LMFDB - Embedded newform 1620.3.t.d.1349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(269,1620)]

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, 1, 3]))

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

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

chi := DirichletCharacter("1620.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.t (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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.154550410641.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 221x^{4} - 60x^{2} + 16 $ x^8 - 15x^6 + 221x^4 - 60*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1349.2
Root			$-3.32360 + 1.91888i$ of defining polynomial
Character	$\chi$	$=$	1620.1349
Dual form			1620.3.t.d.269.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0689865 - 4.99952i) q^{5} +(-2.42096 + 1.39774i) q^{7} +O(q^{10})$	$q+(0.0689865 - 4.99952i) q^{5} +(-2.42096 + 1.39774i) q^{7} +(-15.7154 + 9.07327i) q^{11} +(-19.9416 - 11.5133i) q^{13} +5.72842 q^{17} +23.1852 q^{19} +(-0.135792 + 0.235199i) q^{23} +(-24.9905 - 0.689799i) q^{25} +(34.4674 - 19.8997i) q^{29} +(-23.6852 + 41.0241i) q^{31} +(6.82104 + 12.2001i) q^{35} +34.8712i q^{37} +(-11.4891 - 6.63325i) q^{41} +(40.4571 - 23.3579i) q^{43} +(20.4568 + 35.4323i) q^{47} +(-20.5926 + 35.6675i) q^{49} +91.3705 q^{53} +(44.2779 + 79.1953i) q^{55} +(68.2773 + 39.4199i) q^{59} +(-15.5926 - 27.0072i) q^{61} +(-58.9366 + 98.9042i) q^{65} +(-5.98970 - 3.45816i) q^{67} +81.5870i q^{71} -106.084i q^{73} +(25.3642 - 43.9321i) q^{77} +(-31.7779 - 55.0409i) q^{79} +(0.142081 + 0.246091i) q^{83} +(0.395183 - 28.6394i) q^{85} +28.5210i q^{89} +64.3705 q^{91} +(1.59947 - 115.915i) q^{95} +(80.4657 - 46.4569i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{5}+O(q^{10}) $ 8 * q + 3 * q^5	$ 8 q + 3 q^{5} - 12 q^{17} + 12 q^{19} - 30 q^{23} - 9 q^{25} - 16 q^{31} - 90 q^{35} + 48 q^{47} - 78 q^{49} + 384 q^{53} + 94 q^{55} - 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} - 288 q^{83} + 100 q^{85} + 168 q^{91} + 318 q^{95}+O(q^{100}) $ 8 * q + 3 * q^5 - 12 * q^17 + 12 * q^19 - 30 * q^23 - 9 * q^25 - 16 * q^31 - 90 * q^35 + 48 * q^47 - 78 * q^49 + 384 * q^53 + 94 * q^55 - 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 - 288 * q^83 + 100 * q^85 + 168 * q^91 + 318 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

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
$5$	0.0689865	−	4.99952i	0.0137973	−	0.999905i
$5$	0.0689865	−	4.99952i	0.0137973	−	0.999905i
$6$		0			0
$7$	−2.42096	+	1.39774i	−0.345852	+	0.199678i	−0.662857	−	0.748746i	$-0.730656\pi$
$7$	−2.42096	+	1.39774i	−0.345852	+	0.199678i	0.317005	+	0.948424i	$0.397323\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.7154	+	9.07327i	−1.42867	+	0.824843i	−0.997016	−	0.0771971i	$-0.975403\pi$
$11$	−15.7154	+	9.07327i	−1.42867	+	0.824843i	−0.431653	+	0.902040i	$0.642070\pi$
$12$		0			0
$13$	−19.9416	−	11.5133i	−1.53397	−	0.885637i	−0.999173	−	0.0406560i	$-0.987055\pi$
$13$	−19.9416	−	11.5133i	−1.53397	−	0.885637i	−0.534796	−	0.844981i	$-0.679611\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.72842			0.336966			0.168483	−	0.985705i	$-0.446113\pi$
$17$	5.72842			0.336966			0.168483	+	0.985705i	$0.446113\pi$
$18$		0			0
$19$	23.1852			1.22028			0.610138	−	0.792295i	$-0.291114\pi$
$19$	23.1852			1.22028			0.610138	+	0.792295i	$0.291114\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.135792	+	0.235199i	−0.00590400	+	0.0102260i	−0.868962	−	0.494878i	$-0.835213\pi$
$23$	−0.135792	+	0.235199i	−0.00590400	+	0.0102260i	0.863058	+	0.505104i	$0.168546\pi$
$24$		0			0
$25$	−24.9905	−	0.689799i	−0.999619	−	0.0275920i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	34.4674	−	19.8997i	1.18853	−	0.686198i	0.230558	−	0.973059i	$-0.425945\pi$
$29$	34.4674	−	19.8997i	1.18853	−	0.686198i	0.957972	+	0.286860i	$0.0926116\pi$
$30$		0			0
$31$	−23.6852	+	41.0241i	−0.764040	+	1.32336i	0.176712	+	0.984263i	$0.443454\pi$
$31$	−23.6852	+	41.0241i	−0.764040	+	1.32336i	−0.940752	+	0.339094i	$0.889880\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.82104	+	12.2001i	0.194887	+	0.348574i
$36$		0			0
$37$			34.8712i			0.942465i	0.882009	+	0.471232i	$0.156191\pi$
$37$			34.8712i			0.942465i	−0.882009	+	0.471232i	$0.843809\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.4891	−	6.63325i	−0.280223	−	0.161787i	0.353302	−	0.935509i	$-0.385059\pi$
$41$	−11.4891	−	6.63325i	−0.280223	−	0.161787i	−0.633524	+	0.773723i	$0.718392\pi$
$42$		0			0
$43$	40.4571	−	23.3579i	0.940862	−	0.543207i	0.0506318	−	0.998717i	$-0.483877\pi$
$43$	40.4571	−	23.3579i	0.940862	−	0.543207i	0.890231	+	0.455510i	$0.150543\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	20.4568	+	35.4323i	0.435252	+	0.753878i	0.997316	−	0.0732153i	$-0.0233260\pi$
$47$	20.4568	+	35.4323i	0.435252	+	0.753878i	−0.562064	+	0.827093i	$0.689993\pi$
$48$		0			0
$49$	−20.5926	+	35.6675i	−0.420258	+	0.727908i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	91.3705			1.72397			0.861986	−	0.506932i	$-0.169221\pi$
$53$	91.3705			1.72397			0.861986	+	0.506932i	$0.169221\pi$
$54$		0			0
$55$	44.2779	+	79.1953i	0.805052	+	1.43991i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	68.2773	+	39.4199i	1.15724	+	0.668134i	0.950641	−	0.310292i	$-0.100427\pi$
$59$	68.2773	+	39.4199i	1.15724	+	0.668134i	0.206600	+	0.978425i	$0.433760\pi$
$60$		0			0
$61$	−15.5926	−	27.0072i	−0.255617	−	0.442741i	0.709446	−	0.704760i	$-0.248945\pi$
$61$	−15.5926	−	27.0072i	−0.255617	−	0.442741i	−0.965063	+	0.262018i	$0.915612\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−58.9366	+	98.9042i	−0.906718	+	1.52160i
$66$		0			0
$67$	−5.98970	−	3.45816i	−0.0893986	−	0.0516143i	0.454634	−	0.890678i	$-0.349770\pi$
$67$	−5.98970	−	3.45816i	−0.0893986	−	0.0516143i	−0.544033	+	0.839064i	$0.683103\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			81.5870i			1.14911i	0.818465	+	0.574556i	$0.194825\pi$
$71$			81.5870i			1.14911i	−0.818465	+	0.574556i	$0.805175\pi$
