LMFDB - Embedded newform 1620.4.i.x.1081.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 $ x^12 + 306x^10 + 30777x^8 + 1381040x^6 + 28918584x^4 + 243888288*x^2 + 366645904
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.6
Root			$-7.39017i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.x.541.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{5} +(11.6578 + 20.1920i) q^{7} +O(q^{10})$	$q+(2.50000 - 4.33013i) q^{5} +(11.6578 + 20.1920i) q^{7} +(16.8745 + 29.2275i) q^{11} +(44.4469 - 76.9843i) q^{13} -108.661 q^{17} +21.4799 q^{19} +(42.9612 - 74.4110i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-9.48907 - 16.4355i) q^{29} +(83.1724 - 144.059i) q^{31} +116.578 q^{35} -167.664 q^{37} +(189.143 - 327.605i) q^{41} +(-190.222 - 329.474i) q^{43} +(123.414 + 213.760i) q^{47} +(-100.311 + 173.744i) q^{49} +284.344 q^{53} +168.745 q^{55} +(-363.699 + 629.945i) q^{59} +(-235.434 - 407.783i) q^{61} +(-222.235 - 384.922i) q^{65} +(375.978 - 651.213i) q^{67} +864.963 q^{71} -1003.53 q^{73} +(-393.440 + 681.459i) q^{77} +(419.251 + 726.163i) q^{79} +(238.057 + 412.327i) q^{83} +(-271.653 + 470.517i) q^{85} -554.055 q^{89} +2072.62 q^{91} +(53.6996 - 93.0105i) q^{95} +(523.994 + 907.584i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 30 q^{5} - 12 q^{7}+O(q^{10}) $ 12 * q + 30 * q^5 - 12 * q^7	$ 12 q + 30 q^{5} - 12 q^{7} + 84 q^{13} + 24 q^{17} - 228 q^{19} - 30 q^{23} - 150 q^{25} - 168 q^{29} + 324 q^{31} - 120 q^{35} - 984 q^{37} - 312 q^{41} + 156 q^{43} - 462 q^{47} + 588 q^{49} + 2028 q^{53} - 1008 q^{59} - 36 q^{61} - 420 q^{65} - 144 q^{67} + 2424 q^{71} - 1800 q^{73} - 672 q^{77} + 936 q^{79} - 288 q^{83} + 60 q^{85} - 240 q^{89} + 4572 q^{91} - 570 q^{95} + 1188 q^{97}+O(q^{100}) $ 12 * q + 30 * q^5 - 12 * q^7 + 84 * q^13 + 24 * q^17 - 228 * q^19 - 30 * q^23 - 150 * q^25 - 168 * q^29 + 324 * q^31 - 120 * q^35 - 984 * q^37 - 312 * q^41 + 156 * q^43 - 462 * q^47 + 588 * q^49 + 2028 * q^53 - 1008 * q^59 - 36 * q^61 - 420 * q^65 - 144 * q^67 + 2424 * q^71 - 1800 * q^73 - 672 * q^77 + 936 * q^79 - 288 * q^83 + 60 * q^85 - 240 * q^89 + 4572 * q^91 - 570 * q^95 + 1188 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	11.6578	+	20.1920i	0.629465	+	1.09026i	0.987659	+	0.156618i	$0.0500591\pi$
$7$	11.6578	+	20.1920i	0.629465	+	1.09026i	−0.358195	+	0.933647i	$0.616608\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	16.8745	+	29.2275i	0.462532	+	0.801128i	0.999086	−	0.0427371i	$-0.0136078\pi$
$11$	16.8745	+	29.2275i	0.462532	+	0.801128i	−0.536555	+	0.843866i	$0.680274\pi$
$12$		0			0
$13$	44.4469	−	76.9843i	0.948258	−	1.64243i	0.199166	−	0.979966i	$-0.436177\pi$
$13$	44.4469	−	76.9843i	0.948258	−	1.64243i	0.749092	−	0.662466i	$-0.230490\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−108.661			−1.55025			−0.775124	−	0.631809i	$-0.782313\pi$
$17$	−108.661			−1.55025			−0.775124	+	0.631809i	$0.782313\pi$
$18$		0			0
$19$	21.4799			0.259359			0.129679	−	0.991556i	$-0.458605\pi$
$19$	21.4799			0.259359			0.129679	+	0.991556i	$0.458605\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	42.9612	−	74.4110i	0.389480	−	0.674599i	−0.602900	−	0.797817i	$-0.705988\pi$
$23$	42.9612	−	74.4110i	0.389480	−	0.674599i	0.992380	+	0.123218i	$0.0393214\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.48907	−	16.4355i	−0.0607612	−	0.105242i	0.834045	−	0.551697i	$-0.186019\pi$
$29$	−9.48907	−	16.4355i	−0.0607612	−	0.105242i	−0.894806	+	0.446455i	$0.852686\pi$
$30$		0			0
$31$	83.1724	−	144.059i	0.481877	−	0.834636i	−0.517906	−	0.855437i	$-0.673288\pi$
$31$	83.1724	−	144.059i	0.481877	−	0.834636i	0.999784	+	0.0208014i	$0.00662176\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	116.578			0.563010
$36$		0			0
$37$	−167.664			−0.744966			−0.372483	−	0.928039i	$-0.621493\pi$
$37$	−167.664			−0.744966			−0.372483	+	0.928039i	$0.621493\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	189.143	−	327.605i	0.720466	−	1.24788i	−0.240347	−	0.970687i	$-0.577261\pi$
$41$	189.143	−	327.605i	0.720466	−	1.24788i	0.960813	−	0.277197i	$-0.0894055\pi$
$42$		0			0
$43$	−190.222	−	329.474i	−0.674618	−	1.16847i	−0.976580	−	0.215153i	$-0.930975\pi$
$43$	−190.222	−	329.474i	−0.674618	−	1.16847i	0.301962	−	0.953320i	$-0.402358\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	123.414	+	213.760i	0.383018	+	0.663406i	0.991492	−	0.130167i	$-0.0415515\pi$
$47$	123.414	+	213.760i	0.383018	+	0.663406i	−0.608474	+	0.793574i	$0.708218\pi$
$48$		0			0
$49$	−100.311	+	173.744i	−0.292452	+	0.506541i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	284.344			0.736937			0.368469	−	0.929640i	$-0.379882\pi$
$53$	284.344			0.736937			0.368469	+	0.929640i	$0.379882\pi$
$54$		0			0
$55$	168.745			0.413701
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−363.699	+	629.945i	−0.802534	+	1.39003i	0.115409	+	0.993318i	$0.463182\pi$
$59$	−363.699	+	629.945i	−0.802534	+	1.39003i	−0.917943	+	0.396712i	$0.870151\pi$
$60$		0			0
$61$	−235.434	−	407.783i	−0.494168	−	0.855923i	0.505810	−	0.862645i	$-0.331194\pi$
$61$	−235.434	−	407.783i	−0.494168	−	0.855923i	−0.999977	+	0.00672169i	$0.997860\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−222.235	−	384.922i	−0.424074	−	0.734518i
$66$		0			0
$67$	375.978	−	651.213i	0.685567	−	1.18744i	−0.287691	−	0.957723i	$-0.592888\pi$
$67$	375.978	−	651.213i	0.685567	−	1.18744i	0.973258	−	0.229714i	$-0.0737791\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	864.963			1.44581			0.722903	−	0.690949i	$-0.242807\pi$
$71$	864.963			1.44581			0.722903	+	0.690949i	$0.242807\pi$
$72$		0			0
$73$	−1003.53			−1.60896			−0.804479	−	0.593982i	$-0.797555\pi$
$73$	−1003.53			−1.60896			−0.804479	+	0.593982i	$0.797555\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−393.440	+	681.459i	−0.582295	+	1.00856i
