LMFDB - Embedded newform 1872.2.t.n.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(289,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1872.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.t (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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 10x^{2} + 100 $ x^4 + 10*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$1.58114 + 2.73861i$ of defining polynomial
Character	$\chi$	$=$	1872.289
Dual form			1872.2.t.n.1153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.16228 q^{5} +(1.58114 - 2.73861i) q^{7} +O(q^{10})$	$q-4.16228 q^{5} +(1.58114 - 2.73861i) q^{7} +(0.581139 + 1.00656i) q^{11} +(0.0811388 + 3.60464i) q^{13} +(-1.50000 + 2.59808i) q^{17} +(-2.58114 + 4.47066i) q^{19} +(-3.58114 - 6.20271i) q^{23} +12.3246 q^{25} +(-0.918861 - 1.59151i) q^{29} +6.32456 q^{31} +(-6.58114 + 11.3989i) q^{35} +(-1.91886 - 3.32357i) q^{37} +(1.50000 + 2.59808i) q^{41} +(4.58114 - 7.93477i) q^{43} +4.83772 q^{47} +(-1.50000 - 2.59808i) q^{49} +12.4868 q^{53} +(-2.41886 - 4.18959i) q^{55} +(1.16228 - 2.01312i) q^{59} +(-0.0811388 + 0.140537i) q^{61} +(-0.337722 - 15.0035i) q^{65} +(-1.41886 - 2.45754i) q^{67} +(3.58114 - 6.20271i) q^{71} -1.00000 q^{73} +3.67544 q^{77} +4.00000 q^{79} -3.48683 q^{83} +(6.24342 - 10.8139i) q^{85} +(6.00000 + 10.3923i) q^{89} +(10.0000 + 5.47723i) q^{91} +(10.7434 - 18.6081i) q^{95} +(2.00000 - 3.46410i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} - 4 q^{11} - 6 q^{13} - 6 q^{17} - 4 q^{19} - 8 q^{23} + 24 q^{25} - 10 q^{29} - 20 q^{35} - 14 q^{37} + 6 q^{41} + 12 q^{43} + 32 q^{47} - 6 q^{49} + 12 q^{53} - 16 q^{55} - 8 q^{59} + 6 q^{61} - 14 q^{65} - 12 q^{67} + 8 q^{71} - 4 q^{73} + 40 q^{77} + 16 q^{79} + 24 q^{83} + 6 q^{85} + 24 q^{89} + 40 q^{91} + 24 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^11 - 6 * q^13 - 6 * q^17 - 4 * q^19 - 8 * q^23 + 24 * q^25 - 10 * q^29 - 20 * q^35 - 14 * q^37 + 6 * q^41 + 12 * q^43 + 32 * q^47 - 6 * q^49 + 12 * q^53 - 16 * q^55 - 8 * q^59 + 6 * q^61 - 14 * q^65 - 12 * q^67 + 8 * q^71 - 4 * q^73 + 40 * q^77 + 16 * q^79 + 24 * q^83 + 6 * q^85 + 24 * q^89 + 40 * q^91 + 24 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.16228			−1.86143			−0.930714	−	0.365749i	$-0.880813\pi$
$5$	−4.16228			−1.86143			−0.930714	+	0.365749i	$0.880813\pi$
$6$		0			0
$7$	1.58114	−	2.73861i	0.597614	−	1.03510i	−0.395558	−	0.918441i	$-0.629449\pi$
$7$	1.58114	−	2.73861i	0.597614	−	1.03510i	0.993172	−	0.116657i	$-0.0372179\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.581139	+	1.00656i	0.175220	+	0.303490i	0.940237	−	0.340520i	$-0.110603\pi$
$11$	0.581139	+	1.00656i	0.175220	+	0.303490i	−0.765017	+	0.644010i	$0.777270\pi$
$12$		0			0
$13$	0.0811388	+	3.60464i	0.0225039	+	0.999747i
$13$	0.0811388	+	3.60464i	0.0225039	+	0.999747i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	−2.58114	+	4.47066i	−0.592154	+	1.02564i	0.401788	+	0.915733i	$0.368389\pi$
$19$	−2.58114	+	4.47066i	−0.592154	+	1.02564i	−0.993942	+	0.109908i	$0.964944\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.58114	−	6.20271i	−0.746719	−	1.29336i	−0.949387	−	0.314108i	$-0.898295\pi$
$23$	−3.58114	−	6.20271i	−0.746719	−	1.29336i	0.202668	−	0.979247i	$-0.435039\pi$
$24$		0			0
$25$	12.3246			2.46491
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.918861	−	1.59151i	−0.170628	−	0.295537i	0.768011	−	0.640436i	$-0.221246\pi$
$29$	−0.918861	−	1.59151i	−0.170628	−	0.295537i	−0.938640	+	0.344899i	$0.887913\pi$
$30$		0			0
$31$	6.32456			1.13592			0.567962	−	0.823055i	$-0.307732\pi$
$31$	6.32456			1.13592			0.567962	+	0.823055i	$0.307732\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.58114	+	11.3989i	−1.11242	+	1.92676i
$36$		0			0
$37$	−1.91886	−	3.32357i	−0.315459	−	0.546391i	0.664076	−	0.747665i	$-0.268825\pi$
$37$	−1.91886	−	3.32357i	−0.315459	−	0.546391i	−0.979535	+	0.201274i	$0.935492\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	0.959058	−	0.283211i	$-0.0913998\pi$
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	−0.724797	+	0.688963i	$0.758066\pi$
$42$		0			0
$43$	4.58114	−	7.93477i	0.698617	−	1.21004i	−0.270329	−	0.962768i	$-0.587132\pi$
$43$	4.58114	−	7.93477i	0.698617	−	1.21004i	0.968946	−	0.247272i	$-0.0795342\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.83772			0.705654			0.352827	−	0.935689i	$-0.385220\pi$
$47$	4.83772			0.705654			0.352827	+	0.935689i	$0.385220\pi$
$48$		0			0
$49$	−1.50000	−	2.59808i	−0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.4868			1.71520			0.857599	−	0.514319i	$-0.171955\pi$
$53$	12.4868			1.71520			0.857599	+	0.514319i	$0.171955\pi$
$54$		0			0
$55$	−2.41886	−	4.18959i	−0.326159	−	0.564924i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.16228	−	2.01312i	0.151316	−	0.262086i	−0.780396	−	0.625286i	$-0.784982\pi$
$59$	1.16228	−	2.01312i	0.151316	−	0.262086i	0.931711	+	0.363200i	$0.118316\pi$
$60$		0			0
$61$	−0.0811388	+	0.140537i	−0.0103888	+	0.0179939i	−0.871173	−	0.490976i	$-0.836640\pi$
$61$	−0.0811388	+	0.140537i	−0.0103888	+	0.0179939i	0.860784	+	0.508970i	$0.169974\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.337722	−	15.0035i	−0.0418893	−	1.86096i
$66$		0			0
$67$	−1.41886	−	2.45754i	−0.173341	−	0.300236i	0.766245	−	0.642549i	$-0.222123\pi$
$67$	−1.41886	−	2.45754i	−0.173341	−	0.300236i	−0.939586	+	0.342313i	$0.888790\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.58114	−	6.20271i	0.425003	−	0.736127i	−0.571418	−	0.820659i	$-0.693606\pi$
$71$	3.58114	−	6.20271i	0.425003	−	0.736127i	0.996421	+	0.0845326i	$0.0269397\pi$
$72$		0			0