$72$		0			0
$73$		−	106.084i		−	1.45320i	−0.687060	−	0.726601i	$-0.741099\pi$
$73$		−	106.084i		−	1.45320i	0.687060	−	0.726601i	$-0.258901\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	25.3642	−	43.9321i	0.329405	−	0.570547i
$78$		0			0
$79$	−31.7779	−	55.0409i	−0.402252	−	0.696720i	0.591746	−	0.806125i	$-0.298439\pi$
$79$	−31.7779	−	55.0409i	−0.402252	−	0.696720i	−0.993997	+	0.109405i	$0.965106\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.142081	+	0.246091i	0.00171182	+	0.00296495i	0.866880	−	0.498517i	$-0.166122\pi$
$83$	0.142081	+	0.246091i	0.00171182	+	0.00296495i	−0.865168	+	0.501482i	$0.832788\pi$
$84$		0			0
$85$	0.395183	−	28.6394i	0.00464921	−	0.336934i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			28.5210i			0.320461i	0.987080	+	0.160230i	$0.0512237\pi$
$89$			28.5210i			0.320461i	−0.987080	+	0.160230i	$0.948776\pi$
$90$		0			0
$91$	64.3705			0.707368
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.59947	−	115.915i	0.0168365	−	1.22016i
$96$		0			0
$97$	80.4657	−	46.4569i	0.829543	−	0.478937i	−0.0241531	−	0.999708i	$-0.507689\pi$
$97$	80.4657	−	46.4569i	0.829543	−	0.478937i	0.853696	+	0.520771i	$0.174356\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−40.4153	+	23.3338i	−0.400151	+	0.231027i	−0.686549	−	0.727083i	$-0.740875\pi$
$101$	−40.4153	+	23.3338i	−0.400151	+	0.231027i	0.286398	+	0.958111i	$0.407542\pi$
$102$		0			0
$103$	9.68385	+	5.59098i	0.0940180	+	0.0542813i	0.546272	−	0.837608i	$-0.316047\pi$
$103$	9.68385	+	5.59098i	0.0940180	+	0.0542813i	−0.452254	+	0.891889i	$0.649380\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	147.297			1.37661			0.688303	−	0.725424i	$-0.258356\pi$
$107$	147.297			1.37661			0.688303	+	0.725424i	$0.258356\pi$
$108$		0			0
$109$	−121.556			−1.11519			−0.557595	−	0.830113i	$-0.688276\pi$
$109$	−121.556			−1.11519			−0.557595	+	0.830113i	$0.688276\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−48.8148	+	84.5496i	−0.431989	+	0.748227i	−0.997045	−	0.0768260i	$-0.975521\pi$
$113$	−48.8148	+	84.5496i	−0.431989	+	0.748227i	0.565056	+	0.825053i	$0.308855\pi$
$114$		0			0
$115$	1.16651	+	0.695121i	0.0101436	+	0.00604453i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13.8683	+	8.00686i	−0.116540	+	0.0672845i
$120$		0			0
$121$	104.148	−	180.390i	0.860730	−	1.49083i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.17267	+	124.893i	−0.0413814	+	0.999143i
$126$		0			0
$127$		−	88.6481i		−	0.698017i	−0.937120	−	0.349008i	$-0.886518\pi$
$127$		−	88.6481i		−	0.698017i	0.937120	−	0.349008i	$-0.113482\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	77.9193	+	44.9867i	0.594804	+	0.343410i	0.766995	−	0.641653i	$-0.221751\pi$
$131$	77.9193	+	44.9867i	0.594804	+	0.343410i	−0.172191	+	0.985064i	$0.555085\pi$
$132$		0			0
$133$	−56.1306	+	32.4070i	−0.422035	+	0.243662i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	114.963	+	199.122i	0.839147	+	1.45345i	0.890609	+	0.454770i	$0.150279\pi$
$137$	114.963	+	199.122i	0.839147	+	1.45345i	−0.0514620	+	0.998675i	$0.516388\pi$
$138$		0			0
$139$	−28.8148	+	49.9086i	−0.207300	+	0.359055i	−0.950863	−	0.309611i	$-0.899801\pi$
$139$	−28.8148	+	49.9086i	−0.207300	+	0.359055i	0.743563	+	0.668666i	$0.233134\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	417.852			2.92205
$144$		0			0
$145$	−97.1115	−	173.693i	−0.669734	−	1.19788i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.64205	−	5.56684i	−0.0647117	−	0.0373613i	0.467295	−	0.884101i	$-0.345229\pi$
$149$	−9.64205	−	5.56684i	−0.0647117	−	0.0373613i	−0.532007	+	0.846740i	$0.678562\pi$
$150$		0			0
$151$	11.0926	+	19.2130i	0.0734611	+	0.127238i	0.900416	−	0.435030i	$-0.143262\pi$
$151$	11.0926	+	19.2130i	0.0734611	+	0.127238i	−0.826955	+	0.562268i	$0.809929\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	203.467	+	121.245i	1.31269	+	0.782226i
$156$		0			0
$157$	195.148	+	112.669i	1.24298	+	0.717635i	0.969700	−	0.244299i	$-0.0785579\pi$
$157$	195.148	+	112.669i	1.24298	+	0.717635i	0.273281	+	0.961934i	$0.411891\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.759209i		−	0.00471559i
$162$		0			0
$163$			302.948i			1.85858i	0.369352	+	0.929290i	$0.379580\pi$
$163$			302.948i			1.85858i	−0.369352	+	0.929290i	$0.620420\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	119.334	−	206.692i	0.714573	−	1.23768i	−0.248552	−	0.968619i	$-0.579955\pi$
$167$	119.334	−	206.692i	0.714573	−	1.23768i	0.963124	−	0.269057i	$-0.0867121\pi$
$168$		0			0
$169$	180.611	+	312.828i	1.06871	+	1.85105i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−54.4011	−	94.2254i	−0.314457	−	0.544656i	0.664865	−	0.746964i	$-0.268489\pi$
$173$	−54.4011	−	94.2254i	−0.314457	−	0.544656i	−0.979322	+	0.202308i	$0.935156\pi$
$174$		0			0
$175$	61.4652	−	33.2603i	0.351230	−	0.190059i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			168.054i			0.938849i	0.882973	+	0.469425i	$0.155539\pi$
$179$			168.054i			0.938849i	−0.882973	+	0.469425i	$0.844461\pi$
$180$		0			0
$181$	−154.297			−0.852468			−0.426234	−	0.904613i	$-0.640160\pi$
$181$	−154.297			−0.852468			−0.426234	+	0.904613i	$0.640160\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	174.339	+	2.40564i	0.942375	+	0.0130035i
$186$		0			0
$187$	−90.0241	+	51.9755i	−0.481412	+	0.277944i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−134.175	+	77.4662i	−0.702489	+	0.405582i	−0.808274	−	0.588807i	$-0.799598\pi$
$191$	−134.175	+	77.4662i	−0.702489	+	0.405582i	0.105785	+	0.994389i	$0.466264\pi$
$192$		0			0
$193$	55.4313	+	32.0033i	0.287209	+	0.165820i	0.636683	−	0.771126i	$-0.280306\pi$
$193$	55.4313	+	32.0033i	0.287209	+	0.165820i	−0.349473	+	0.936946i	$0.613640\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.56832			0.0485702			0.0242851	−	0.999705i	$-0.492269\pi$
$197$	9.56832			0.0485702			0.0242851	+	0.999705i	$0.492269\pi$
$198$		0			0
$199$	−117.815			−0.592034			−0.296017	−	0.955183i	$-0.595658\pi$
$199$	−117.815			−0.592034			−0.296017	+	0.955183i	$0.595658\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−55.6295	+	96.3531i	−0.274037	+	0.474646i