$78$		0			0
$79$	419.251	+	726.163i	0.597081	+	1.03417i	0.993250	+	0.115996i	$0.0370061\pi$
$79$	419.251	+	726.163i	0.597081	+	1.03417i	−0.396169	+	0.918178i	$0.629661\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	238.057	+	412.327i	0.314821	+	0.545287i	0.979399	−	0.201932i	$-0.0647221\pi$
$83$	238.057	+	412.327i	0.314821	+	0.545287i	−0.664578	+	0.747219i	$0.731389\pi$
$84$		0			0
$85$	−271.653	+	470.517i	−0.346646	+	0.600408i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−554.055			−0.659884			−0.329942	−	0.944001i	$-0.607029\pi$
$89$	−554.055			−0.659884			−0.329942	+	0.944001i	$0.607029\pi$
$90$		0			0
$91$	2072.62			2.38758
$92$		0			0
$93$		0			0
$94$		0			0
$95$	53.6996	−	93.0105i	0.0579944	−	0.100449i
$96$		0			0
$97$	523.994	+	907.584i	0.548490	+	0.950012i	0.998378	+	0.0569275i	$0.0181304\pi$
$97$	523.994	+	907.584i	0.548490	+	0.950012i	−0.449888	+	0.893085i	$0.648536\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−209.199	−	362.343i	−0.206100	−	0.356975i	0.744383	−	0.667753i	$-0.232744\pi$
$101$	−209.199	−	362.343i	−0.206100	−	0.356975i	−0.950483	+	0.310778i	$0.899410\pi$
$102$		0			0
$103$	640.394	−	1109.19i	0.612620	−	1.06109i	−0.378177	−	0.925733i	$-0.623449\pi$
$103$	640.394	−	1109.19i	0.612620	−	1.06109i	0.990797	−	0.135356i	$-0.0432178\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−351.116			−0.317231			−0.158615	−	0.987340i	$-0.550703\pi$
$107$	−351.116			−0.317231			−0.158615	+	0.987340i	$0.550703\pi$
$108$		0			0
$109$	1762.43			1.54872			0.774360	−	0.632746i	$-0.218072\pi$
$109$	1762.43			1.54872			0.774360	+	0.632746i	$0.218072\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−500.931	+	867.638i	−0.417023	+	0.722306i	−0.995639	−	0.0932949i	$-0.970260\pi$
$113$	−500.931	+	867.638i	−0.417023	+	0.722306i	0.578615	+	0.815601i	$0.303593\pi$
$114$		0			0
$115$	−214.806	−	372.055i	−0.174181	−	0.301690i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1266.76	−	2194.09i	−0.975826	−	1.69018i
$120$		0			0
$121$	96.0034	−	166.283i	0.0721288	−	0.124931i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	852.753			0.595824			0.297912	−	0.954593i	$-0.403710\pi$
$127$	852.753			0.595824			0.297912	+	0.954593i	$0.403710\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1339.12	−	2319.42i	0.893123	−	1.54693i	0.0570131	−	0.998373i	$-0.481842\pi$
$131$	1339.12	−	2319.42i	0.893123	−	1.54693i	0.836110	−	0.548562i	$-0.184824\pi$
$132$		0			0
$133$	250.409	+	433.721i	0.163257	+	0.282770i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−211.997	−	367.189i	−0.132205	−	0.228986i	0.792321	−	0.610104i	$-0.208872\pi$
$137$	−211.997	−	367.189i	−0.132205	−	0.228986i	−0.924526	+	0.381118i	$0.875539\pi$
$138$		0			0
$139$	335.757	−	581.549i	0.204882	−	0.354866i	−0.745213	−	0.666826i	$-0.767652\pi$
$139$	335.757	−	581.549i	0.204882	−	0.354866i	0.950095	+	0.311960i	$0.100986\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3000.08			1.75440
$144$		0			0
$145$	−94.8907			−0.0543465
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1198.72	+	2076.24i	−0.659080	+	1.14156i	0.321774	+	0.946816i	$0.395721\pi$
$149$	−1198.72	+	2076.24i	−0.659080	+	1.14156i	−0.980854	+	0.194743i	$0.937613\pi$
$150$		0			0
$151$	151.005	+	261.547i	0.0813813	+	0.140957i	0.903843	−	0.427863i	$-0.140734\pi$
$151$	151.005	+	261.547i	0.0813813	+	0.140957i	−0.822462	+	0.568820i	$0.807400\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−415.862	−	720.294i	−0.215502	−	0.373261i
$156$		0			0
$157$	1117.50	−	1935.56i	0.568063	−	0.983914i	−0.428695	−	0.903449i	$-0.641026\pi$
$157$	1117.50	−	1935.56i	0.568063	−	0.983914i	0.996757	−	0.0804642i	$-0.0256403\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2003.34			0.980655
$162$		0			0
$163$	−45.7276			−0.0219734			−0.0109867	−	0.999940i	$-0.503497\pi$
$163$	−45.7276			−0.0219734			−0.0109867	+	0.999940i	$0.503497\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1238.07	+	2144.40i	−0.573681	+	0.993645i	0.422502	+	0.906362i	$0.361152\pi$
$167$	−1238.07	+	2144.40i	−0.573681	+	0.993645i	−0.996184	+	0.0872833i	$0.972181\pi$
$168$		0			0
$169$	−2852.56	−	4940.77i	−1.29839	−	2.24887i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1574.72	−	2727.49i	−0.692044	−	1.19866i	−0.971167	−	0.238400i	$-0.923377\pi$
$173$	−1574.72	−	2727.49i	−0.692044	−	1.19866i	0.279123	−	0.960255i	$-0.409956\pi$
$174$		0			0
$175$	291.446	−	504.800i	0.125893	−	0.218053i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4111.57			−1.71683			−0.858417	−	0.512953i	$-0.828552\pi$
$179$	−4111.57			−1.71683			−0.858417	+	0.512953i	$0.828552\pi$
$180$		0			0
$181$	−1344.52			−0.552138			−0.276069	−	0.961138i	$-0.589032\pi$
$181$	−1344.52			−0.552138			−0.276069	+	0.961138i	$0.589032\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−419.159	+	726.005i	−0.166579	+	0.288524i
$186$		0			0
$187$	−1833.60	−	3175.89i	−0.717039	−	1.24195i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1264.78	+	2190.67i	0.479144	+	0.829901i	0.999714	−	0.0239175i	$-0.00761390\pi$
$191$	1264.78	+	2190.67i	0.479144	+	0.829901i	−0.520570	+	0.853819i	$0.674281\pi$
$192$		0			0
$193$	2004.82	−	3472.45i	0.747722	−	1.29509i	−0.201191	−	0.979552i	$-0.564481\pi$
$193$	2004.82	−	3472.45i	0.747722	−	1.29509i	0.948912	−	0.315540i	$-0.102186\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2489.65			0.900408			0.450204	−	0.892926i	$-0.351351\pi$
$197$	2489.65			0.900408			0.450204	+	0.892926i	$0.351351\pi$
$198$		0			0
$199$	1358.91			0.484073			0.242037	−	0.970267i	$-0.422185\pi$
$199$	1358.91			0.484073			0.242037	+	0.970267i	$0.422185\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	221.244	−	383.206i	0.0764941	−	0.132492i
$204$		0			0