$73$	−1.00000			−0.117041			−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000			−0.117041			−0.0585206	+	0.998286i	$0.518638\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.67544			0.418856
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.48683			−0.382730			−0.191365	−	0.981519i	$-0.561291\pi$
$83$	−3.48683			−0.382730			−0.191365	+	0.981519i	$0.561291\pi$
$84$		0			0
$85$	6.24342	−	10.8139i	0.677194	−	1.17293i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000	+	10.3923i	0.635999	+	1.10158i	0.986303	+	0.164946i	$0.0527450\pi$
$89$	6.00000	+	10.3923i	0.635999	+	1.10158i	−0.350304	+	0.936636i	$0.613922\pi$
$90$		0			0
$91$	10.0000	+	5.47723i	1.04828	+	0.574169i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.7434	−	18.6081i	1.10225	−	1.90916i
$96$		0			0
$97$	2.00000	−	3.46410i	0.203069	−	0.351726i	−0.746447	−	0.665445i	$-0.768242\pi$
$97$	2.00000	−	3.46410i	0.203069	−	0.351726i	0.949516	+	0.313719i	$0.101575\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.91886	+	6.78767i	0.389941	+	0.675398i	0.992441	−	0.122721i	$-0.0391620\pi$
$101$	3.91886	+	6.78767i	0.389941	+	0.675398i	−0.602500	+	0.798119i	$0.705829\pi$
$102$		0			0
$103$	15.8114			1.55794			0.778971	−	0.627060i	$-0.215742\pi$
$103$	15.8114			1.55794			0.778971	+	0.627060i	$0.215742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.90569	+	15.4251i	0.860946	+	1.49120i	0.871017	+	0.491253i	$0.163461\pi$
$107$	8.90569	+	15.4251i	0.860946	+	1.49120i	−0.0100711	+	0.999949i	$0.503206\pi$
$108$		0			0
$109$	6.64911			0.636869			0.318435	−	0.947945i	$-0.396843\pi$
$109$	6.64911			0.636869			0.318435	+	0.947945i	$0.396843\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.33772	−	5.78110i	0.313987	−	0.543841i	−0.665235	−	0.746634i	$-0.731669\pi$
$113$	3.33772	−	5.78110i	0.313987	−	0.543841i	0.979222	+	0.202793i	$0.0650020\pi$
$114$		0			0
$115$	14.9057	+	25.8174i	1.38996	+	2.40749i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.74342	+	8.21584i	0.434828	+	0.753145i
$120$		0			0
$121$	4.82456	−	8.35637i	0.438596	−	0.759670i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−30.4868			−2.72683
$126$		0			0
$127$	−9.16228	−	15.8695i	−0.813021	−	1.40819i	−0.910740	−	0.412979i	$-0.864488\pi$
$127$	−9.16228	−	15.8695i	−0.813021	−	1.40819i	0.0977198	−	0.995214i	$-0.468845\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	16.6491			1.45464			0.727320	−	0.686299i	$-0.240766\pi$
$131$	16.6491			1.45464			0.727320	+	0.686299i	$0.240766\pi$
$132$		0			0
$133$	8.16228	+	14.1375i	0.707759	+	1.22587i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	−0.922961	−	0.384893i	$-0.874238\pi$
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	0.794808	+	0.606861i	$0.207572\pi$
$138$		0			0
$139$	−3.16228	+	5.47723i	−0.268221	+	0.464572i	−0.968402	−	0.249393i	$-0.919769\pi$
$139$	−3.16228	+	5.47723i	−0.268221	+	0.464572i	0.700182	+	0.713965i	$0.253102\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.58114	+	2.17647i	−0.299470	+	0.182005i
$144$		0			0
$145$	3.82456	+	6.62432i	0.317612	+	0.550120i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.243416	−	0.421610i	0.0199415	−	0.0345396i	−0.855882	−	0.517170i	$-0.826985\pi$
$149$	0.243416	−	0.421610i	0.0199415	−	0.0345396i	0.875824	+	0.482631i	$0.160319\pi$
$150$		0			0
$151$	−0.837722			−0.0681729			−0.0340864	−	0.999419i	$-0.510852\pi$
$151$	−0.837722			−0.0681729			−0.0340864	+	0.999419i	$0.510852\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−26.3246			−2.11444
$156$		0			0
$157$	−10.4868			−0.836940			−0.418470	−	0.908231i	$-0.637434\pi$
$157$	−10.4868			−0.836940			−0.418470	+	0.908231i	$0.637434\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−22.6491			−1.78500
$162$		0			0
$163$	−8.00000	+	13.8564i	−0.626608	+	1.08532i	0.361619	+	0.932326i	$0.382224\pi$
$163$	−8.00000	+	13.8564i	−0.626608	+	1.08532i	−0.988227	+	0.152992i	$0.951109\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	0.534875	−	0.844931i	$-0.320359\pi$
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	−0.999169	+	0.0407502i	$0.987025\pi$
$168$		0			0
$169$	−12.9868	+	0.584952i	−0.998987	+	0.0449963i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.3246	−	19.6147i	0.860990	−	1.49128i	−0.00998448	−	0.999950i	$-0.503178\pi$
$173$	11.3246	−	19.6147i	0.860990	−	1.49128i	0.870974	−	0.491328i	$-0.163488\pi$
$174$		0			0
$175$	19.4868	−	33.7522i	1.47307	−	2.55143i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.74342	−	13.4120i	−0.578770	−	1.00246i	−0.995621	−	0.0934843i	$-0.970200\pi$
$179$	−7.74342	−	13.4120i	−0.578770	−	1.00246i	0.416851	−	0.908975i	$-0.363134\pi$
$180$		0			0
$181$	3.83772			0.285256			0.142628	−	0.989776i	$-0.454445\pi$
$181$	3.83772			0.285256			0.142628	+	0.989776i	$0.454445\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.98683	+	13.8336i	0.587204	+	1.01707i
$186$		0			0
$187$	−3.48683			−0.254982
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.16228	+	12.4054i	−0.518244	+	0.897625i	0.481531	+	0.876429i	$0.340081\pi$
$191$	−7.16228	+	12.4054i	−0.518244	+	0.897625i	−0.999775	+	0.0211963i	$0.993252\pi$
$192$		0			0
$193$	9.98683	+	17.2977i	0.718868	+	1.24512i	0.961449	+	0.274985i	$0.0886728\pi$
$193$	9.98683	+	17.2977i	0.718868	+	1.24512i	−0.242581	+	0.970131i	$0.577994\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.48683	+	16.4317i	0.675909	+	1.17071i	0.976202	+	0.216862i	$0.0695820\pi$
$197$	9.48683	+	16.4317i	0.675909	+	1.17071i	−0.300293	+	0.953847i	$0.597085\pi$
$198$		0			0
$199$	−3.25658	+	5.64057i	−0.230853	+	0.399849i	−0.958059	−	0.286570i	$-0.907485\pi$