$204$		0			0
$205$	−33.9557	+	56.9826i	−0.165637	+	0.277964i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−364.365	+	210.366i	−1.74337	+	1.00654i
$210$		0			0
$211$	24.4074	−	42.2748i	0.115675	−	0.200355i	−0.802375	−	0.596821i	$-0.796430\pi$
$211$	24.4074	−	42.2748i	0.115675	−	0.200355i	0.918049	+	0.396466i	$0.129764\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−113.987	−	203.878i	−0.530174	−	0.948268i
$216$		0			0
$217$		−	132.424i		−	0.610247i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−114.234	−	65.9529i	−0.516895	−	0.298429i
$222$		0			0
$223$	−276.887	+	159.861i	−1.24164	+	0.716864i	−0.969429	−	0.245372i	$-0.921090\pi$
$223$	−276.887	+	159.861i	−1.24164	+	0.716864i	−0.272216	+	0.962236i	$0.587756\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.13579	+	15.8237i	0.0402458	+	0.0697077i	0.885447	−	0.464741i	$-0.153853\pi$
$227$	9.13579	+	15.8237i	0.0402458	+	0.0697077i	−0.845201	+	0.534449i	$0.820519\pi$
$228$		0			0
$229$	−67.0369	+	116.111i	−0.292737	+	0.507036i	−0.974456	−	0.224578i	$-0.927900\pi$
$229$	−67.0369	+	116.111i	−0.292737	+	0.507036i	0.681719	+	0.731615i	$0.261233\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	284.173			1.21963			0.609813	−	0.792546i	$-0.291245\pi$
$233$	284.173			1.21963			0.609813	+	0.792546i	$0.291245\pi$
$234$		0			0
$235$	178.556	−	99.8301i	0.759812	−	0.424809i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−179.068	−	103.385i	−0.749237	−	0.432572i	0.0761810	−	0.997094i	$-0.475727\pi$
$239$	−179.068	−	103.385i	−0.749237	−	0.432572i	−0.825418	+	0.564522i	$0.809061\pi$
$240$		0			0
$241$	81.3336	+	140.874i	0.337484	+	0.584539i	0.983959	−	0.178396i	$-0.0570908\pi$
$241$	81.3336	+	140.874i	0.337484	+	0.584539i	−0.646475	+	0.762935i	$0.723757\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	176.900	+	105.414i	0.722040	+	0.430261i
$246$		0			0
$247$	−462.351	−	266.938i	−1.87187	−	1.08072i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			347.387i			1.38401i	0.721893	+	0.692005i	$0.243272\pi$
$251$			347.387i			1.38401i	−0.721893	+	0.692005i	$0.756728\pi$
$252$		0			0
$253$		−	4.92831i		−	0.0194795i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.5189	+	18.2192i	−0.0409294	+	0.0708919i	−0.885764	−	0.464135i	$-0.846365\pi$
$257$	−10.5189	+	18.2192i	−0.0409294	+	0.0708919i	0.844835	+	0.535027i	$0.179699\pi$
$258$		0			0
$259$	−48.7410	−	84.4219i	−0.188189	−	0.325953i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−107.728	−	186.591i	−0.409614	−	0.709472i	0.585233	−	0.810865i	$-0.301003\pi$
$263$	−107.728	−	186.591i	−0.409614	−	0.709472i	−0.994846	+	0.101394i	$0.967670\pi$
$264$		0			0
$265$	6.30333	−	456.809i	0.0237861	−	1.72381i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			180.237i			0.670024i	0.942214	+	0.335012i	$0.108740\pi$
$269$			180.237i			0.670024i	−0.942214	+	0.335012i	$0.891260\pi$
$270$		0			0
$271$	231.741			0.855133			0.427566	−	0.903984i	$-0.359371\pi$
$271$	231.741			0.855133			0.427566	+	0.903984i	$0.359371\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	398.993	−	215.905i	1.45088	−	0.785109i
$276$		0			0
$277$	−91.4227	+	52.7829i	−0.330046	+	0.190552i	−0.655862	−	0.754881i	$-0.727694\pi$
$277$	−91.4227	+	52.7829i	−0.330046	+	0.190552i	0.325816	+	0.945433i	$0.394361\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	436.993	−	252.298i	1.55514	−	0.897859i	0.557426	−	0.830226i	$-0.311789\pi$
$281$	436.993	−	252.298i	1.55514	−	0.897859i	0.997710	−	0.0676323i	$-0.0215445\pi$
$282$		0			0
$283$	131.306	+	75.8095i	0.463978	+	0.267878i	0.713716	−	0.700436i	$-0.247011\pi$
$283$	131.306	+	75.8095i	0.463978	+	0.267878i	−0.249737	+	0.968314i	$0.580344\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	37.0863			0.129221
$288$		0			0
$289$	−256.185			−0.886454
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−274.519	+	475.481i	−0.936924	+	1.62280i	−0.165759	+	0.986166i	$0.553007\pi$
$293$	−274.519	+	475.481i	−0.936924	+	1.62280i	−0.771166	+	0.636634i	$0.780326\pi$
$294$		0			0
$295$	201.791	−	338.634i	0.684037	−	1.14791i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.41582	−	3.12682i	0.0181131	−	0.0104576i
$300$		0			0
$301$	−65.2967	+	113.097i	−0.216933	+	0.375738i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−136.099	+	76.0926i	−0.446226	+	0.249484i
$306$		0			0
$307$			146.401i			0.476876i	0.971158	+	0.238438i	$0.0766355\pi$
$307$			146.401i			0.476876i	−0.971158	+	0.238438i	$0.923365\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	422.468	+	243.912i	1.35842	+	0.784282i	0.989410	−	0.145144i	$-0.0463647\pi$
$311$	422.468	+	243.912i	1.35842	+	0.784282i	0.369006	+	0.929427i	$0.379698\pi$
$312$		0			0
$313$	94.9915	−	54.8433i	0.303487	−	0.175218i	−0.340521	−	0.940237i	$-0.610604\pi$
$313$	94.9915	−	54.8433i	0.303487	−	0.175218i	0.644008	+	0.765018i	$0.277270\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	181.784	+	314.859i	0.573452	+	0.993247i	0.996208	+	0.0870039i	$0.0277293\pi$
$317$	181.784	+	314.859i	0.573452	+	0.993247i	−0.422756	+	0.906243i	$0.638937\pi$
$318$		0			0
$319$	−361.111	+	625.463i	−1.13201	+	1.96070i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	132.815			0.411191
$324$		0			0
$325$	490.408	+	301.478i	1.50895	+	0.927625i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−99.0505	−	57.1868i	−0.301065	−	0.173820i
$330$		0			0
$331$	−150.111	−	260.001i	−0.453509	−	0.785501i	0.545092	−	0.838376i	$-0.316495\pi$
$331$	−150.111	−	260.001i	−0.453509	−	0.785501i	−0.998601	+	0.0528755i	$0.983161\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.7023	+	29.7071i	−0.0528428	+	0.0886779i
$336$		0			0
$337$	323.334	+	186.677i	0.959447	+	0.553937i	0.896003	−	0.444049i	$-0.146458\pi$
$337$	323.334	+	186.677i	0.959447	+	0.553937i	0.0634441	+	0.997985i	$0.479792\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	859.610i		−	2.52085i
$342$		0			0