$205$	−945.713	−	1638.02i	−0.322202	−	0.558071i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	362.462	+	627.802i	0.119962	+	0.207780i
$210$		0			0
$211$	1897.31	−	3286.24i	0.619034	−	1.07220i	−0.370628	−	0.928781i	$-0.620858\pi$
$211$	1897.31	−	3286.24i	0.619034	−	1.07220i	0.989662	−	0.143417i	$-0.0458090\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1902.22			−0.603397
$216$		0			0
$217$	3878.44			1.21330
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4829.66	+	8365.21i	−1.47004	+	2.54618i
$222$		0			0
$223$	664.359	+	1150.70i	0.199501	+	0.345546i	0.948367	−	0.317176i	$-0.102734\pi$
$223$	664.359	+	1150.70i	0.199501	+	0.345546i	−0.748866	+	0.662722i	$0.769401\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−495.789	−	858.731i	−0.144963	−	0.251084i	0.784396	−	0.620260i	$-0.212973\pi$
$227$	−495.789	−	858.731i	−0.144963	−	0.251084i	−0.929359	+	0.369177i	$0.879640\pi$
$228$		0			0
$229$	1987.89	−	3443.13i	0.573640	−	0.993574i	−0.422547	−	0.906341i	$-0.638864\pi$
$229$	1987.89	−	3443.13i	0.573640	−	0.993574i	0.996188	−	0.0872336i	$-0.0278026\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4850.18			1.36372			0.681858	−	0.731484i	$-0.261172\pi$
$233$	4850.18			1.36372			0.681858	+	0.731484i	$0.261172\pi$
$234$		0			0
$235$	1234.14			0.342582
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1835.50	−	3179.18i	0.496773	−	0.860437i	−0.503220	−	0.864159i	$-0.667851\pi$
$239$	1835.50	−	3179.18i	0.496773	−	0.860437i	0.999993	+	0.00372178i	$0.00118468\pi$
$240$		0			0
$241$	1137.89	+	1970.88i	0.304140	+	0.526786i	0.977069	−	0.212921i	$-0.0682976\pi$
$241$	1137.89	+	1970.88i	0.304140	+	0.526786i	−0.672929	+	0.739707i	$0.734964\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	501.555	+	868.718i	0.130788	+	0.226532i
$246$		0			0
$247$	954.713	−	1653.61i	0.245939	−	0.425979i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−816.764			−0.205393			−0.102697	−	0.994713i	$-0.532747\pi$
$251$	−816.764			−0.205393			−0.102697	+	0.994713i	$0.532747\pi$
$252$		0			0
$253$	2899.80			0.720587
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3147.07	−	5450.88i	0.763847	−	1.32302i	−0.177008	−	0.984209i	$-0.556642\pi$
$257$	3147.07	−	5450.88i	0.763847	−	1.32302i	0.940854	−	0.338812i	$-0.110025\pi$
$258$		0			0
$259$	−1954.60	−	3385.46i	−0.468930	−	0.812210i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	927.493	+	1606.47i	0.217459	+	0.376650i	0.954030	−	0.299710i	$-0.0968899\pi$
$263$	927.493	+	1606.47i	0.217459	+	0.376650i	−0.736572	+	0.676360i	$0.763557\pi$
$264$		0			0
$265$	710.861	−	1231.25i	0.164784	−	0.285415i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3602.16			0.816460			0.408230	−	0.912879i	$-0.366146\pi$
$269$	3602.16			0.816460			0.408230	+	0.912879i	$0.366146\pi$
$270$		0			0
$271$	2920.88			0.654727			0.327363	−	0.944898i	$-0.393840\pi$
$271$	2920.88			0.654727			0.327363	+	0.944898i	$0.393840\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	421.862	−	730.687i	0.0925063	−	0.160226i
$276$		0			0
$277$	268.940	+	465.817i	0.0583358	+	0.101041i	0.893718	−	0.448628i	$-0.148087\pi$
$277$	268.940	+	465.817i	0.0583358	+	0.101041i	−0.835383	+	0.549669i	$0.814754\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2035.26	−	3525.18i	−0.432077	−	0.748379i	0.564975	−	0.825108i	$-0.308886\pi$
$281$	−2035.26	−	3525.18i	−0.432077	−	0.748379i	−0.997052	+	0.0767289i	$0.975552\pi$
$282$		0			0
$283$	805.835	−	1395.75i	0.169265	−	0.293175i	−0.768897	−	0.639373i	$-0.779194\pi$
$283$	805.835	−	1395.75i	0.169265	−	0.293175i	0.938162	+	0.346198i	$0.112527\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8819.98			1.81403
$288$		0			0
$289$	6894.26			1.40327
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2228.29	−	3859.51i	0.444293	−	0.769539i	−0.553709	−	0.832710i	$-0.686788\pi$
$293$	2228.29	−	3859.51i	0.444293	−	0.769539i	0.998003	+	0.0631712i	$0.0201214\pi$
$294$		0			0
$295$	1818.49	+	3149.72i	0.358904	+	0.621640i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3818.99	−	6614.68i	−0.738655	−	1.27939i
$300$		0			0
$301$	4435.16	−	7681.92i	0.849297	−	1.47102i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2354.34			−0.441997
$306$		0			0
$307$	−11.9533			−0.00222219			−0.00111109	−	0.999999i	$-0.500354\pi$
$307$	−11.9533			−0.00222219			−0.00111109	+	0.999999i	$0.500354\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4672.56	+	8093.12i	−0.851951	+	1.47562i	0.0274941	+	0.999622i	$0.491247\pi$
$311$	−4672.56	+	8093.12i	−0.851951	+	1.47562i	−0.879445	+	0.476000i	$0.842086\pi$
$312$		0			0
$313$	4384.12	+	7593.51i	0.791709	+	1.37128i	0.924908	+	0.380191i	$0.124142\pi$
$313$	4384.12	+	7593.51i	0.791709	+	1.37128i	−0.133199	+	0.991089i	$0.542525\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−988.087	−	1711.42i	−0.175068	−	0.303226i	0.765117	−	0.643891i	$-0.222681\pi$
$317$	−988.087	−	1711.42i	−0.175068	−	0.303226i	−0.940185	+	0.340665i	$0.889348\pi$
$318$		0			0
$319$	320.246	−	554.683i	0.0562080	−	0.0973551i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2334.03			−0.402070
$324$		0			0
$325$	−2222.35			−0.379303
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2877.49	+	4983.96i	−0.482192	+	0.835182i
$330$		0			0
$331$	−3644.28	−	6312.08i	−0.605159	−	1.04817i	−0.992026	−	0.126031i	$-0.959776\pi$
$331$	−3644.28	−	6312.08i	−0.605159	−	1.04817i	0.386867	−	0.922135i	$-0.373557\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1879.89	−	3256.06i	−0.306595	−	0.531038i
$336$		0			0
$337$	−3649.89	+	6321.80i	−0.589977	+	1.02187i	0.404258	+	0.914645i	$0.367530\pi$
$337$	−3649.89	+	6321.80i	−0.589977	+	1.02187i	−0.994235	+	0.107225i	$0.965803\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5613.96			0.891534
$342$		0			0
$343$	3319.65			0.522577