$199$	−3.25658	+	5.64057i	−0.230853	+	0.399849i	0.727206	+	0.686419i	$0.240818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.81139			−0.407879
$204$		0			0
$205$	−6.24342	−	10.8139i	−0.436059	−	0.755277i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	−2.00000	−	3.46410i	−0.137686	−	0.238479i	0.788935	−	0.614477i	$-0.210633\pi$
$211$	−2.00000	−	3.46410i	−0.137686	−	0.238479i	−0.926620	+	0.375999i	$0.877300\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−19.0680	+	33.0267i	−1.30042	+	2.25240i
$216$		0			0
$217$	10.0000	−	17.3205i	0.678844	−	1.17579i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.48683	−	5.19615i	−0.638153	−	0.349531i
$222$		0			0
$223$	−0.837722	−	1.45098i	−0.0560980	−	0.0971647i	0.836613	−	0.547795i	$-0.184533\pi$
$223$	−0.837722	−	1.45098i	−0.0560980	−	0.0971647i	−0.892711	+	0.450630i	$0.851199\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.74342	−	13.4120i	0.513949	−	0.890185i	−0.485920	−	0.874003i	$-0.661516\pi$
$227$	7.74342	−	13.4120i	0.513949	−	0.890185i	0.999869	−	0.0161821i	$-0.00515113\pi$
$228$		0			0
$229$	22.3246			1.47525			0.737624	−	0.675212i	$-0.235948\pi$
$229$	22.3246			1.47525			0.737624	+	0.675212i	$0.235948\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.6491			1.09072			0.545360	−	0.838202i	$-0.316393\pi$
$233$	16.6491			1.09072			0.545360	+	0.838202i	$0.316393\pi$
$234$		0			0
$235$	−20.1359			−1.31352
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.4868			−1.38987			−0.694934	−	0.719074i	$-0.744566\pi$
$239$	−21.4868			−1.38987			−0.694934	+	0.719074i	$0.744566\pi$
$240$		0			0
$241$	−6.66228	+	11.5394i	−0.429155	+	0.743318i	−0.996798	−	0.0799563i	$-0.974522\pi$
$241$	−6.66228	+	11.5394i	−0.429155	+	0.743318i	0.567643	+	0.823275i	$0.307855\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.24342	+	10.8139i	0.398877	+	0.690876i
$246$		0			0
$247$	−16.3246	−	8.94133i	−1.03871	−	0.568923i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.00000	−	10.3923i	0.378717	−	0.655956i	−0.612159	−	0.790735i	$-0.709699\pi$
$251$	6.00000	−	10.3923i	0.378717	−	0.655956i	0.990876	+	0.134778i	$0.0430322\pi$
$252$		0			0
$253$	4.16228	−	7.20928i	0.261680	−	0.453243i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.9868	−	19.0298i	−0.685340	−	1.18704i	−0.973330	−	0.229410i	$-0.926320\pi$
$257$	−10.9868	−	19.0298i	−0.685340	−	1.18704i	0.287990	−	0.957633i	$-0.407013\pi$
$258$		0			0
$259$	−12.1359			−0.754091
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.25658	+	2.17647i	0.0774843	+	0.134207i	0.902164	−	0.431393i	$-0.141978\pi$
$263$	1.25658	+	2.17647i	0.0774843	+	0.134207i	−0.824680	+	0.565600i	$0.808645\pi$
$264$		0			0
$265$	−51.9737			−3.19272
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000	−	5.19615i	0.182913	−	0.316815i	−0.759958	−	0.649972i	$-0.774781\pi$
$269$	3.00000	−	5.19615i	0.182913	−	0.316815i	0.942871	+	0.333157i	$0.108114\pi$
$270$		0			0
$271$	−3.16228	−	5.47723i	−0.192095	−	0.332718i	0.753850	−	0.657047i	$-0.228195\pi$
$271$	−3.16228	−	5.47723i	−0.192095	−	0.332718i	−0.945944	+	0.324329i	$0.894861\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.16228	+	12.4054i	0.431902	+	0.748076i
$276$		0			0
$277$	−5.40569	+	9.36294i	−0.324797	+	0.562564i	−0.981471	−	0.191610i	$-0.938629\pi$
$277$	−5.40569	+	9.36294i	−0.324797	+	0.562564i	0.656675	+	0.754174i	$0.271963\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.675445			0.0402937			0.0201468	−	0.999797i	$-0.493587\pi$
$281$	0.675445			0.0402937			0.0201468	+	0.999797i	$0.493587\pi$
$282$		0			0
$283$	−1.41886	−	2.45754i	−0.0843425	−	0.146086i	0.820768	−	0.571261i	$-0.193546\pi$
$283$	−1.41886	−	2.45754i	−0.0843425	−	0.146086i	−0.905111	+	0.425176i	$0.860212\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.48683			0.559990
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.918861	−	1.59151i	0.0536804	−	0.0929773i	−0.837937	−	0.545768i	$-0.816238\pi$
$293$	0.918861	−	1.59151i	0.0536804	−	0.0929773i	0.891617	+	0.452790i	$0.149571\pi$
$294$		0			0
$295$	−4.83772	+	8.37918i	−0.281663	+	0.487855i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.0680	−	13.4120i	1.27622	−	0.775635i
$300$		0			0
$301$	−14.4868	−	25.0919i	−0.835007	−	1.44627i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.337722	−	0.584952i	0.0193379	−	0.0334943i
$306$		0			0
$307$	−11.4868			−0.655588			−0.327794	−	0.944749i	$-0.606305\pi$
$307$	−11.4868			−0.655588			−0.327794	+	0.944749i	$0.606305\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	21.4868			1.21841			0.609203	−	0.793014i	$-0.291489\pi$
$311$	21.4868			1.21841			0.609203	+	0.793014i	$0.291489\pi$
$312$		0			0
$313$	−4.00000			−0.226093			−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000			−0.226093			−0.113047	+	0.993590i	$0.536061\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.4868			−0.701330			−0.350665	−	0.936501i	$-0.614044\pi$
$317$	−12.4868			−0.701330			−0.350665	+	0.936501i	$0.614044\pi$
$318$		0			0
$319$	1.06797	−	1.84978i	0.0597949	−	0.103568i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.74342	−	13.4120i	−0.430855	−	0.746263i
$324$		0			0
$325$	1.00000	+	44.4256i	0.0554700	+	2.46429i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.64911	−	13.2486i	0.421709	−	0.730422i
$330$		0			0
$331$	−5.48683	+	9.50347i	−0.301584	+	0.522358i	−0.976495	−	0.215541i	$-0.930849\pi$
$331$	−5.48683	+	9.50347i	−0.301584	+	0.522358i	0.674911	+	0.737899i	$0.264182\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.90569	+	10.2290i	0.322663	+	0.558868i
$336$		0			0
