	−	0.390086i	$-0.872445\pi$
$911$	−950.485	+	548.763i	−1.04334	+	0.602374i	−0.122564	+	0.992461i	$0.539112\pi$
$912$		0			0
$913$	356.444	−	617.378i	0.390409	−	0.676208i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1731.33	+	999.582i	1.88803	+	1.09006i
$918$		0			0
$919$	1001.55			1.08983			0.544914	−	0.838492i	$-0.316562\pi$
$919$	1001.55			1.08983			0.544914	+	0.838492i	$0.316562\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−114.025	−	65.8323i	−0.123537	−	0.0713242i
$924$		0			0
$925$	171.889	−	297.721i	0.185826	−	0.321861i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	672.987i		−	0.724421i	−0.932096	−	0.362210i	$-0.882022\pi$
$929$		−	672.987i		−	0.724421i	0.932096	−	0.362210i	$-0.117978\pi$
$930$		0			0
$931$	896.575	+	1237.41i	0.963023	+	1.32911i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	726.821	−	419.630i	0.777348	−	0.448802i
$936$		0			0
$937$	244.680	+	423.797i	0.261131	+	0.452292i	0.966543	−	0.256506i	$-0.0825713\pi$
$937$	244.680	+	423.797i	0.261131	+	0.452292i	−0.705412	+	0.708798i	$0.749238\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1127.40i		−	1.19809i	−0.800717	−	0.599043i	$-0.795548\pi$
$941$		−	1127.40i		−	1.19809i	0.800717	−	0.599043i	$-0.204452\pi$
$942$		0			0
$943$	−775.182	−	1342.65i	−0.822038	−	1.42381i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			916.672i			0.967974i	0.875075	+	0.483987i	$0.160812\pi$
$947$			916.672i			0.967974i	−0.875075	+	0.483987i	$0.839188\pi$
$948$		0			0
$949$	863.345	−	1495.36i	0.909742	−	1.57572i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1302.50	+	752.000i	1.36674	+	0.789087i	0.990510	−	0.137440i	$-0.0438875\pi$
$953$	1302.50	+	752.000i	1.36674	+	0.789087i	0.376228	+	0.926527i	$0.377221\pi$
$954$		0			0
$955$	−635.012	+	1099.87i	−0.664934	+	1.15170i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			2199.38i			2.29341i
$960$		0			0
$961$	−375.755	−	650.827i	−0.391004	−	0.677240i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	329.341	+	190.145i	0.341286	+	0.197042i
$966$		0			0
$967$	191.925	+	332.424i	0.198474	+	0.343768i	0.948034	−	0.318169i	$-0.103068\pi$
$967$	191.925	+	332.424i	0.198474	+	0.343768i	−0.749560	+	0.661937i	$0.769735\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	564.271	+	325.782i	0.581124	+	0.335512i	0.761580	−	0.648071i	$-0.224424\pi$
$971$	564.271	+	325.782i	0.581124	+	0.335512i	−0.180456	+	0.983583i	$0.557757\pi$
$972$		0			0
$973$	830.907	+	1439.17i	0.853964	+	1.47911i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	243.785	+	140.750i	0.249525	+	0.144063i	0.619547	−	0.784960i	$-0.287317\pi$
$977$	243.785	+	140.750i	0.249525	+	0.144063i	−0.370022	+	0.929023i	$0.620650\pi$
$978$		0			0
$979$	−251.223	+	435.130i	−0.256612	+	0.444464i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1626.89i		−	1.65503i	−0.561445	−	0.827514i	$-0.689754\pi$
$983$		−	1626.89i		−	1.65503i	0.561445	−	0.827514i	$-0.310246\pi$
$984$		0			0
$985$	20.9673	+	36.3164i	0.0212866	+	0.0368695i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1670.71i			1.68929i
$990$		0			0
$991$	616.114	−	1067.14i	0.621710	−	1.07683i	−0.367458	−	0.930040i	$-0.619772\pi$
$991$	616.114	−	1067.14i	0.621710	−	1.07683i	0.989167	−	0.146792i	$-0.0468949\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−599.016	−	345.842i	−0.602026	−	0.347580i
$996$		0			0
$997$	730.976	−	1266.09i	0.733175	−	1.26990i	−0.222344	−	0.974968i	$-0.571371\pi$
$997$	730.976	−	1266.09i	0.733175	−	1.26990i	0.955519	−	0.294928i	$-0.0952958\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.be.a.125.10		80
3.2	odd	2		684.3.be.a.581.26	yes	80
9.2	odd	6		2052.3.m.a.1493.31		80
9.7	even	3		684.3.m.a.353.28	✓	80
19.7	even	3		2052.3.m.a.881.10		80
57.26	odd	6		684.3.m.a.653.28	yes	80
171.7	even	3		684.3.be.a.425.26	yes	80
171.83	odd	6	inner	2052.3.be.a.197.10		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.28	✓	80	9.7	even	3
684.3.m.a.653.28	yes	80	57.26	odd	6
684.3.be.a.425.26	yes	80	171.7	even	3
684.3.be.a.581.26	yes	80	3.2	odd	2
2052.3.m.a.881.10		80	19.7	even	3
2052.3.m.a.1493.31		80	9.2	odd	6
2052.3.be.a.125.10		80	1.1	even	1	trivial
2052.3.be.a.197.10		80	171.83	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-4.96437 + 2.86618i) q^{5} +(5.68826 + 9.85235i) q^{7} +O(q^{10})\)	\(q+(-4.96437 + 2.86618i) q^{5} +(5.68826 + 9.85235i) q^{7} +(7.26930 - 4.19693i) q^{11} +15.9504 q^{13} +(-15.1054 - 8.72110i) q^{17} +(7.75053 - 17.3473i) q^{19} -29.6650i q^{23} +(3.92997 - 6.80691i) q^{25} +(-13.4267 - 7.75191i) q^{29} +(20.6912 - 35.8383i) q^{31} +(-56.4772 - 32.6071i) q^{35} +43.7381 q^{37} +(45.2606 - 26.1312i) q^{41} -56.3193 q^{43} +(50.4293 + 29.1153i) q^{47} +(-40.2126 + 69.6502i) q^{49} +(8.05606 - 4.65117i) q^{53} +(-24.0583 + 41.6702i) q^{55} +(-58.7003 + 33.8906i) q^{59} +(48.9248 - 84.7403i) q^{61} +(-79.1838 + 45.7168i) q^{65} -77.1899 q^{67} +(-7.14871 - 4.12731i) q^{71} +(54.1268 - 93.7503i) q^{73} +(82.6993 + 47.7465i) q^{77} +14.8260 q^{79} +(73.5512 - 42.4648i) q^{83} +99.9850 q^{85} +(-51.8391 + 29.9293i) q^{89} +(90.7301 + 157.149i) q^{91} +(11.2440 + 108.333i) q^{95} -42.9047 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Introduction

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