$343$		−	252.112i		−	0.735020i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.72213	+	15.1072i	−0.0251358	+	0.0435365i	−0.878320	−	0.478074i	$-0.841335\pi$
$347$	−8.72213	+	15.1072i	−0.0251358	+	0.0435365i	0.853184	+	0.521610i	$0.174668\pi$
$348$		0			0
$349$	−117.075	−	202.779i	−0.335457	−	0.581029i	0.648115	−	0.761542i	$-0.275558\pi$
$349$	−117.075	−	202.779i	−0.335457	−	0.581029i	−0.983573	+	0.180513i	$0.942224\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−190.642	−	330.202i	−0.540063	−	0.935416i	−0.998900	−	0.0468954i	$-0.985067\pi$
$353$	−190.642	−	330.202i	−0.540063	−	0.935416i	0.458837	−	0.888520i	$-0.348266\pi$
$354$		0			0
$355$	407.896	+	5.62840i	1.14900	+	0.0158546i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	222.024i		−	0.618451i	−0.950989	−	0.309226i	$-0.899930\pi$
$359$		−	222.024i		−	0.618451i	0.950989	−	0.309226i	$-0.100070\pi$
$360$		0			0
$361$	176.556			0.489074
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−530.368	−	7.31834i	−1.45306	−	0.0200502i
$366$		0			0
$367$	389.023	−	224.602i	1.06001	−	0.611995i	0.134573	−	0.990904i	$-0.457034\pi$
$367$	389.023	−	224.602i	1.06001	−	0.611995i	0.925434	+	0.378908i	$0.123700\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−221.205	+	127.713i	−0.596239	+	0.344239i
$372$		0			0
$373$	−278.932	−	161.041i	−0.747806	−	0.431746i	0.0770948	−	0.997024i	$-0.475436\pi$
$373$	−278.932	−	161.041i	−0.747806	−	0.431746i	−0.824901	+	0.565278i	$0.808769\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−916.446			−2.43089
$378$		0			0
$379$	−216.074			−0.570115			−0.285058	−	0.958510i	$-0.592013\pi$
$379$	−216.074			−0.570115			−0.285058	+	0.958510i	$0.592013\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	186.988	−	323.873i	0.488220	−	0.845622i	−0.511688	−	0.859171i	$-0.670980\pi$
$383$	186.988	−	323.873i	0.488220	−	0.845622i	0.999908	+	0.0135493i	$0.00431302\pi$
$384$		0			0
$385$	−217.890	−	129.840i	−0.565948	−	0.337246i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	193.217	−	111.554i	0.496702	−	0.286771i	−0.230648	−	0.973037i	$-0.574085\pi$
$389$	193.217	−	111.554i	0.496702	−	0.286771i	0.727351	+	0.686266i	$0.240751\pi$
$390$		0			0
$391$	−0.777873	+	1.34731i	−0.00198944	+	0.00344582i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−277.370	+	155.077i	−0.702204	+	0.392600i
$396$		0			0
$397$			610.535i			1.53787i	0.639325	+	0.768936i	$0.279214\pi$
$397$			610.535i			1.53787i	−0.639325	+	0.768936i	$0.720786\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−356.319	−	205.721i	−0.888575	−	0.513019i	−0.0150992	−	0.999886i	$-0.504806\pi$
$401$	−356.319	−	205.721i	−0.888575	−	0.513019i	−0.873476	+	0.486867i	$0.838140\pi$
$402$		0			0
$403$	944.643	−	545.390i	2.34403	−	1.35333i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−316.396	−	548.013i	−0.777385	−	1.34647i
$408$		0			0
$409$	−295.315	+	511.500i	−0.722041	+	1.25061i	0.238139	+	0.971231i	$0.423462\pi$
$409$	−295.315	+	511.500i	−0.722041	+	1.25061i	−0.960180	+	0.279381i	$0.909871\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−220.396			−0.533646
$414$		0			0
$415$	1.24014	−	0.693359i	0.00298829	−	0.00167074i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	207.462	+	119.778i	0.495135	+	0.285867i	0.726702	−	0.686952i	$-0.241052\pi$
$419$	207.462	+	119.778i	0.495135	+	0.285867i	−0.231567	+	0.972819i	$0.574385\pi$
$420$		0			0
$421$	−296.704	−	513.907i	−0.704760	−	1.22068i	−0.966778	−	0.255618i	$-0.917721\pi$
$421$	−296.704	−	513.907i	−0.704760	−	1.22068i	0.262018	−	0.965063i	$-0.415612\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−143.156	−	3.95146i	−0.336837	−	0.00929754i
$426$		0			0
$427$	75.4983	+	43.5890i	0.176811	+	0.102082i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			436.566i			1.01291i	0.862265	+	0.506457i	$0.169045\pi$
$431$			436.566i			1.01291i	−0.862265	+	0.506457i	$0.830955\pi$
$432$		0			0
$433$		−	231.736i		−	0.535187i	−0.963532	−	0.267593i	$-0.913772\pi$
$433$		−	231.736i		−	0.535187i	0.963532	−	0.267593i	$-0.0862284\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.14837	+	5.45314i	−0.00720451	+	0.0124786i
$438$		0			0
$439$	209.611	+	363.058i	0.477475	+	0.827011i	0.999667	−	0.0258173i	$-0.00821881\pi$
$439$	209.611	+	363.058i	0.477475	+	0.827011i	−0.522192	+	0.852828i	$0.674885\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−223.260	−	386.697i	−0.503973	−	0.872906i	−0.999989	−	0.00459325i	$-0.998538\pi$
$443$	−223.260	−	386.697i	−0.503973	−	0.872906i	0.496017	−	0.868313i	$-0.334795\pi$
$444$		0			0
$445$	142.591	+	1.96756i	0.320430	+	0.00442149i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	495.776i		−	1.10418i	−0.833785	−	0.552089i	$-0.813831\pi$
$449$		−	495.776i		−	1.10418i	0.833785	−	0.552089i	$-0.186169\pi$
$450$		0			0
$451$	240.741			0.533794
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.44069	−	321.822i	0.00975976	−	0.707301i
$456$		0			0
$457$	−591.810	+	341.681i	−1.29499	+	0.747662i	−0.979534	−	0.201279i	$-0.935490\pi$
$457$	−591.810	+	341.681i	−1.29499	+	0.747662i	−0.315454	+	0.948941i	$0.602157\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−346.521	+	200.064i	−0.751672	+	0.433978i	−0.826298	−	0.563234i	$-0.809557\pi$
$461$	−346.521	+	200.064i	−0.751672	+	0.433978i	0.0746257	+	0.997212i	$0.476224\pi$
$462$		0			0
$463$	−273.892	−	158.132i	−0.591559	−	0.341537i	0.174155	−	0.984718i	$-0.444281\pi$
$463$	−273.892	−	158.132i	−0.591559	−	0.341537i	−0.765714	+	0.643181i	$0.777614\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	419.741			0.898803			0.449401	−	0.893330i	$-0.351637\pi$
$467$	419.741			0.898803			0.449401	+	0.893330i	$0.351637\pi$
$468$		0			0
$469$	19.3345			0.0412249
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−423.865	+	734.156i	−0.896121	+	1.55213i
$474$		0			0
$475$	−579.411	−	15.9932i	−1.21981	−	0.0336698i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	731.766	−	422.485i	1.52769	−	0.882015i	0.528237	−	0.849097i	$-0.322853\pi$
$479$	731.766	−	422.485i	1.52769	−	0.882015i	0.999458	−	0.0329177i	$-0.0104799\pi$
$480$		0			0