$337$	11.0000			0.599208			0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000			0.599208			0.299604	+	0.954064i	$0.403145\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.67544	+	6.36606i	0.199036	+	0.344741i
$342$		0			0
$343$	12.6491			0.682988
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.74342	−	13.4120i	0.415688	−	0.719993i	−0.579812	−	0.814750i	$-0.696874\pi$
$347$	7.74342	−	13.4120i	0.415688	−	0.719993i	0.995500	+	0.0947569i	$0.0302074\pi$
$348$		0			0
$349$	2.67544	+	4.63401i	0.143213	+	0.248053i	0.928705	−	0.370820i	$-0.120923\pi$
$349$	2.67544	+	4.63401i	0.143213	+	0.248053i	−0.785492	+	0.618872i	$0.787590\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.6623	+	25.3958i	0.780394	+	1.35168i	0.931712	+	0.363197i	$0.118315\pi$
$353$	14.6623	+	25.3958i	0.780394	+	1.35168i	−0.151318	+	0.988485i	$0.548352\pi$
$354$		0			0
$355$	−14.9057	+	25.8174i	−0.791112	+	1.37025i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.4605			1.50209			0.751044	−	0.660252i	$-0.229551\pi$
$359$	28.4605			1.50209			0.751044	+	0.660252i	$0.229551\pi$
$360$		0			0
$361$	−3.82456	−	6.62432i	−0.201292	−	0.348649i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.16228			0.217864
$366$		0			0
$367$	−12.7434	−	22.0722i	−0.665201	−	1.15216i	−0.979231	−	0.202749i	$-0.935013\pi$
$367$	−12.7434	−	22.0722i	−0.665201	−	1.15216i	0.314030	−	0.949413i	$-0.398321\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.7434	−	34.1966i	1.02503	−	1.77540i
$372$		0			0
$373$	8.24342	−	14.2780i	0.426828	−	0.739288i	−0.569761	−	0.821810i	$-0.692964\pi$
$373$	8.24342	−	14.2780i	0.426828	−	0.739288i	0.996589	+	0.0825226i	$0.0262977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.66228	−	3.44130i	0.291622	−	0.177236i
$378$		0			0
$379$	8.83772	+	15.3074i	0.453963	+	0.786288i	0.998628	−	0.0523664i	$-0.0166764\pi$
$379$	8.83772	+	15.3074i	0.453963	+	0.786288i	−0.544665	+	0.838654i	$0.683343\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.4868	+	26.8240i	−0.791340	+	1.37064i	0.133797	+	0.991009i	$0.457283\pi$
$383$	−15.4868	+	26.8240i	−0.791340	+	1.37064i	−0.925137	+	0.379633i	$0.876050\pi$
$384$		0			0
$385$	−15.2982			−0.779670
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.4868			−0.633108			−0.316554	−	0.948575i	$-0.602526\pi$
$389$	−12.4868			−0.633108			−0.316554	+	0.948575i	$0.602526\pi$
$390$		0			0
$391$	21.4868			1.08664
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.6491			−0.837708
$396$		0			0
$397$	−13.0000	+	22.5167i	−0.652451	+	1.13008i	0.330075	+	0.943955i	$0.392926\pi$
$397$	−13.0000	+	22.5167i	−0.652451	+	1.13008i	−0.982526	+	0.186124i	$0.940407\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.50000	−	12.9904i	−0.374532	−	0.648709i	0.615725	−	0.787961i	$-0.288863\pi$
$401$	−7.50000	−	12.9904i	−0.374532	−	0.648709i	−0.990257	+	0.139253i	$0.955530\pi$
$402$		0			0
$403$	0.513167	+	22.7977i	0.0255627	+	1.13564i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.23025	−	3.86291i	0.110549	−	0.191477i
$408$		0			0
$409$	−7.33772	+	12.7093i	−0.362827	+	0.628435i	−0.988425	−	0.151711i	$-0.951522\pi$
$409$	−7.33772	+	12.7093i	−0.362827	+	0.628435i	0.625598	+	0.780146i	$0.284855\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.67544	−	6.36606i	−0.180857	−	0.313253i
$414$		0			0
$415$	14.5132			0.712423
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15.4868	+	26.8240i	0.756581	+	1.31044i	0.944584	+	0.328269i	$0.106465\pi$
$419$	15.4868	+	26.8240i	0.756581	+	1.31044i	−0.188003	+	0.982168i	$0.560201\pi$
$420$		0			0
$421$	−23.8377			−1.16178			−0.580890	−	0.813982i	$-0.697295\pi$
$421$	−23.8377			−1.16178			−0.580890	+	0.813982i	$0.697295\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18.4868	+	32.0201i	−0.896743	+	1.55320i
$426$		0			0
$427$	0.256584	+	0.444416i	0.0124169	+	0.0215068i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.74342	+	8.21584i	0.228482	+	0.395743i	0.957359	−	0.288903i	$-0.0932904\pi$
$431$	4.74342	+	8.21584i	0.228482	+	0.395743i	−0.728876	+	0.684646i	$0.759957\pi$
$432$		0			0
$433$	4.66228	−	8.07530i	0.224055	−	0.388074i	−0.731981	−	0.681325i	$-0.761404\pi$
$433$	4.66228	−	8.07530i	0.224055	−	0.388074i	0.956035	+	0.293251i	$0.0947372\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	36.9737			1.76869
$438$		0			0
$439$	−12.7434	−	22.0722i	−0.608210	−	1.05345i	−0.991535	−	0.129837i	$-0.958554\pi$
$439$	−12.7434	−	22.0722i	−0.608210	−	1.05345i	0.383325	−	0.923613i	$-0.374779\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	−24.9737	−	43.2557i	−1.18387	−	2.05051i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.67544	−	6.36606i	0.173455	−	0.300433i	−0.766171	−	0.642637i	$-0.777840\pi$
$449$	3.67544	−	6.36606i	0.173455	−	0.300433i	0.939625	+	0.342205i	$0.111174\pi$
$450$		0			0
$451$	−1.74342	+	3.01969i	−0.0820943	+	0.142191i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−41.6228	−	22.7977i	−1.95131	−	1.06877i
$456$		0			0
$457$	1.66228	+	2.87915i	0.0777581	+	0.134681i	0.902282	−	0.431146i	$-0.141890\pi$
$457$	1.66228	+	2.87915i	0.0777581	+	0.134681i	−0.824524	+	0.565827i	$0.808557\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.24342	−	16.0101i	0.430509	−	0.745663i	−0.566408	−	0.824125i	$-0.691667\pi$
$461$	9.24342	−	16.0101i	0.430509	−	0.745663i	0.996917	+	0.0784616i	$0.0250008\pi$
$462$		0			0
$463$	−15.1623			−0.704651			−0.352325	−	0.935878i	$-0.614609\pi$
$463$	−15.1623			−0.704651			−0.352325	+	0.935878i	$0.614609\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.18861			0.286375			0.143187	−	0.989696i	$-0.454265\pi$
$467$	6.18861			0.286375			0.143187	+	0.989696i	$0.454265\pi$
$468$		0			0
$469$	−8.97367			−0.414365