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3381.18	+	5856.37i	−0.523087	+	0.906013i	0.476552	+	0.879146i	$0.341886\pi$
$347$	−3381.18	+	5856.37i	−0.523087	+	0.906013i	−0.999639	+	0.0268668i	$0.991447\pi$
$348$		0			0
$349$	3436.48	+	5952.16i	0.527079	+	0.912928i	0.999502	+	0.0315556i	$0.0100461\pi$
$349$	3436.48	+	5952.16i	0.527079	+	0.912928i	−0.472423	+	0.881372i	$0.656621\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4805.59	−	8323.53i	−0.724578	−	1.25501i	−0.959148	−	0.282906i	$-0.908702\pi$
$353$	−4805.59	−	8323.53i	−0.724578	−	1.25501i	0.234570	−	0.972099i	$-0.424632\pi$
$354$		0			0
$355$	2162.41	−	3745.40i	0.323292	−	0.559958i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7951.69			1.16901			0.584504	−	0.811391i	$-0.301289\pi$
$359$	7951.69			1.16901			0.584504	+	0.811391i	$0.301289\pi$
$360$		0			0
$361$	−6397.62			−0.932733
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2508.82	+	4345.40i	−0.359774	+	0.623146i
$366$		0			0
$367$	−3194.49	−	5533.02i	−0.454362	−	0.786978i	0.544289	−	0.838898i	$-0.316799\pi$
$367$	−3194.49	−	5533.02i	−0.454362	−	0.786978i	−0.998651	+	0.0519193i	$0.983466\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3314.84	+	5741.48i	0.463876	+	0.803457i
$372$		0			0
$373$	4581.58	−	7935.52i	0.635992	−	1.10157i	−0.350312	−	0.936633i	$-0.613924\pi$
$373$	4581.58	−	7935.52i	0.635992	−	1.10157i	0.986304	−	0.164938i	$-0.0527423\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1687.04			−0.230469
$378$		0			0
$379$	−10303.5			−1.39646			−0.698228	−	0.715875i	$-0.746028\pi$
$379$	−10303.5			−1.39646			−0.698228	+	0.715875i	$0.746028\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1859.95	+	3221.53i	−0.248144	+	0.429798i	−0.963011	−	0.269463i	$-0.913154\pi$
$383$	−1859.95	+	3221.53i	−0.248144	+	0.429798i	0.714867	+	0.699260i	$0.246487\pi$
$384$		0			0
$385$	1967.20	+	3407.29i	0.260410	+	0.451044i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2235.23	+	3871.54i	0.291339	+	0.504614i	0.974127	−	0.226003i	$-0.0725659\pi$
$389$	2235.23	+	3871.54i	0.291339	+	0.504614i	−0.682788	+	0.730617i	$0.739233\pi$
$390$		0			0
$391$	−4668.22	+	8085.59i	−0.603790	+	1.04580i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4192.51			0.534045
$396$		0			0
$397$	4848.35			0.612927			0.306463	−	0.951882i	$-0.400854\pi$
$397$	4848.35			0.612927			0.306463	+	0.951882i	$0.400854\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	74.7667	−	129.500i	0.00931090	−	0.0161269i	−0.861332	−	0.508042i	$-0.830370\pi$
$401$	74.7667	−	129.500i	0.00931090	−	0.0161269i	0.870643	+	0.491915i	$0.163703\pi$
$402$		0			0
$403$	−7393.51	−	12805.9i	−0.913888	−	1.58290i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2829.24	−	4900.38i	−0.344570	−	0.596813i
$408$		0			0
$409$	−5324.84	+	9222.89i	−0.643756	+	1.11502i	0.340831	+	0.940125i	$0.389292\pi$
$409$	−5324.84	+	9222.89i	−0.643756	+	1.11502i	−0.984587	+	0.174894i	$0.944042\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16959.8			−2.02067
$414$		0			0
$415$	2380.57			0.281585
$416$		0			0
$417$		0			0
$418$		0			0
$419$	169.897	−	294.269i	0.0198090	−	0.0343103i	−0.855951	−	0.517057i	$-0.827028\pi$
$419$	169.897	−	294.269i	0.0198090	−	0.0343103i	0.875760	+	0.482747i	$0.160361\pi$
$420$		0			0
$421$	−1066.02	−	1846.40i	−0.123408	−	0.213748i	0.797702	−	0.603052i	$-0.206049\pi$
$421$	−1066.02	−	1846.40i	−0.123408	−	0.213748i	−0.921109	+	0.389304i	$0.872716\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1358.27	+	2352.58i	0.155025	+	0.268511i
$426$		0			0
$427$	5489.31	−	9507.76i	0.622122	−	1.07755i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2509.67			0.280480			0.140240	−	0.990118i	$-0.455213\pi$
$431$	2509.67			0.280480			0.140240	+	0.990118i	$0.455213\pi$
$432$		0			0
$433$	2792.17			0.309891			0.154946	−	0.987923i	$-0.450480\pi$
$433$	2792.17			0.309891			0.154946	+	0.987923i	$0.450480\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	922.801	−	1598.34i	0.101015	−	0.174963i
$438$		0			0
$439$	3278.50	+	5678.53i	0.356433	+	0.617361i	0.987362	−	0.158480i	$-0.0506594\pi$
$439$	3278.50	+	5678.53i	0.356433	+	0.617361i	−0.630929	+	0.775841i	$0.717326\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	949.888	+	1645.25i	0.101875	+	0.176452i	0.912457	−	0.409172i	$-0.134183\pi$
$443$	949.888	+	1645.25i	0.101875	+	0.176452i	−0.810582	+	0.585625i	$0.800849\pi$
$444$		0			0
$445$	−1385.14	+	2399.13i	−0.147555	+	0.255572i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9468.83			0.995238			0.497619	−	0.867396i	$-0.334208\pi$
$449$	9468.83			0.995238			0.497619	+	0.867396i	$0.334208\pi$
$450$		0			0
$451$	12766.7			1.33295
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5181.55	−	8974.72i	0.533879	−	0.924706i
$456$		0			0
$457$	−1449.85	−	2511.21i	−0.148405	−	0.257045i	0.782233	−	0.622986i	$-0.214081\pi$
$457$	−1449.85	−	2511.21i	−0.148405	−	0.257045i	−0.930638	+	0.365941i	$0.880747\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2758.81	+	4778.39i	0.278721	+	0.482759i	0.971067	−	0.238806i	$-0.0767562\pi$
$461$	2758.81	+	4778.39i	0.278721	+	0.482759i	−0.692346	+	0.721566i	$0.743423\pi$
$462$		0			0
$463$	4485.15	−	7768.50i	0.450200	−	0.779769i	−0.548198	−	0.836348i	$-0.684686\pi$
$463$	4485.15	−	7768.50i	0.450200	−	0.779769i	0.998398	+	0.0565795i	$0.0180194\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12314.8			−1.22026			−0.610130	−	0.792302i	$-0.708883\pi$
$467$	−12314.8			−1.22026			−0.610130	+	0.792302i	$0.708883\pi$
$468$		0			0
$469$	17532.4			1.72616
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6419.80	−	11119.4i	0.624065	−	1.08091i
$474$		0			0
$475$	−268.498	−	465.053i	−0.0259359	−	0.0449223i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6364.47	+	11023.6i	0.607098	+	1.05153i	0.991716	+	0.128449i	$0.0409999\pi$