$481$	401.482	−	695.387i	0.834682	−	1.44571i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−226.711	−	405.495i	−0.467446	−	0.836072i
$486$		0			0
$487$		−	546.384i		−	1.12194i	−0.827837	−	0.560969i	$-0.810429\pi$
$487$		−	546.384i		−	1.12194i	0.827837	−	0.560969i	$-0.189571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−560.713	−	323.728i	−1.14198	−	0.659324i	−0.195062	−	0.980791i	$-0.562491\pi$
$491$	−560.713	−	323.728i	−1.14198	−	0.659324i	−0.946921	+	0.321467i	$0.895824\pi$
$492$		0			0
$493$	197.443	−	113.994i	0.400494	−	0.231225i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−114.038	−	197.519i	−0.229452	−	0.397423i
$498$		0			0
$499$	−291.407	+	504.732i	−0.583983	+	1.01149i	0.411019	+	0.911627i	$0.365173\pi$
$499$	−291.407	+	504.732i	−0.583983	+	1.01149i	−0.995001	+	0.0998608i	$0.968160\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−752.520			−1.49606			−0.748032	−	0.663663i	$-0.769001\pi$
$503$	−752.520			−1.49606			−0.748032	+	0.663663i	$0.769001\pi$
$504$		0			0
$505$	113.870	+	203.667i	0.225484	+	0.403301i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	252.479	+	145.769i	0.496030	+	0.286383i	0.727073	−	0.686560i	$-0.240880\pi$
$509$	252.479	+	145.769i	0.496030	+	0.286383i	−0.231042	+	0.972944i	$0.574214\pi$
$510$		0			0
$511$	148.278	+	256.825i	0.290172	+	0.502593i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	28.6203	−	48.0290i	0.0555733	−	0.0932601i
$516$		0			0
$517$	−642.973	−	371.221i	−1.24366	−	0.718028i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			137.690i			0.264280i	0.991231	+	0.132140i	$0.0421848\pi$
$521$			137.690i			0.264280i	−0.991231	+	0.132140i	$0.957815\pi$
$522$		0			0
$523$		−	330.987i		−	0.632862i	−0.948616	−	0.316431i	$-0.897515\pi$
$523$		−	330.987i		−	0.632862i	0.948616	−	0.316431i	$-0.102485\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−135.679	+	235.003i	−0.257455	+	0.445926i
$528$		0			0
$529$	264.463	+	458.064i	0.499930	+	0.865905i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	152.741	+	264.555i	0.286568	+	0.496351i
$534$		0			0
$535$	10.1615	−	736.414i	0.0189934	−	1.37647i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	747.370i		−	1.38659i
$540$		0			0
$541$	556.593			1.02882			0.514412	−	0.857543i	$-0.328010\pi$
$541$	556.593			1.02882			0.514412	+	0.857543i	$0.328010\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.38570	+	607.721i	−0.0153866	+	1.11508i
$546$		0			0
$547$	−402.849	+	232.585i	−0.736470	+	0.425201i	−0.820784	−	0.571238i	$-0.806463\pi$
$547$	−402.849	+	232.585i	−0.736470	+	0.425201i	0.0843144	+	0.996439i	$0.473130\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	799.135	−	461.381i	1.45034	−	0.837351i
$552$		0			0
$553$	153.866	+	88.8347i	0.278239	+	0.160641i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1059.37			−1.90192			−0.950962	−	0.309306i	$-0.899903\pi$
$557$	−1059.37			−1.90192			−0.950962	+	0.309306i	$0.899903\pi$
$558$		0			0
$559$	−1075.70			−1.92434
$560$		0			0
$561$		0			0
$562$		0			0
$563$	193.080	−	334.424i	0.342949	−	0.594004i	−0.642030	−	0.766679i	$-0.721908\pi$
$563$	193.080	−	334.424i	0.342949	−	0.594004i	0.984979	+	0.172675i	$0.0552410\pi$
$564$		0			0
$565$	419.340	+	249.883i	0.742195	+	0.442271i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	594.147	−	343.031i	1.04420	−	0.602866i	0.123176	−	0.992385i	$-0.460692\pi$
$569$	594.147	−	343.031i	1.04420	−	0.602866i	0.921019	+	0.389518i	$0.127359\pi$
$570$		0			0
$571$	410.816	−	711.554i	0.719467	−	1.24615i	−0.241744	−	0.970340i	$-0.577720\pi$
$571$	410.816	−	711.554i	0.719467	−	1.24615i	0.961211	−	0.275813i	$-0.0889471\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.55575	−	5.78406i	0.00618390	−	0.0100592i
$576$		0			0
$577$			183.840i			0.318613i	0.987229	+	0.159306i	$0.0509258\pi$
$577$			183.840i			0.318613i	−0.987229	+	0.159306i	$0.949074\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.687945	−	0.397185i	−0.00118407	−	0.000683623i
$582$		0			0
$583$	−1435.92	+	829.029i	−2.46299	+	1.42201i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−160.203	−	277.480i	−0.272919	−	0.472709i	0.696689	−	0.717373i	$-0.254656\pi$
$587$	−160.203	−	277.480i	−0.272919	−	0.472709i	−0.969608	+	0.244664i	$0.921322\pi$
$588$		0			0
$589$	−549.148	+	951.153i	−0.932340	+	1.61486i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1061.34			−1.78977			−0.894887	−	0.446292i	$-0.852744\pi$
$593$	−1061.34			−1.78977			−0.894887	+	0.446292i	$0.852744\pi$
$594$		0			0
$595$	39.0738	+	69.8872i	0.0656702	+	0.117457i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	846.155	+	488.528i	1.41261	+	0.815573i	0.995634	−	0.0933433i	$-0.0297554\pi$
$599$	846.155	+	488.528i	1.41261	+	0.815573i	0.416979	+	0.908916i	$0.363089\pi$
$600$		0			0
$601$	130.908	+	226.740i	0.217817	+	0.377271i	0.954140	−	0.299359i	$-0.0967730\pi$
$601$	130.908	+	226.740i	0.217817	+	0.377271i	−0.736323	+	0.676630i	$0.763440\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−894.681	−	533.137i	−1.47881	−	0.881218i
$606$		0			0
$607$	579.203	+	334.403i	0.954206	+	0.550911i	0.894385	−	0.447298i	$-0.147614\pi$
$607$	579.203	+	334.403i	0.954206	+	0.550911i	0.0598212	+	0.998209i	$0.480947\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	942.101i		−	1.54190i
$612$		0			0
$613$			220.119i			0.359086i	0.983750	+	0.179543i	$0.0574618\pi$
$613$			220.119i			0.359086i	−0.983750	+	0.179543i	$0.942538\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−160.531	+	278.049i	−0.260181	+	0.450646i	−0.966290	−	0.257457i	$-0.917115\pi$
$617$	−160.531	+	278.049i	−0.260181	+	0.450646i	0.706109	+	0.708103i	$0.250449\pi$
$618$		0			0
$619$	445.593	+	771.791i	0.719860	+	1.24683i	0.961055	+	0.276358i	$0.0891275\pi$
$619$	445.593	+	771.791i	0.719860	+	1.24683i	−0.241195	+	0.970477i	$0.577539\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−39.8651	−	69.0483i	−0.0639889	−	0.110832i
$624$		0			0
$625$	624.048	+	34.4768i	0.998477	+	0.0551629i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			199.757i			0.317578i