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.6491			0.489647
$474$		0			0
$475$	−31.8114	+	55.0989i	−1.45961	+	2.52811i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		0			0
$481$	11.8246	−	7.18647i	0.539153	−	0.327675i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.32456	+	14.4186i	−0.377999	+	0.654713i
$486$		0			0
$487$	−6.74342	+	11.6799i	−0.305573	+	0.529269i	−0.977389	−	0.211450i	$-0.932181\pi$
$487$	−6.74342	+	11.6799i	−0.305573	+	0.529269i	0.671815	+	0.740719i	$0.265515\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.90569	−	5.03281i	−0.131132	−	0.227128i	0.792981	−	0.609246i	$-0.208528\pi$
$491$	−2.90569	−	5.03281i	−0.131132	−	0.227128i	−0.924113	+	0.382119i	$0.875195\pi$
$492$		0			0
$493$	5.51317			0.248301
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.3246	−	19.6147i	−0.507976	−	0.879840i
$498$		0			0
$499$	−12.6491			−0.566252			−0.283126	−	0.959083i	$-0.591371\pi$
$499$	−12.6491			−0.566252			−0.283126	+	0.959083i	$0.591371\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.0943058	−	0.163343i	0.00420489	−	0.00728308i	−0.863915	−	0.503637i	$-0.831995\pi$
$503$	0.0943058	−	0.163343i	0.00420489	−	0.00728308i	0.868120	+	0.496354i	$0.165328\pi$
$504$		0			0
$505$	−16.3114	−	28.2522i	−0.725847	−	1.25720i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.0811	−	24.3892i	−0.624136	−	1.08103i	−0.988707	−	0.149859i	$-0.952118\pi$
$509$	−14.0811	−	24.3892i	−0.624136	−	1.08103i	0.364572	−	0.931175i	$-0.381215\pi$
$510$		0			0
$511$	−1.58114	+	2.73861i	−0.0699455	+	0.121149i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−65.8114			−2.90000
$516$		0			0
$517$	2.81139	+	4.86947i	0.123645	+	0.214159i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.0000			1.18289			0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000			1.18289			0.591446	+	0.806345i	$0.298557\pi$
$522$		0			0
$523$	−2.58114	−	4.47066i	−0.112865	−	0.195488i	0.804059	−	0.594549i	$-0.202670\pi$
$523$	−2.58114	−	4.47066i	−0.112865	−	0.195488i	−0.916924	+	0.399061i	$0.869336\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.48683	+	16.4317i	−0.413253	+	0.715775i
$528$		0			0
$529$	−14.1491	+	24.5070i	−0.615179	+	1.06552i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.24342	+	5.61776i	−0.400377	+	0.243332i
$534$		0			0
$535$	−37.0680	−	64.2036i	−1.60259	−	2.77576i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.74342	−	3.01969i	0.0750943	−	0.130067i
$540$		0			0
$541$	−15.5132			−0.666963			−0.333482	−	0.942757i	$-0.608223\pi$
$541$	−15.5132			−0.666963			−0.333482	+	0.942757i	$0.608223\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−27.6754			−1.18549
$546$		0			0
$547$	33.8114			1.44567			0.722835	−	0.691020i	$-0.242839\pi$
$547$	33.8114			1.44567			0.722835	+	0.691020i	$0.242839\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.48683			0.404153
$552$		0			0
$553$	6.32456	−	10.9545i	0.268947	−	0.465831i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.24342	−	16.0101i	−0.391656	−	0.678368i	0.601012	−	0.799240i	$-0.294764\pi$
$557$	−9.24342	−	16.0101i	−0.391656	−	0.678368i	−0.992668	+	0.120872i	$0.961431\pi$
$558$		0			0
$559$	28.9737	+	15.8695i	1.22546	+	0.671210i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.3246	+	35.2032i	−0.856578	+	1.48364i	0.0185956	+	0.999827i	$0.494081\pi$
$563$	−20.3246	+	35.2032i	−0.856578	+	1.48364i	−0.875173	+	0.483809i	$0.839253\pi$
$564$		0			0
$565$	−13.8925	+	24.0626i	−0.584463	+	1.01232i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.6491	+	23.6410i	0.572200	+	0.991080i	0.996340	+	0.0854833i	$0.0272434\pi$
$569$	13.6491	+	23.6410i	0.572200	+	0.991080i	−0.424139	+	0.905597i	$0.639423\pi$
$570$		0			0
$571$	36.1359			1.51224			0.756121	−	0.654432i	$-0.227092\pi$
$571$	36.1359			1.51224			0.756121	+	0.654432i	$0.227092\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−44.1359	−	76.4457i	−1.84060	−	3.18801i
$576$		0			0
$577$	0.350889			0.0146077			0.00730386	−	0.999973i	$-0.497675\pi$
$577$	0.350889			0.0146077			0.00730386	+	0.999973i	$0.497675\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.51317	+	9.54909i	−0.228725	+	0.396163i
$582$		0			0
$583$	7.25658	+	12.5688i	0.300537	+	0.520545i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.3246	−	24.8109i	−0.591238	−	1.02405i	−0.994066	−	0.108778i	$-0.965306\pi$
$587$	−14.3246	−	24.8109i	−0.591238	−	1.02405i	0.402828	−	0.915276i	$-0.368027\pi$
$588$		0			0
$589$	−16.3246	+	28.2750i	−0.672642	+	1.16505i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.675445			−0.0277372			−0.0138686	−	0.999904i	$-0.504415\pi$
$593$	−0.675445			−0.0277372			−0.0138686	+	0.999904i	$0.504415\pi$
$594$		0			0
$595$	−19.7434	−	34.1966i	−0.809401	−	1.40192i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−23.6228			−0.965200			−0.482600	−	0.875841i	$-0.660308\pi$
$599$	−23.6228			−0.965200			−0.482600	+	0.875841i	$0.660308\pi$
$600$		0			0
$601$	17.1491	+	29.7031i	0.699527	+	1.21162i	0.968631	+	0.248505i	$0.0799392\pi$
$601$	17.1491	+	29.7031i	0.699527	+	1.21162i	−0.269104	+	0.963111i	$0.586727\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−20.0811	+	34.7816i	−0.816414	+	1.41407i
$606$		0			0
$607$	8.64911	−	14.9807i	0.351057	−	0.608048i	−0.635378	−	0.772201i	$-0.719156\pi$
$607$	8.64911	−	14.9807i	0.351057	−	0.608048i	0.986435	+	0.164153i	$0.0524891\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.392527	+	17.4382i	0.0158799	+	0.705476i
$612$		0			0
$613$	−10.2434	−	17.7421i	−0.413728	−	0.716597i	0.581566	−	0.813499i	$-0.302440\pi$