$479$	6364.47	+	11023.6i	0.607098	+	1.05153i	−0.384618	+	0.923076i	$0.625667\pi$
$480$		0			0
$481$	−7452.13	+	12907.5i	−0.706420	+	1.22356i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5239.94			0.490584
$486$		0			0
$487$	−7230.67			−0.672798			−0.336399	−	0.941719i	$-0.609209\pi$
$487$	−7230.67			−0.672798			−0.336399	+	0.941719i	$0.609209\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8803.45	−	15248.0i	0.809153	−	1.40149i	−0.104298	−	0.994546i	$-0.533259\pi$
$491$	8803.45	−	15248.0i	0.809153	−	1.40149i	0.913451	−	0.406949i	$-0.133407\pi$
$492$		0			0
$493$	1031.09	+	1785.91i	0.0941950	+	0.163150i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10083.6	+	17465.3i	0.910084	+	1.57631i
$498$		0			0
$499$	10175.8	−	17625.0i	0.912886	−	1.58116i	0.102918	−	0.994690i	$-0.467182\pi$
$499$	10175.8	−	17625.0i	0.912886	−	1.58116i	0.809967	−	0.586475i	$-0.199485\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14477.1			−1.28331			−0.641654	−	0.766995i	$-0.721751\pi$
$503$	−14477.1			−1.28331			−0.641654	+	0.766995i	$0.721751\pi$
$504$		0			0
$505$	−2091.99			−0.184341
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1841.81	+	3190.11i	−0.160387	+	0.277798i	−0.935007	−	0.354628i	$-0.884607\pi$
$509$	−1841.81	+	3190.11i	−0.160387	+	0.277798i	0.774621	+	0.632426i	$0.217941\pi$
$510$		0			0
$511$	−11699.0	−	20263.2i	−1.01278	−	1.75419i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3201.97	−	5545.97i	−0.273972	−	0.474534i
$516$		0			0
$517$	−4165.11	+	7214.18i	−0.354316	+	0.613693i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20684.0			−1.73931			−0.869655	−	0.493660i	$-0.835659\pi$
$521$	−20684.0			−1.73931			−0.869655	+	0.493660i	$0.835659\pi$
$522$		0			0
$523$	−18781.8			−1.57031			−0.785153	−	0.619302i	$-0.787416\pi$
$523$	−18781.8			−1.57031			−0.785153	+	0.619302i	$0.787416\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9037.61	+	15653.6i	−0.747029	+	1.29389i
$528$		0			0
$529$	2392.16	+	4143.35i	0.196611	+	0.340540i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−16813.6	−	29122.0i	−1.36638	−	2.36663i
$534$		0			0
$535$	−877.791	+	1520.38i	−0.0709349	+	0.122863i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6770.78			−0.541073
$540$		0			0
$541$	9322.54			0.740864			0.370432	−	0.928860i	$-0.379210\pi$
$541$	9322.54			0.740864			0.370432	+	0.928860i	$0.379210\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4406.08	−	7631.56i	0.346304	−	0.599816i
$546$		0			0
$547$	4072.48	+	7053.74i	0.318330	+	0.551364i	0.980140	−	0.198308i	$-0.0635446\pi$
$547$	4072.48	+	7053.74i	0.318330	+	0.551364i	−0.661809	+	0.749672i	$0.730211\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−203.824	−	353.033i	−0.0157590	−	0.0272953i
$552$		0			0
$553$	−9775.12	+	16931.0i	−0.751682	+	1.30195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1025.83			0.0780355			0.0390177	−	0.999239i	$-0.487577\pi$
$557$	1025.83			0.0780355			0.0390177	+	0.999239i	$0.487577\pi$
$558$		0			0
$559$	−33819.1			−2.55885
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4406.31	−	7631.95i	0.329847	−	0.571311i	−0.652634	−	0.757673i	$-0.726336\pi$
$563$	4406.31	−	7631.95i	0.329847	−	0.571311i	0.982481	+	0.186362i	$0.0596696\pi$
$564$		0			0
$565$	2504.66	+	4338.19i	0.186499	+	0.323025i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3186.43	+	5519.07i	0.234767	+	0.406628i	0.959205	−	0.282712i	$-0.0912340\pi$
$569$	3186.43	+	5519.07i	0.234767	+	0.406628i	−0.724438	+	0.689340i	$0.757901\pi$
$570$		0			0
$571$	−5420.99	+	9389.44i	−0.397306	+	0.688154i	−0.993393	−	0.114766i	$-0.963388\pi$
$571$	−5420.99	+	9389.44i	−0.397306	+	0.688154i	0.596087	+	0.802920i	$0.296721\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2148.06			−0.155792
$576$		0			0
$577$	−1494.00			−0.107792			−0.0538959	−	0.998547i	$-0.517164\pi$
$577$	−1494.00			−0.107792			−0.0538959	+	0.998547i	$0.517164\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5550.47	+	9613.70i	−0.396338	+	0.686477i
$582$		0			0
$583$	4798.16	+	8310.66i	0.340857	+	0.590382i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7829.53	+	13561.1i	0.550527	+	0.953541i	0.998237	+	0.0593617i	$0.0189065\pi$
$587$	7829.53	+	13561.1i	0.550527	+	0.953541i	−0.447710	+	0.894179i	$0.647760\pi$
$588$		0			0
$589$	1786.53	−	3094.36i	0.124979	−	0.216470i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1722.59			0.119289			0.0596444	−	0.998220i	$-0.481003\pi$
$593$	1722.59			0.119289			0.0596444	+	0.998220i	$0.481003\pi$
$594$		0			0
$595$	−12667.6			−0.872806
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3301.86	−	5718.98i	0.225226	−	0.390102i	−0.731161	−	0.682205i	$-0.761021\pi$
$599$	3301.86	−	5718.98i	0.225226	−	0.390102i	0.956387	+	0.292102i	$0.0943547\pi$
$600$		0			0
$601$	3237.98	+	5608.35i	0.219767	+	0.380648i	0.954737	−	0.297452i	$-0.0961369\pi$
$601$	3237.98	+	5608.35i	0.219767	+	0.380648i	−0.734970	+	0.678100i	$0.762804\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−480.017	−	831.414i	−0.0322570	−	0.0558707i
$606$		0			0
$607$	3965.93	−	6869.20i	0.265193	−	0.459328i	−0.702421	−	0.711762i	$-0.747898\pi$
$607$	3965.93	−	6869.20i	0.265193	−	0.459328i	0.967614	+	0.252434i	$0.0812309\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	21941.6			1.45280
$612$		0			0
$613$	−16332.6			−1.07613			−0.538065	−	0.842903i	$-0.680845\pi$
$613$	−16332.6			−1.07613			−0.538065	+	0.842903i	$0.680845\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5835.76	+	10107.8i	−0.380776	+	0.659524i	−0.991173	−	0.132572i	$-0.957676\pi$
$617$	−5835.76	+	10107.8i	−0.380776	+	0.659524i	0.610397	+	0.792095i	$0.291010\pi$
$618$		0			0
$619$	5361.24	+	9285.93i	0.348120	+	0.602961i	0.985915	−	0.167244i	$-0.0534869\pi$