$630$		0			0
$631$	−583.669			−0.924990			−0.462495	−	0.886622i	$-0.653046\pi$
$631$	−583.669			−0.924990			−0.462495	+	0.886622i	$0.653046\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−443.198	−	6.11552i	−0.697950	−	0.00963074i
$636$		0			0
$637$	821.300	−	474.178i	1.28932	−	0.744392i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−603.006	+	348.146i	−0.940727	+	0.543129i	−0.890188	−	0.455593i	$-0.849427\pi$
$641$	−603.006	+	348.146i	−0.940727	+	0.543129i	−0.0505391	+	0.998722i	$0.516094\pi$
$642$		0			0
$643$	694.262	+	400.832i	1.07972	+	0.623378i	0.930822	−	0.365474i	$-0.119093\pi$
$643$	694.262	+	400.832i	1.07972	+	0.623378i	0.148901	+	0.988852i	$0.452426\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	216.221			0.334191			0.167095	−	0.985941i	$-0.446561\pi$
$647$	216.221			0.334191			0.167095	+	0.985941i	$0.446561\pi$
$648$		0			0
$649$	−1430.67			−2.20442
$650$		0			0
$651$		0			0
$652$		0			0
$653$	219.401	−	380.014i	0.335989	−	0.581951i	−0.647685	−	0.761908i	$-0.724263\pi$
$653$	219.401	−	380.014i	0.335989	−	0.581951i	0.983674	+	0.179958i	$0.0575960\pi$
$654$		0			0
$655$	230.288	−	386.456i	0.351584	−	0.590009i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−317.063	+	183.056i	−0.481127	+	0.277779i	−0.720886	−	0.693054i	$-0.756265\pi$
$659$	−317.063	+	183.056i	−0.481127	+	0.277779i	0.239759	+	0.970832i	$0.422932\pi$
$660$		0			0
$661$	−79.0377	+	136.897i	−0.119573	+	0.207106i	−0.919599	−	0.392859i	$-0.871486\pi$
$661$	−79.0377	+	136.897i	−0.119573	+	0.207106i	0.800026	+	0.599966i	$0.204819\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	158.148	+	282.862i	0.237816	+	0.425357i
$666$		0			0
$667$			10.8089i			0.0162052i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	490.087	+	282.952i	0.730384	+	0.421687i
$672$		0			0
$673$	−561.683	+	324.288i	−0.834595	+	0.481854i	−0.855423	−	0.517929i	$-0.826703\pi$
$673$	−561.683	+	324.288i	−0.834595	+	0.481854i	0.0208282	+	0.999783i	$0.493370\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	101.173	+	175.236i	0.149443	+	0.258842i	0.931022	−	0.364964i	$-0.118919\pi$
$677$	101.173	+	175.236i	0.149443	+	0.258842i	−0.781579	+	0.623806i	$0.785585\pi$
$678$		0			0
$679$	−129.870	+	224.941i	−0.191266	+	0.331283i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	173.829			0.254508			0.127254	−	0.991870i	$-0.459384\pi$
$683$	173.829			0.254508			0.127254	+	0.991870i	$0.459384\pi$
$684$		0			0
$685$	1003.45	−	561.024i	1.46488	−	0.819013i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1822.07	−	1051.97i	−2.64452	−	1.52681i
$690$		0			0
$691$	423.371	+	733.301i	0.612694	+	1.06122i	0.990784	+	0.135448i	$0.0432474\pi$
$691$	423.371	+	733.301i	0.612694	+	1.06122i	−0.378091	+	0.925769i	$0.623419\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	247.531	+	147.503i	0.356160	+	0.212235i
$696$		0			0
$697$	−65.8145	−	37.9980i	−0.0944254	−	0.0545165i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			72.2614i			0.103083i	0.998671	+	0.0515416i	$0.0164135\pi$
$701$			72.2614i			0.103083i	−0.998671	+	0.0515416i	$0.983587\pi$
$702$		0			0
$703$			808.497i			1.15007i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	65.2293	−	112.980i	0.0922621	−	0.159803i
$708$		0			0
$709$	−143.779	−	249.032i	−0.202791	−	0.351244i	0.746636	−	0.665233i	$-0.231668\pi$
$709$	−143.779	−	249.032i	−0.202791	−	0.351244i	−0.949427	+	0.313989i	$0.898334\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.43253	−	11.1415i	−0.00902178	−	0.0156262i
$714$		0			0
$715$	28.8262	−	2089.06i	0.0403163	−	2.92177i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1015.27i			1.41205i	0.708185	+	0.706027i	$0.249514\pi$
$719$			1015.27i			1.41205i	−0.708185	+	0.706027i	$0.750486\pi$
$720$		0			0
$721$	−31.2590			−0.0433551
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−875.083	+	473.529i	−1.20701	+	0.653143i
$726$		0			0
$727$	−269.696	+	155.709i	−0.370971	+	0.214180i	−0.673883	−	0.738838i	$-0.735375\pi$
$727$	−269.696	+	155.709i	−0.370971	+	0.214180i	0.302911	+	0.953019i	$0.402041\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	231.755	−	133.804i	0.317038	−	0.183042i
$732$		0			0
$733$	278.035	+	160.523i	0.379310	+	0.218995i	0.677518	−	0.735506i	$-0.263055\pi$
$733$	278.035	+	160.523i	0.379310	+	0.218995i	−0.298208	+	0.954501i	$0.596389\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	125.507			0.170295
$738$		0			0
$739$	−657.185			−0.889290			−0.444645	−	0.895707i	$-0.646670\pi$
$739$	−657.185			−0.889290			−0.444645	+	0.895707i	$0.646670\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	512.482	−	887.645i	0.689747	−	1.19468i	−0.282173	−	0.959364i	$-0.591055\pi$
$743$	512.482	−	887.645i	0.689747	−	1.19468i	0.971920	−	0.235313i	$-0.0756116\pi$
$744$		0			0
$745$	−28.4967	+	47.8216i	−0.0382506	+	0.0641901i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−356.600	+	205.883i	−0.476101	+	0.274877i
$750$		0			0
$751$	31.5738	−	54.6874i	0.0420423	−	0.0728194i	−0.844239	−	0.535968i	$-0.819947\pi$
$751$	31.5738	−	54.6874i	0.0420423	−	0.0728194i	0.886281	+	0.463148i	$0.153280\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	96.8210	−	54.1324i	0.128240	−	0.0716986i
$756$		0			0
$757$		−	1452.21i		−	1.91837i	−0.282781	−	0.959185i	$-0.591257\pi$
$757$		−	1452.21i		−	1.91837i	0.282781	−	0.959185i	$-0.408743\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	422.468	+	243.912i	0.555148	+	0.320515i	0.751196	−	0.660079i	$-0.229477\pi$
$761$	422.468	+	243.912i	0.555148	+	0.320515i	−0.196048	+	0.980594i	$0.562811\pi$
$762$		0			0
$763$	294.282	−	169.904i	0.385691	−	0.222679i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−907.705	−	1572.19i	−1.18345	−	2.04979i
$768$		0			0
$769$	−338.574	+	586.427i	−0.440278	+	0.762584i	−0.997710	−	0.0676389i	$-0.978453\pi$
$769$	−338.574	+	586.427i	−0.440278	+	0.762584i	0.557432	+	0.830223i	$0.311787\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−949.926			−1.22888			−0.614441	−	0.788963i	$-0.710619\pi$