$613$	−10.2434	−	17.7421i	−0.413728	−	0.716597i	−0.995294	+	0.0969016i	$0.969107\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.3114	+	23.0560i	−0.535896	+	0.928200i	0.463223	+	0.886242i	$0.346693\pi$
$617$	−13.3114	+	23.0560i	−0.535896	+	0.928200i	−0.999119	+	0.0419579i	$0.986640\pi$
$618$		0			0
$619$	−24.6491			−0.990731			−0.495366	−	0.868685i	$-0.664966\pi$
$619$	−24.6491			−0.990731			−0.495366	+	0.868685i	$0.664966\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	37.9473			1.52033
$624$		0			0
$625$	65.2719			2.61088
$626$		0			0
$627$		0			0
$628$		0			0
$629$	11.5132			0.459060
$630$		0			0
$631$	−12.6491	+	21.9089i	−0.503553	+	0.872180i	0.496438	+	0.868072i	$0.334641\pi$
$631$	−12.6491	+	21.9089i	−0.503553	+	0.872180i	−0.999992	+	0.00410769i	$0.998692\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	38.1359	+	66.0534i	1.51338	+	2.62125i
$636$		0			0
$637$	9.24342	−	5.61776i	0.366237	−	0.222584i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.17544	−	14.1603i	0.322911	−	0.559298i	−0.658177	−	0.752864i	$-0.728672\pi$
$641$	8.17544	−	14.1603i	0.322911	−	0.559298i	0.981087	+	0.193566i	$0.0620053\pi$
$642$		0			0
$643$	10.0000	−	17.3205i	0.394362	−	0.683054i	−0.598658	−	0.801005i	$-0.704299\pi$
$643$	10.0000	−	17.3205i	0.394362	−	0.683054i	0.993019	+	0.117951i	$0.0376325\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.3246	+	35.2032i	0.799041	+	1.38398i	0.920241	+	0.391351i	$0.127992\pi$
$647$	20.3246	+	35.2032i	0.799041	+	1.38398i	−0.121201	+	0.992628i	$0.538674\pi$
$648$		0			0
$649$	2.70178			0.106054
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	0.994623	+	0.103558i	$0.0330227\pi$
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	−0.407628	+	0.913148i	$0.633644\pi$
$654$		0			0
$655$	−69.2982			−2.70771
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.51317	−	4.35293i	0.0978991	−	0.169566i	−0.812916	−	0.582381i	$-0.802121\pi$
$659$	2.51317	−	4.35293i	0.0978991	−	0.169566i	0.910815	+	0.412815i	$0.135454\pi$
$660$		0			0
$661$	−13.2434	−	22.9383i	−0.515109	−	0.892195i	−0.999846	−	0.0175354i	$-0.994418\pi$
$661$	−13.2434	−	22.9383i	−0.515109	−	0.892195i	0.484737	−	0.874660i	$-0.338915\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−33.9737	−	58.8441i	−1.31744	−	2.28188i
$666$		0			0
$667$	−6.58114	+	11.3989i	−0.254823	+	0.441366i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.188612			−0.00728127
$672$		0			0
$673$	−7.33772	−	12.7093i	−0.282848	−	0.489908i	0.689237	−	0.724536i	$-0.257946\pi$
$673$	−7.33772	−	12.7093i	−0.282848	−	0.489908i	−0.972085	+	0.234628i	$0.924613\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−6.32456	−	10.9545i	−0.242714	−	0.420393i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17.8114	−	30.8502i	0.681534	−	1.18045i	−0.292979	−	0.956119i	$-0.594647\pi$
$683$	17.8114	−	30.8502i	0.681534	−	1.18045i	0.974513	−	0.224332i	$-0.0720201\pi$
$684$		0			0
$685$	6.24342	−	10.8139i	0.238549	−	0.413178i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.01317	+	45.0105i	0.0385986	+	1.71476i
$690$		0			0
$691$	4.58114	+	7.93477i	0.174275	+	0.301853i	0.939910	−	0.341422i	$-0.110909\pi$
$691$	4.58114	+	7.93477i	0.174275	+	0.301853i	−0.765635	+	0.643275i	$0.777575\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.1623	−	22.7977i	0.499274	−	0.864767i
$696$		0			0
$697$	−9.00000			−0.340899
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	19.8114			0.747201
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.7851			0.932138
$708$		0			0
$709$	−22.7302	+	39.3699i	−0.853652	+	1.47857i	0.0242371	+	0.999706i	$0.492284\pi$
$709$	−22.7302	+	39.3699i	−0.853652	+	1.47857i	−0.877890	+	0.478863i	$0.841049\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−22.6491	−	39.2294i	−0.848216	−	1.46915i
$714$		0			0
$715$	14.9057	−	9.05906i	0.557441	−	0.338790i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.1623	−	33.1900i	0.714632	−	1.23778i	−0.248469	−	0.968640i	$-0.579927\pi$
$719$	19.1623	−	33.1900i	0.714632	−	1.23778i	0.963101	−	0.269140i	$-0.0867393\pi$
$720$		0			0
$721$	25.0000	−	43.3013i	0.931049	−	1.61262i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11.3246	−	19.6147i	−0.420583	−	0.728472i
$726$		0			0
$727$	8.83772			0.327773			0.163886	−	0.986479i	$-0.447597\pi$
$727$	8.83772			0.327773			0.163886	+	0.986479i	$0.447597\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13.7434	+	23.8043i	0.508319	+	0.880434i
$732$		0			0
$733$	21.4605			0.792662			0.396331	−	0.918108i	$-0.370283\pi$
$733$	21.4605			0.792662			0.396331	+	0.918108i	$0.370283\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.64911	−	2.85634i	0.0607458	−	0.105215i
$738$		0			0
$739$	−19.8114	−	34.3143i	−0.728774	−	1.26227i	−0.957402	−	0.288759i	$-0.906757\pi$
$739$	−19.8114	−	34.3143i	−0.728774	−	1.26227i	0.228628	−	0.973514i	$-0.426576\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.16228	−	2.01312i	−0.0426398	−	0.0738544i	0.843918	−	0.536472i	$-0.180243\pi$
$743$	−1.16228	−	2.01312i	−0.0426398	−	0.0738544i	−0.886558	+	0.462618i	$0.846910\pi$
$744$		0			0
$745$	−1.01317	+	1.75486i	−0.0371196	+	0.0642930i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	56.3246			2.05805
$750$		0			0
$751$	19.3925	+	33.5888i	0.707643	+	1.22567i	0.965729	+	0.259552i	$0.0835750\pi$
$751$	19.3925	+	33.5888i	0.707643	+	1.22567i	−0.258086	+	0.966122i	$0.583092\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.48683			0.126899
$756$		0			0
$757$	13.3246	+	23.0788i	0.484289	+	0.838814i	0.999837	−	0.0180472i	$-0.00574493\pi$
$757$	13.3246	+	23.0788i	0.484289	+	0.838814i	−0.515548	+	0.856861i	$0.672412\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000	+	10.3923i	−0.217500	+	0.376721i	−0.954043	−	0.299670i	$-0.903123\pi$