$619$	5361.24	+	9285.93i	0.348120	+	0.602961i	−0.637796	+	0.770206i	$0.720154\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6459.09	−	11187.5i	−0.415374	−	0.719448i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18218.5			1.15488
$630$		0			0
$631$	9754.15			0.615383			0.307691	−	0.951486i	$-0.400444\pi$
$631$	9754.15			0.615383			0.307691	+	0.951486i	$0.400444\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2131.88	−	3692.53i	0.133230	−	0.230761i
$636$		0			0
$637$	8917.02	+	15444.7i	0.554639	+	0.960664i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9682.73	+	16771.0i	0.596638	+	1.03341i	0.993313	+	0.115448i	$0.0368305\pi$
$641$	9682.73	+	16771.0i	0.596638	+	1.03341i	−0.396676	+	0.917959i	$0.629836\pi$
$642$		0			0
$643$	−14537.7	+	25180.0i	−0.891616	+	1.54432i	−0.0536786	+	0.998558i	$0.517095\pi$
$643$	−14537.7	+	25180.0i	−0.891616	+	1.54432i	−0.837938	+	0.545766i	$0.816239\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17400.3			−1.05730			−0.528651	−	0.848839i	$-0.677302\pi$
$647$	−17400.3			−1.05730			−0.528651	+	0.848839i	$0.677302\pi$
$648$		0			0
$649$	−24548.9			−1.48479
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4233.84	+	7333.23i	−0.253726	+	0.439466i	−0.964549	−	0.263905i	$-0.914989\pi$
$653$	−4233.84	+	7333.23i	−0.253726	+	0.439466i	0.710823	+	0.703371i	$0.248323\pi$
$654$		0			0
$655$	−6695.58	−	11597.1i	−0.399417	−	0.691810i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11991.3	+	20769.5i	0.708822	+	1.22772i	0.965294	+	0.261164i	$0.0841064\pi$
$659$	11991.3	+	20769.5i	0.708822	+	1.22772i	−0.256472	+	0.966552i	$0.582560\pi$
$660$		0			0
$661$	−13361.5	+	23142.9i	−0.786238	+	1.36180i	0.142018	+	0.989864i	$0.454641\pi$
$661$	−13361.5	+	23142.9i	−0.786238	+	1.36180i	−0.928256	+	0.371941i	$0.878692\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2504.09			0.146022
$666$		0			0
$667$	−1630.65			−0.0946611
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7945.65	−	13762.3i	0.457136	−	0.791783i
$672$		0			0
$673$	7560.71	+	13095.5i	0.433052	+	0.750068i	0.997134	−	0.0756503i	$-0.0241033\pi$
$673$	7560.71	+	13095.5i	0.433052	+	0.750068i	−0.564082	+	0.825719i	$0.690770\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7468.82	−	12936.4i	−0.424003	−	0.734395i	0.572324	−	0.820028i	$-0.306042\pi$
$677$	−7468.82	−	12936.4i	−0.424003	−	0.734395i	−0.996327	+	0.0856328i	$0.972709\pi$
$678$		0			0
$679$	−12217.3	+	21161.0i	−0.690510	+	1.19600i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11141.0			−0.624158			−0.312079	−	0.950056i	$-0.601025\pi$
$683$	−11141.0			−0.624158			−0.312079	+	0.950056i	$0.601025\pi$
$684$		0			0
$685$	−2119.97			−0.118248
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12638.2	−	21890.0i	0.698807	−	1.21037i
$690$		0			0
$691$	5981.42	+	10360.1i	0.329297	+	0.570359i	0.982373	−	0.186934i	$-0.0598550\pi$
$691$	5981.42	+	10360.1i	0.329297	+	0.570359i	−0.653076	+	0.757293i	$0.726522\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1678.79	−	2907.74i	−0.0916259	−	0.158701i
$696$		0			0
$697$	−20552.5	+	35597.9i	−1.11690	+	1.93453i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17436.0			0.939442			0.469721	−	0.882815i	$-0.344354\pi$
$701$	17436.0			0.939442			0.469721	+	0.882815i	$0.344354\pi$
$702$		0			0
$703$	−3601.39			−0.193213
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4877.62	−	8448.28i	0.259465	−	0.449406i
$708$		0			0
$709$	−5664.21	−	9810.70i	−0.300034	−	0.519673i	0.676110	−	0.736801i	$-0.263665\pi$
$709$	−5664.21	−	9810.70i	−0.300034	−	0.519673i	−0.976143	+	0.217128i	$0.930331\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7146.37	−	12377.9i	−0.375363	−	0.650148i
$714$		0			0
$715$	7500.19	−	12990.7i	0.392295	−	0.679475i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36889.9			−1.91344			−0.956718	−	0.291015i	$-0.906007\pi$
$719$	−36889.9			−1.91344			−0.956718	+	0.291015i	$0.906007\pi$
$720$		0			0
$721$	29862.5			1.54249
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−237.227	+	410.889i	−0.0121522	+	0.0210483i
$726$		0			0
$727$	−5477.89	−	9487.99i	−0.279455	−	0.484030i	0.691794	−	0.722095i	$-0.256820\pi$
$727$	−5477.89	−	9487.99i	−0.279455	−	0.484030i	−0.971249	+	0.238064i	$0.923487\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	20669.7	+	35801.1i	1.04583	+	1.81142i
$732$		0			0
$733$	11191.0	−	19383.4i	0.563915	−	0.976730i	−0.433234	−	0.901281i	$-0.642628\pi$
$733$	11191.0	−	19383.4i	0.563915	−	0.976730i	0.997150	−	0.0754488i	$-0.0240390\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25377.7			1.26839
$738$		0			0
$739$	27800.0			1.38382			0.691908	−	0.721985i	$-0.256770\pi$
$739$	27800.0			1.38382			0.691908	+	0.721985i	$0.256770\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8963.13	−	15524.6i	0.442564	−	0.766544i	−0.555315	−	0.831640i	$-0.687402\pi$
$743$	8963.13	−	15524.6i	0.442564	−	0.766544i	0.997879	+	0.0650965i	$0.0207355\pi$
$744$		0			0
$745$	5993.59	+	10381.2i	0.294749	+	0.510521i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4093.26	−	7089.74i	−0.199686	−	0.345866i
$750$		0			0
$751$	−12059.9	+	20888.4i	−0.585982	+	1.01495i	0.408770	+	0.912637i	$0.365958\pi$
$751$	−12059.9	+	20888.4i	−0.585982	+	1.01495i	−0.994752	+	0.102313i	$0.967376\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1510.05			0.0727896
$756$		0			0
$757$	10582.4			0.508088			0.254044	−	0.967193i	$-0.418239\pi$
$757$	10582.4			0.508088			0.254044	+	0.967193i	$0.418239\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18291.5	+	31681.9i	−0.871312	+	1.50916i	−0.0106710	+	0.999943i	$0.503397\pi$
$761$	−18291.5	+	31681.9i	−0.871312	+	1.50916i	−0.860641	+	0.509213i	$0.829937\pi$
$762$		0			0