$773$	−949.926			−1.22888			−0.614441	+	0.788963i	$0.710619\pi$
$774$		0			0
$775$	620.204	−	1008.87i	0.800263	−	1.30177i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−266.378	−	153.794i	−0.341949	−	0.197424i
$780$		0			0
$781$	−740.261	−	1282.17i	−0.947837	−	1.64170i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	576.752	−	967.874i	0.734717	−	1.23296i
$786$		0			0
$787$	991.018	+	572.164i	1.25923	+	0.727020i	0.972926	−	0.231118i	$-0.0742383\pi$
$787$	991.018	+	572.164i	1.25923	+	0.727020i	0.286309	+	0.958137i	$0.407572\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	272.922i		−	0.345034i
$792$		0			0
$793$			718.089i			0.905535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−325.550	+	563.870i	−0.408470	+	0.707490i	−0.994718	−	0.102641i	$-0.967271\pi$
$797$	−325.550	+	563.870i	−0.408470	+	0.707490i	0.586249	+	0.810131i	$0.300604\pi$
$798$		0			0
$799$	117.185	+	202.971i	0.146665	+	0.254031i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	962.526	+	1667.14i	1.19866	+	2.07614i
$804$		0			0
$805$	−3.79569	−	0.0523752i	−0.00471514	−	6.50623e-5i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1296.03i		−	1.60202i	−0.598653	−	0.801008i	$-0.704297\pi$
$809$		−	1296.03i		−	1.60202i	0.598653	−	0.801008i	$-0.295703\pi$
$810$		0			0
$811$	69.9245			0.0862201			0.0431101	−	0.999070i	$-0.486273\pi$
$811$	69.9245			0.0862201			0.0431101	+	0.999070i	$0.486273\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1514.60	+	20.8993i	1.85840	+	0.0256434i
$816$		0			0
$817$	938.007	−	541.559i	1.14811	−	0.662863i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	809.810	−	467.544i	0.986371	−	0.569481i	0.0821833	−	0.996617i	$-0.473811\pi$
$821$	809.810	−	467.544i	0.986371	−	0.569481i	0.904187	+	0.427136i	$0.140477\pi$
$822$		0			0
$823$	−935.659	−	540.203i	−1.13689	−	0.656383i	−0.191230	−	0.981545i	$-0.561248\pi$
$823$	−935.659	−	540.203i	−1.13689	−	0.656383i	−0.945658	+	0.325163i	$0.894581\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1105.43			1.33668			0.668339	−	0.743856i	$-0.267005\pi$
$827$	1105.43			1.33668			0.668339	+	0.743856i	$0.267005\pi$
$828$		0			0
$829$	899.633			1.08520			0.542601	−	0.839990i	$-0.317439\pi$
$829$	899.633			1.08520			0.542601	+	0.839990i	$0.317439\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−117.963	+	204.318i	−0.141612	+	0.245280i
$834$		0			0
$835$	−1025.13	−	610.870i	−1.22770	−	0.731581i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−987.814	+	570.315i	−1.17737	+	0.679755i	−0.955405	−	0.295299i	$-0.904581\pi$
$839$	−987.814	+	570.315i	−1.17737	+	0.679755i	−0.221966	+	0.975054i	$0.571247\pi$
$840$		0			0
$841$	371.500	−	643.457i	0.441736	−	0.765109i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1576.45	−	881.391i	1.86562	−	1.04307i
$846$		0			0
$847$			582.291i			0.687475i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.20165	−	4.73523i	−0.00963766	−	0.00556431i
$852$		0			0
$853$	1059.52	−	611.716i	1.24211	−	0.717135i	0.272590	−	0.962130i	$-0.412120\pi$
$853$	1059.52	−	611.716i	1.24211	−	0.717135i	0.969524	+	0.244996i	$0.0787865\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	835.039	+	1446.33i	0.974374	+	1.68767i	0.681985	+	0.731366i	$0.261117\pi$
$857$	835.039	+	1446.33i	0.974374	+	1.68767i	0.292389	+	0.956300i	$0.405550\pi$
$858$		0			0
$859$	−218.111	+	377.779i	−0.253912	+	0.439789i	−0.964600	−	0.263719i	$-0.915051\pi$
$859$	−218.111	+	377.779i	−0.253912	+	0.439789i	0.710687	+	0.703508i	$0.248384\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−883.333			−1.02356			−0.511780	−	0.859116i	$-0.671014\pi$
$863$	−883.333			−1.02356			−0.511780	+	0.859116i	$0.671014\pi$
$864$		0			0
$865$	−474.835	+	265.479i	−0.548943	+	0.306912i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	998.801	+	576.658i	1.14937	+	0.663588i
$870$		0			0
$871$	79.6295	+	137.922i	0.0914231	+	0.158349i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−162.045	−	309.591i	−0.185195	−	0.353819i
$876$		0			0
$877$	−1305.39	−	753.665i	−1.48847	−	0.859367i	−0.488554	−	0.872534i	$-0.662476\pi$
$877$	−1305.39	−	753.665i	−1.48847	−	0.859367i	−0.999913	+	0.0131665i	$0.995809\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			626.273i			0.710866i	0.934702	+	0.355433i	$0.115667\pi$
$881$			626.273i			0.710866i	−0.934702	+	0.355433i	$0.884333\pi$
$882$		0			0
$883$		−	23.3162i		−	0.0264057i	−0.999913	−	0.0132028i	$-0.995797\pi$
$883$		−	23.3162i		−	0.0264057i	0.999913	−	0.0132028i	$-0.00420271\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−264.666	+	458.416i	−0.298384	+	0.516816i	−0.975766	−	0.218815i	$-0.929781\pi$
$887$	−264.666	+	458.416i	−0.298384	+	0.516816i	0.677383	+	0.735631i	$0.263114\pi$
$888$		0			0
$889$	123.907	+	214.614i	0.139378	+	0.241410i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	474.297	+	821.506i	0.531127	+	0.919940i
$894$		0			0
$895$	840.190	+	11.5935i	0.938760	+	0.0129536i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1885.32i			2.09713i
$900$		0			0
$901$	523.408			0.580919
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.6444	+	771.410i	−0.0117618	+	0.852387i
$906$		0			0
$907$	329.897	−	190.466i	0.363723	−	0.209996i	−0.306989	−	0.951713i	$-0.599322\pi$
$907$	329.897	−	190.466i	0.363723	−	0.209996i	0.670713	+	0.741717i	$0.265988\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−99.7080	+	57.5664i	−0.109449	+	0.0631904i	−0.553725	−	0.832699i	$-0.686794\pi$
$911$	−99.7080	+	57.5664i	−0.109449	+	0.0631904i	0.444276	+	0.895890i	$0.353461\pi$
$912$		0			0
$913$	−4.46570	−	2.57827i	−0.00489124	−	0.00282396i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−251.520			−0.274285
$918$		0			0
$919$	−31.5163			−0.0342941			−0.0171471	−	0.999853i	$-0.505458\pi$
$919$	−31.5163			−0.0342941			−0.0171471	+	0.999853i	$0.505458\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	939.334	−	1626.98i	1.01770	−	1.76270i
$924$		0			0
$925$	24.0541	−	871.448i	0.0260044	−	0.942106i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−941.451	+	543.547i	−1.01340	+	0.585088i	−0.912186	−	0.409776i	$-0.865607\pi$