$761$	−6.00000	+	10.3923i	−0.217500	+	0.376721i	0.736543	+	0.676391i	$0.236457\pi$
$762$		0			0
$763$	10.5132	−	18.2093i	0.380602	−	0.659222i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.35089	+	4.02625i	0.265425	+	0.145379i
$768$		0			0
$769$	−1.67544	−	2.90196i	−0.0604181	−	0.104647i	0.834234	−	0.551410i	$-0.185910\pi$
$769$	−1.67544	−	2.90196i	−0.0604181	−	0.104647i	−0.894652	+	0.446763i	$0.852577\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.3246	−	19.6147i	0.407316	−	0.705492i	−0.587272	−	0.809390i	$-0.699798\pi$
$773$	11.3246	−	19.6147i	0.407316	−	0.705492i	0.994588	+	0.103898i	$0.0331315\pi$
$774$		0			0
$775$	77.9473			2.79995
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.4868			−0.554873
$780$		0			0
$781$	8.32456			0.297876
$782$		0			0
$783$		0			0
$784$		0			0
$785$	43.6491			1.55790
$786$		0			0
$787$	−4.51317	+	7.81703i	−0.160877	+	0.278647i	−0.935183	−	0.354164i	$-0.884766\pi$
$787$	−4.51317	+	7.81703i	−0.160877	+	0.278647i	0.774306	+	0.632811i	$0.218099\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10.5548	−	18.2815i	−0.375286	−	0.650014i
$792$		0			0
$793$	−0.513167	−	0.281073i	−0.0182231	−	0.00998120i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.48683	+	16.4317i	−0.336041	+	0.582040i	−0.983684	−	0.179904i	$-0.942421\pi$
$797$	−9.48683	+	16.4317i	−0.336041	+	0.582040i	0.647643	+	0.761944i	$0.275755\pi$
$798$		0			0
$799$	−7.25658	+	12.5688i	−0.256719	+	0.444651i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.581139	−	1.00656i	−0.0205079	−	0.0355208i
$804$		0			0
$805$	94.2719			3.32265
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.50000	+	2.59808i	0.0527372	+	0.0913435i	0.891189	−	0.453632i	$-0.149872\pi$
$809$	1.50000	+	2.59808i	0.0527372	+	0.0913435i	−0.838452	+	0.544976i	$0.816539\pi$
$810$		0			0
$811$	−1.02633			−0.0360395			−0.0180197	−	0.999838i	$-0.505736\pi$
$811$	−1.02633			−0.0360395			−0.0180197	+	0.999838i	$0.505736\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	33.2982	−	57.6742i	1.16639	−	2.02024i
$816$		0			0
$817$	23.6491	+	40.9615i	0.827378	+	1.43306i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.32456	+	9.22240i	0.185828	+	0.321864i	0.943855	−	0.330359i	$-0.107170\pi$
$821$	5.32456	+	9.22240i	0.185828	+	0.321864i	−0.758027	+	0.652223i	$0.773837\pi$
$822$		0			0
$823$	26.6491	−	46.1576i	0.928930	−	1.60895i	0.143813	−	0.989605i	$-0.454064\pi$
$823$	26.6491	−	46.1576i	0.928930	−	1.60895i	0.785116	−	0.619348i	$-0.212603\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.97367			−0.242498			−0.121249	−	0.992622i	$-0.538690\pi$
$827$	−6.97367			−0.242498			−0.121249	+	0.992622i	$0.538690\pi$
$828$		0			0
$829$	−6.08114	−	10.5328i	−0.211207	−	0.365821i	0.740886	−	0.671631i	$-0.234406\pi$
$829$	−6.08114	−	10.5328i	−0.211207	−	0.365821i	−0.952092	+	0.305810i	$0.901073\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.00000			0.311832
$834$		0			0
$835$	24.9737	+	43.2557i	0.864249	+	1.49692i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.32456	+	4.02625i	−0.0802526	+	0.139002i	−0.903358	−	0.428887i	$-0.858906\pi$
$839$	−2.32456	+	4.02625i	−0.0802526	+	0.139002i	0.823106	+	0.567888i	$0.192239\pi$
$840$		0			0
$841$	12.8114	−	22.1900i	0.441772	−	0.765172i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	54.0548	−	2.43473i	1.85954	−	0.0837574i
$846$		0			0
$847$	−15.2566	−	26.4252i	−0.524222	−	0.907980i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.7434	+	23.8043i	−0.471118	+	0.816001i
$852$		0			0
$853$	55.1359			1.88782			0.943909	−	0.330205i	$-0.107118\pi$
$853$	55.1359			1.88782			0.943909	+	0.330205i	$0.107118\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.6491			0.876157			0.438078	−	0.898937i	$-0.355659\pi$
$857$	25.6491			0.876157			0.438078	+	0.898937i	$0.355659\pi$
$858$		0			0
$859$	−28.5132			−0.972857			−0.486428	−	0.873720i	$-0.661701\pi$
$859$	−28.5132			−0.972857			−0.486428	+	0.873720i	$0.661701\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	47.8114			1.62752			0.813759	−	0.581202i	$-0.197417\pi$
$863$	47.8114			1.62752			0.813759	+	0.581202i	$0.197417\pi$
$864$		0			0
$865$	−47.1359	+	81.6418i	−1.60267	+	2.77591i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.32456	+	4.02625i	0.0788551	+	0.136581i
$870$		0			0
$871$	8.74342	−	5.31388i	0.296259	−	0.180054i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−48.2039	+	83.4916i	−1.62959	+	2.82253i
$876$		0			0
$877$	22.5680	−	39.0889i	0.762066	−	1.31994i	−0.179717	−	0.983718i	$-0.557518\pi$
$877$	22.5680	−	39.0889i	0.762066	−	1.31994i	0.941784	−	0.336219i	$-0.109148\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.98683	+	8.63745i	0.168011	+	0.291003i	0.937720	−	0.347391i	$-0.112932\pi$
$881$	4.98683	+	8.63745i	0.168011	+	0.291003i	−0.769710	+	0.638394i	$0.779599\pi$
$882$		0			0
$883$	11.3509			0.381988			0.190994	−	0.981591i	$-0.438829\pi$
$883$	11.3509			0.381988			0.190994	+	0.981591i	$0.438829\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−19.1623	−	33.1900i	−0.643406	−	1.11441i	−0.984667	−	0.174444i	$-0.944187\pi$
$887$	−19.1623	−	33.1900i	−0.643406	−	1.11441i	0.341261	−	0.939969i	$-0.389146\pi$
$888$		0			0
$889$	−57.9473			−1.94349
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.4868	+	21.6278i	−0.417856	+	0.723748i
$894$		0			0
$895$	32.2302	+	55.8244i	1.07734	+	1.86600i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.81139	−	10.0656i	−0.193821	−	0.335707i
$900$		0			0
$901$	−18.7302	+	32.4417i	−0.623995	+	1.08079i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.9737			−0.530983