$763$	20546.2	+	35587.0i	0.974864	+	1.68851i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32330.6	+	55998.2i	1.52202	+	2.63622i
$768$		0			0
$769$	−7388.24	+	12796.8i	−0.346459	+	0.600084i	−0.985618	−	0.168991i	$-0.945949\pi$
$769$	−7388.24	+	12796.8i	−0.346459	+	0.600084i	0.639159	+	0.769075i	$0.279283\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3195.12			−0.148668			−0.0743340	−	0.997233i	$-0.523683\pi$
$773$	−3195.12			−0.148668			−0.0743340	+	0.997233i	$0.523683\pi$
$774$		0			0
$775$	−4158.62			−0.192751
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4062.76	−	7036.90i	0.186859	−	0.323650i
$780$		0			0
$781$	14595.8	+	25280.7i	0.668731	+	1.15828i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5587.48	−	9677.80i	−0.254045	−	0.440020i
$786$		0			0
$787$	−15493.6	+	26835.6i	−0.701761	+	1.21549i	0.266087	+	0.963949i	$0.414269\pi$
$787$	−15493.6	+	26835.6i	−0.701761	+	1.21549i	−0.967848	+	0.251537i	$0.919064\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−23359.1			−1.05001
$792$		0			0
$793$	−41857.2			−1.87439
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18541.5	−	32114.9i	0.824059	−	1.42731i	−0.0785767	−	0.996908i	$-0.525038\pi$
$797$	18541.5	−	32114.9i	0.824059	−	1.42731i	0.902636	−	0.430405i	$-0.141629\pi$
$798$		0			0
$799$	−13410.4	−	23227.4i	−0.593772	−	1.02844i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16934.0	−	29330.5i	−0.744194	−	1.28898i
$804$		0			0
$805$	5008.36	−	8674.73i	0.219281	−	0.379806i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	648.034			0.0281627			0.0140814	−	0.999901i	$-0.495518\pi$
$809$	648.034			0.0281627			0.0140814	+	0.999901i	$0.495518\pi$
$810$		0			0
$811$	−36017.4			−1.55948			−0.779742	−	0.626101i	$-0.784650\pi$
$811$	−36017.4			−1.55948			−0.779742	+	0.626101i	$0.784650\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−114.319	+	198.006i	−0.00491340	+	0.00851025i
$816$		0			0
$817$	−4085.94	−	7077.06i	−0.174968	−	0.303054i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14404.5	+	24949.3i	0.612326	+	1.06058i	0.990847	+	0.134987i	$0.0430993\pi$
$821$	14404.5	+	24949.3i	0.612326	+	1.06058i	−0.378522	+	0.925592i	$0.623567\pi$
$822$		0			0
$823$	13777.3	−	23863.0i	0.583533	−	1.01071i	−0.411524	−	0.911399i	$-0.635003\pi$
$823$	13777.3	−	23863.0i	0.583533	−	1.01071i	0.995057	−	0.0993094i	$-0.0316634\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−34914.4			−1.46807			−0.734035	−	0.679111i	$-0.762365\pi$
$827$	−34914.4			−1.46807			−0.734035	+	0.679111i	$0.762365\pi$
$828$		0			0
$829$	749.124			0.0313850			0.0156925	−	0.999877i	$-0.495005\pi$
$829$	749.124			0.0313850			0.0156925	+	0.999877i	$0.495005\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10899.9	−	18879.2i	0.453373	−	0.785264i
$834$		0			0
$835$	6190.35	+	10722.0i	0.256558	+	0.444372i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18727.9	+	32437.6i	0.770629	+	1.33477i	0.937219	+	0.348742i	$0.113391\pi$
$839$	18727.9	+	32437.6i	0.770629	+	1.33477i	−0.166590	+	0.986026i	$0.553276\pi$
$840$		0			0
$841$	12014.4	−	20809.6i	0.492616	−	0.853236i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−28525.6			−1.16131
$846$		0			0
$847$	4476.77			0.181610
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7203.04	+	12476.0i	−0.290149	+	0.502553i
$852$		0			0
$853$	5903.02	+	10224.3i	0.236947	+	0.410404i	0.959837	−	0.280559i	$-0.0905199\pi$
$853$	5903.02	+	10224.3i	0.236947	+	0.410404i	−0.722890	+	0.690963i	$0.757187\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12034.6	−	20844.5i	−0.479688	−	0.830844i	0.520041	−	0.854141i	$-0.325917\pi$
$857$	−12034.6	−	20844.5i	−0.479688	−	0.830844i	−0.999729	+	0.0232978i	$0.992583\pi$
$858$		0			0
$859$	5683.98	−	9844.94i	0.225768	−	0.391042i	−0.730782	−	0.682611i	$-0.760844\pi$
$859$	5683.98	−	9844.94i	0.225768	−	0.391042i	0.956550	+	0.291570i	$0.0941775\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10733.7			−0.423382			−0.211691	−	0.977337i	$-0.567897\pi$
$863$	−10733.7			−0.423382			−0.211691	+	0.977337i	$0.567897\pi$
$864$		0			0
$865$	−15747.2			−0.618983
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14149.3	+	24507.3i	−0.552338	+	0.956677i
$870$		0			0
$871$	−33422.1	−	57888.8i	−1.30019	−	2.25199i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1457.23	−	2524.00i	−0.0563010	−	0.0975163i
$876$		0			0
$877$	−9775.48	+	16931.6i	−0.376391	+	0.651928i	−0.990534	−	0.137267i	$-0.956168\pi$
$877$	−9775.48	+	16931.6i	−0.376391	+	0.651928i	0.614143	+	0.789194i	$0.289502\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−35047.7			−1.34028			−0.670140	−	0.742235i	$-0.733766\pi$
$881$	−35047.7			−1.34028			−0.670140	+	0.742235i	$0.733766\pi$
$882$		0			0
$883$	−35769.3			−1.36323			−0.681615	−	0.731711i	$-0.738722\pi$
$883$	−35769.3			−1.36323			−0.681615	+	0.731711i	$0.738722\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9019.69	−	15622.6i	0.341434	−	0.591380i	−0.643266	−	0.765643i	$-0.722421\pi$
$887$	9019.69	−	15622.6i	0.341434	−	0.591380i	0.984699	+	0.174263i	$0.0557542\pi$
$888$		0			0
$889$	9941.27	+	17218.8i	0.375050	+	0.649606i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2650.92	+	4591.53i	0.0993390	+	0.172060i
$894$		0			0
$895$	−10278.9	+	17803.6i	−0.383896	+	0.664927i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3156.91			−0.117118
$900$		0			0
$901$	−30897.2			−1.14244
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3361.29	+	5821.92i	−0.123462	+	0.213842i
$906$		0			0
$907$	−14525.1	−	25158.2i	−0.531750	−	0.921019i	−0.999313	−	0.0370589i	$-0.988201\pi$
$907$	−14525.1	−	25158.2i	−0.531750	−	0.921019i	0.467563	−	0.883960i	$-0.345132\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5273.31	+	9133.64i	0.191781	+	0.332174i	0.945841	−	0.324632i	$-0.105240\pi$