$929$	−941.451	+	543.547i	−1.01340	+	0.585088i	−0.101216	+	0.994864i	$0.532273\pi$
$930$		0			0
$931$	−477.445	+	826.959i	−0.512830	+	0.888248i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	253.642	+	453.663i	0.271275	+	0.485201i
$936$		0			0
$937$		−	40.9800i		−	0.0437354i	−0.999761	−	0.0218677i	$-0.993039\pi$
$937$		−	40.9800i		−	0.0437354i	0.999761	−	0.0218677i	$-0.00696125\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1165.03	−	672.633i	−1.23808	−	0.714807i	−0.269380	−	0.963034i	$-0.586819\pi$
$941$	−1165.03	−	672.633i	−1.23808	−	0.714807i	−0.968702	+	0.248227i	$0.920152\pi$
$942$		0			0
$943$	3.12026	−	1.80148i	0.00330887	−	0.00191037i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	182.390	+	315.909i	0.192598	+	0.333589i	0.946110	−	0.323844i	$-0.104975\pi$
$947$	182.390	+	315.909i	0.192598	+	0.333589i	−0.753513	+	0.657434i	$0.771642\pi$
$948$		0			0
$949$	−1221.37	+	2115.48i	−1.28701	+	2.22917i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1529.14			1.60455			0.802277	−	0.596952i	$-0.203622\pi$
$953$	1529.14			1.60455			0.802277	+	0.596952i	$0.203622\pi$
$954$		0			0
$955$	378.038	+	676.157i	0.395851	+	0.708018i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−556.643	−	321.378i	−0.580441	−	0.335118i
$960$		0			0
$961$	−641.482	−	1111.08i	−0.667515	−	1.15617i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	163.825	−	274.923i	0.169767	−	0.284894i
$966$		0			0
$967$	604.937	+	349.260i	0.625581	+	0.361179i	0.779039	−	0.626976i	$-0.215708\pi$
$967$	604.937	+	349.260i	0.625581	+	0.361179i	−0.153458	+	0.988155i	$0.549041\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			554.481i			0.571041i	0.958373	+	0.285521i	$0.0921665\pi$
$971$			554.481i			0.571041i	−0.958373	+	0.285521i	$0.907834\pi$
$972$		0			0
$973$		−	161.103i		−	0.165573i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	785.223	−	1360.05i	0.803708	−	1.39206i	−0.113451	−	0.993544i	$-0.536191\pi$
$977$	785.223	−	1360.05i	0.803708	−	1.39206i	0.917160	−	0.398520i	$-0.130476\pi$
$978$		0			0
$979$	−258.779	−	448.218i	−0.264330	−	0.457832i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−145.828	−	252.582i	−0.148350	−	0.256950i	0.782268	−	0.622942i	$-0.214063\pi$
$983$	−145.828	−	252.582i	−0.148350	−	0.256950i	−0.930618	+	0.365992i	$0.880730\pi$
$984$		0			0
$985$	0.660085	−	47.8371i	0.000670137	−	0.0485655i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			12.6873i			0.0128284i
$990$		0			0
$991$	1554.11			1.56823			0.784114	−	0.620616i	$-0.213117\pi$
$991$	1554.11			1.56823			0.784114	+	0.620616i	$0.213117\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.12762	+	589.018i	−0.00816846	+	0.591978i
$996$		0			0
$997$	−771.337	+	445.332i	−0.773658	+	0.446672i	−0.834178	−	0.551495i	$-0.814057\pi$
$997$	−771.337	+	445.332i	−0.773658	+	0.446672i	0.0605199	+	0.998167i	$0.480724\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.t.d.1349.2		8
3.2	odd	2		1620.3.t.a.1349.3		8
5.4	even	2		1620.3.t.a.1349.1		8
9.2	odd	6		1620.3.t.a.269.1		8
9.4	even	3		540.3.b.a.269.2	yes	4
9.5	odd	6		540.3.b.b.269.3	yes	4
9.7	even	3	inner	1620.3.t.d.269.4		8
15.14	odd	2	inner	1620.3.t.d.1349.4		8
36.23	even	6		2160.3.c.l.1889.3		4
36.31	odd	6		2160.3.c.h.1889.2		4
45.4	even	6		540.3.b.b.269.4	yes	4
45.13	odd	12		2700.3.g.s.701.3		8
45.14	odd	6		540.3.b.a.269.1	✓	4
45.22	odd	12		2700.3.g.s.701.5		8
45.23	even	12		2700.3.g.s.701.4		8
45.29	odd	6	inner	1620.3.t.d.269.2		8
45.32	even	12		2700.3.g.s.701.6		8
45.34	even	6		1620.3.t.a.269.3		8
180.59	even	6		2160.3.c.h.1889.1		4
180.139	odd	6		2160.3.c.l.1889.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.b.a.269.1	✓	4	45.14	odd	6
540.3.b.a.269.2	yes	4	9.4	even	3
540.3.b.b.269.3	yes	4	9.5	odd	6
540.3.b.b.269.4	yes	4	45.4	even	6
1620.3.t.a.269.1		8	9.2	odd	6
1620.3.t.a.269.3		8	45.34	even	6
1620.3.t.a.1349.1		8	5.4	even	2
1620.3.t.a.1349.3		8	3.2	odd	2
1620.3.t.d.269.2		8	45.29	odd	6	inner
1620.3.t.d.269.4		8	9.7	even	3	inner
1620.3.t.d.1349.2		8	1.1	even	1	trivial
1620.3.t.d.1349.4		8	15.14	odd	2	inner
2160.3.c.h.1889.1		4	180.59	even	6
2160.3.c.h.1889.2		4	36.31	odd	6
2160.3.c.l.1889.3		4	36.23	even	6
2160.3.c.l.1889.4		4	180.139	odd	6
2700.3.g.s.701.3		8	45.13	odd	12
2700.3.g.s.701.4		8	45.23	even	12
2700.3.g.s.701.5		8	45.22	odd	12
2700.3.g.s.701.6		8	45.32	even	12

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0689865 - 4.99952i) q^{5} +(-2.42096 + 1.39774i) q^{7} +O(q^{10})\)	\(q+(0.0689865 - 4.99952i) q^{5} +(-2.42096 + 1.39774i) q^{7} +(-15.7154 + 9.07327i) q^{11} +(-19.9416 - 11.5133i) q^{13} +5.72842 q^{17} +23.1852 q^{19} +(-0.135792 + 0.235199i) q^{23} +(-24.9905 - 0.689799i) q^{25} +(34.4674 - 19.8997i) q^{29} +(-23.6852 + 41.0241i) q^{31} +(6.82104 + 12.2001i) q^{35} +34.8712i q^{37} +(-11.4891 - 6.63325i) q^{41} +(40.4571 - 23.3579i) q^{43} +(20.4568 + 35.4323i) q^{47} +(-20.5926 + 35.6675i) q^{49} +91.3705 q^{53} +(44.2779 + 79.1953i) q^{55} +(68.2773 + 39.4199i) q^{59} +(-15.5926 - 27.0072i) q^{61} +(-58.9366 + 98.9042i) q^{65} +(-5.98970 - 3.45816i) q^{67} +81.5870i q^{71} -106.084i q^{73} +(25.3642 - 43.9321i) q^{77} +(-31.7779 - 55.0409i) q^{79} +(0.142081 + 0.246091i) q^{83} +(0.395183 - 28.6394i) q^{85} +28.5210i q^{89} +64.3705 q^{91} +(1.59947 - 115.915i) q^{95} +(80.4657 - 46.4569i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{5}+O(q^{10}) \) 8 * q + 3 * q^5	\( 8 q + 3 q^{5} - 12 q^{17} + 12 q^{19} - 30 q^{23} - 9 q^{25} - 16 q^{31} - 90 q^{35} + 48 q^{47} - 78 q^{49} + 384 q^{53} + 94 q^{55} - 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} - 288 q^{83} + 100 q^{85} + 168 q^{91} + 318 q^{95}+O(q^{100}) \) 8 * q + 3 * q^5 - 12 * q^17 + 12 * q^19 - 30 * q^23 - 9 * q^25 - 16 * q^31 - 90 * q^35 + 48 * q^47 - 78 * q^49 + 384 * q^53 + 94 * q^55 - 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 - 288 * q^83 + 100 * q^85 + 168 * q^91 + 318 * q^95

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