$911$	5273.31	+	9133.64i	0.191781	+	0.332174i	−0.754060	+	0.656806i	$0.771907\pi$
$912$		0			0
$913$	−8034.19	+	13915.6i	−0.291230	+	0.504425i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	62444.9			2.24876
$918$		0			0
$919$	2975.12			0.106790			0.0533951	−	0.998573i	$-0.482996\pi$
$919$	2975.12			0.106790			0.0533951	+	0.998573i	$0.482996\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	38444.9	−	66588.6i	1.37100	−	2.37464i
$924$		0			0
$925$	2095.80	+	3630.02i	0.0744966	+	0.129032i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25152.3	−	43565.1i	−0.888290	−	1.53856i	−0.841896	−	0.539640i	$-0.818560\pi$
$929$	−25152.3	−	43565.1i	−0.888290	−	1.53856i	−0.0463940	−	0.998923i	$-0.514773\pi$
$930$		0			0
$931$	−2154.66	+	3731.99i	−0.0758499	+	0.131376i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−18336.0			−0.641339
$936$		0			0
$937$	−44253.7			−1.54291			−0.771454	−	0.636285i	$-0.780470\pi$
$937$	−44253.7			−1.54291			−0.771454	+	0.636285i	$0.780470\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23676.1	+	41008.1i	−0.820210	+	1.42064i	0.0853163	+	0.996354i	$0.472810\pi$
$941$	−23676.1	+	41008.1i	−0.820210	+	1.42064i	−0.905526	+	0.424291i	$0.860523\pi$
$942$		0			0
$943$	−16251.6	−	28148.6i	−0.561214	−	0.972051i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11982.1	−	20753.5i	−0.411156	−	0.712143i	0.583861	−	0.811854i	$-0.301541\pi$
$947$	−11982.1	−	20753.5i	−0.411156	−	0.712143i	−0.995016	+	0.0997113i	$0.968208\pi$
$948$		0			0
$949$	−44603.7	+	77255.8i	−1.52571	+	2.64260i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6697.41			0.227650			0.113825	−	0.993501i	$-0.463690\pi$
$953$	6697.41			0.227650			0.113825	+	0.993501i	$0.463690\pi$
$954$		0			0
$955$	12647.8			0.428559
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4942.85	−	8561.26i	0.166437	−	0.288277i
$960$		0			0
$961$	1060.22	+	1836.35i	0.0355885	+	0.0616411i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10024.1	−	17362.3i	−0.334391	−	0.579183i
$966$		0			0
$967$	18497.8	−	32039.2i	0.615150	−	1.06547i	−0.375208	−	0.926941i	$-0.622429\pi$
$967$	18497.8	−	32039.2i	0.615150	−	1.06547i	0.990358	−	0.138530i	$-0.0442379\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32987.1			1.09022			0.545111	−	0.838364i	$-0.316487\pi$
$971$	32987.1			1.09022			0.545111	+	0.838364i	$0.316487\pi$
$972$		0			0
$973$	15656.8			0.515864
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9924.79	+	17190.2i	−0.324997	+	0.562911i	−0.981512	−	0.191402i	$-0.938697\pi$
$977$	−9924.79	+	17190.2i	−0.324997	+	0.562911i	0.656515	+	0.754313i	$0.272030\pi$
$978$		0			0
$979$	−9349.39	−	16193.6i	−0.305217	−	0.528652i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6503.35	−	11264.1i	−0.211012	−	0.365483i	0.741020	−	0.671483i	$-0.234343\pi$
$983$	−6503.35	−	11264.1i	−0.211012	−	0.365483i	−0.952031	+	0.306000i	$0.901009\pi$
$984$		0			0
$985$	6224.13	−	10780.5i	0.201337	−	0.348727i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32688.7			−1.05100
$990$		0			0
$991$	14119.3			0.452588			0.226294	−	0.974059i	$-0.427339\pi$
$991$	14119.3			0.452588			0.226294	+	0.974059i	$0.427339\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3397.28	−	5884.25i	0.108242	−	0.187481i
$996$		0			0
$997$	−9958.43	−	17248.5i	−0.316336	−	0.547910i	0.663385	−	0.748278i	$-0.269119\pi$
$997$	−9958.43	−	17248.5i	−0.316336	−	0.547910i	−0.979721	+	0.200369i	$0.935786\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.x.1081.6		12
3.2	odd	2		1620.4.i.w.1081.6		12
9.2	odd	6		1620.4.i.w.541.6		12
9.4	even	3		1620.4.a.i.1.1	✓	6
9.5	odd	6		1620.4.a.j.1.1	yes	6
9.7	even	3	inner	1620.4.i.x.541.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.1	✓	6	9.4	even	3
1620.4.a.j.1.1	yes	6	9.5	odd	6
1620.4.i.w.541.6		12	9.2	odd	6
1620.4.i.w.1081.6		12	3.2	odd	2
1620.4.i.x.541.6		12	9.7	even	3	inner
1620.4.i.x.1081.6		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 - 4.33013i) q^{5} +(11.6578 + 20.1920i) q^{7} +O(q^{10})\)	\(q+(2.50000 - 4.33013i) q^{5} +(11.6578 + 20.1920i) q^{7} +(16.8745 + 29.2275i) q^{11} +(44.4469 - 76.9843i) q^{13} -108.661 q^{17} +21.4799 q^{19} +(42.9612 - 74.4110i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-9.48907 - 16.4355i) q^{29} +(83.1724 - 144.059i) q^{31} +116.578 q^{35} -167.664 q^{37} +(189.143 - 327.605i) q^{41} +(-190.222 - 329.474i) q^{43} +(123.414 + 213.760i) q^{47} +(-100.311 + 173.744i) q^{49} +284.344 q^{53} +168.745 q^{55} +(-363.699 + 629.945i) q^{59} +(-235.434 - 407.783i) q^{61} +(-222.235 - 384.922i) q^{65} +(375.978 - 651.213i) q^{67} +864.963 q^{71} -1003.53 q^{73} +(-393.440 + 681.459i) q^{77} +(419.251 + 726.163i) q^{79} +(238.057 + 412.327i) q^{83} +(-271.653 + 470.517i) q^{85} -554.055 q^{89} +2072.62 q^{91} +(53.6996 - 93.0105i) q^{95} +(523.994 + 907.584i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 30 q^{5} - 12 q^{7}+O(q^{10}) \) 12 * q + 30 * q^5 - 12 * q^7	\( 12 q + 30 q^{5} - 12 q^{7} + 84 q^{13} + 24 q^{17} - 228 q^{19} - 30 q^{23} - 150 q^{25} - 168 q^{29} + 324 q^{31} - 120 q^{35} - 984 q^{37} - 312 q^{41} + 156 q^{43} - 462 q^{47} + 588 q^{49} + 2028 q^{53} - 1008 q^{59} - 36 q^{61} - 420 q^{65} - 144 q^{67} + 2424 q^{71} - 1800 q^{73} - 672 q^{77} + 936 q^{79} - 288 q^{83} + 60 q^{85} - 240 q^{89} + 4572 q^{91} - 570 q^{95} + 1188 q^{97}+O(q^{100}) \) 12 * q + 30 * q^5 - 12 * q^7 + 84 * q^13 + 24 * q^17 - 228 * q^19 - 30 * q^23 - 150 * q^25 - 168 * q^29 + 324 * q^31 - 120 * q^35 - 984 * q^37 - 312 * q^41 + 156 * q^43 - 462 * q^47 + 588 * q^49 + 2028 * q^53 - 1008 * q^59 - 36 * q^61 - 420 * q^65 - 144 * q^67 + 2424 * q^71 - 1800 * q^73 - 672 * q^77 + 936 * q^79 - 288 * q^83 + 60 * q^85 - 240 * q^89 + 4572 * q^91 - 570 * q^95 + 1188 * q^97

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