$906$		0			0
$907$	18.1359	+	31.4124i	0.602194	+	1.04303i	0.992488	+	0.122341i	$0.0390400\pi$
$907$	18.1359	+	31.4124i	0.602194	+	1.04303i	−0.390294	+	0.920690i	$0.627627\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−49.9473			−1.65483			−0.827414	−	0.561592i	$-0.810189\pi$
$911$	−49.9473			−1.65483			−0.827414	+	0.561592i	$0.810189\pi$
$912$		0			0
$913$	−2.02633	−	3.50971i	−0.0670619	−	0.116155i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	26.3246	−	45.5955i	0.869313	−	1.50569i
$918$		0			0
$919$	1.67544	−	2.90196i	0.0552678	−	0.0957267i	−0.837068	−	0.547099i	$-0.815732\pi$
$919$	1.67544	−	2.90196i	0.0552678	−	0.0957267i	0.892336	+	0.451372i	$0.149065\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	22.6491	+	12.4054i	0.745505	+	0.408330i
$924$		0			0
$925$	−23.6491	−	40.9615i	−0.777578	−	1.34680i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.4737	−	25.0691i	0.474866	−	0.822491i	−0.524720	−	0.851275i	$-0.675830\pi$
$929$	14.4737	−	25.0691i	0.474866	−	0.822491i	0.999586	+	0.0287835i	$0.00916334\pi$
$930$		0			0
$931$	15.4868			0.507560
$932$		0			0
$933$		0			0
$934$		0			0
$935$	14.5132			0.474631
$936$		0			0
$937$	14.2982			0.467103			0.233551	−	0.972344i	$-0.424965\pi$
$937$	14.2982			0.467103			0.233551	+	0.972344i	$0.424965\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	48.5964			1.58420			0.792099	−	0.610392i	$-0.208988\pi$
$941$	48.5964			1.58420			0.792099	+	0.610392i	$0.208988\pi$
$942$		0			0
$943$	10.7434	−	18.6081i	0.349854	−	0.605965i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.48683	−	16.4317i	−0.308281	−	0.533958i	0.669706	−	0.742627i	$-0.266420\pi$
$947$	−9.48683	−	16.4317i	−0.308281	−	0.533958i	−0.977986	+	0.208669i	$0.933087\pi$
$948$		0			0
$949$	−0.0811388	−	3.60464i	−0.00263388	−	0.117012i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−13.6491	+	23.6410i	−0.442138	+	0.765806i	−0.997848	−	0.0655707i	$-0.979113\pi$
$953$	−13.6491	+	23.6410i	−0.442138	+	0.765806i	0.555710	+	0.831376i	$0.312447\pi$
$954$		0			0
$955$	29.8114	−	51.6348i	0.964674	−	1.67086i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.74342	+	8.21584i	0.153173	+	0.265303i
$960$		0			0
$961$	9.00000			0.290323
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−41.5680	−	71.9978i	−1.33812	−	2.31769i
$966$		0			0
$967$	−34.5132			−1.10987			−0.554934	−	0.831894i	$-0.687257\pi$
$967$	−34.5132			−1.10987			−0.554934	+	0.831894i	$0.687257\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.48683	−	16.4317i	0.304447	−	0.527318i	−0.672691	−	0.739923i	$-0.734862\pi$
$971$	9.48683	−	16.4317i	0.304447	−	0.527318i	0.977138	+	0.212606i	$0.0681950\pi$
$972$		0			0
$973$	10.0000	+	17.3205i	0.320585	+	0.555270i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.5000	+	18.1865i	0.335925	+	0.581839i	0.983662	−	0.180025i	$-0.0576179\pi$
$977$	10.5000	+	18.1865i	0.335925	+	0.581839i	−0.647737	+	0.761864i	$0.724285\pi$
$978$		0			0
$979$	−6.97367	+	12.0787i	−0.222879	+	0.386038i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23.6228			0.753450			0.376725	−	0.926325i	$-0.377050\pi$
$983$	23.6228			0.753450			0.376725	+	0.926325i	$0.377050\pi$
$984$		0			0
$985$	−39.4868	−	68.3932i	−1.25816	−	2.17919i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−65.6228			−2.08668
$990$		0			0
$991$	13.3925	+	23.1965i	0.425428	+	0.736862i	0.996460	−	0.0840651i	$-0.0267904\pi$
$991$	13.3925	+	23.1965i	0.425428	+	0.736862i	−0.571033	+	0.820927i	$0.693457\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	13.5548	−	23.4776i	0.429716	−	0.744290i
$996$		0			0
$997$	29.7302	−	51.4943i	0.941566	−	1.63084i	0.179082	−	0.983834i	$-0.442687\pi$
$997$	29.7302	−	51.4943i	0.941566	−	1.63084i	0.762484	−	0.647007i	$-0.223979\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.t.n.289.1		4
3.2	odd	2		1872.2.t.p.289.2		4
4.3	odd	2		234.2.h.e.55.1	yes	4
12.11	even	2		234.2.h.d.55.2	✓	4
13.9	even	3	inner	1872.2.t.n.1153.1		4
39.35	odd	6		1872.2.t.p.1153.2		4
52.3	odd	6		3042.2.a.r.1.1		2
52.11	even	12		3042.2.b.j.1351.1		4
52.15	even	12		3042.2.b.j.1351.4		4
52.23	odd	6		3042.2.a.x.1.2		2
52.35	odd	6		234.2.h.e.217.1	yes	4
156.11	odd	12		3042.2.b.k.1351.4		4
156.23	even	6		3042.2.a.q.1.1		2
156.35	even	6		234.2.h.d.217.2	yes	4
156.107	even	6		3042.2.a.w.1.2		2
156.119	odd	12		3042.2.b.k.1351.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.h.d.55.2	✓	4	12.11	even	2
234.2.h.d.217.2	yes	4	156.35	even	6
234.2.h.e.55.1	yes	4	4.3	odd	2
234.2.h.e.217.1	yes	4	52.35	odd	6
1872.2.t.n.289.1		4	1.1	even	1	trivial
1872.2.t.n.1153.1		4	13.9	even	3	inner
1872.2.t.p.289.2		4	3.2	odd	2
1872.2.t.p.1153.2		4	39.35	odd	6
3042.2.a.q.1.1		2	156.23	even	6
3042.2.a.r.1.1		2	52.3	odd	6
3042.2.a.w.1.2		2	156.107	even	6
3042.2.a.x.1.2		2	52.23	odd	6
3042.2.b.j.1351.1		4	52.11	even	12
3042.2.b.j.1351.4		4	52.15	even	12
3042.2.b.k.1351.1		4	156.119	odd	12
3042.2.b.k.1351.4		4	156.11	odd	12

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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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.t (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-4.16228 q^{5} +(1.58114 - 2.73861i) q^{7} +O(q^{10})\)	\(q-4.16228 q^{5} +(1.58114 - 2.73861i) q^{7} +(0.581139 + 1.00656i) q^{11} +(0.0811388 + 3.60464i) q^{13} +(-1.50000 + 2.59808i) q^{17} +(-2.58114 + 4.47066i) q^{19} +(-3.58114 - 6.20271i) q^{23} +12.3246 q^{25} +(-0.918861 - 1.59151i) q^{29} +6.32456 q^{31} +(-6.58114 + 11.3989i) q^{35} +(-1.91886 - 3.32357i) q^{37} +(1.50000 + 2.59808i) q^{41} +(4.58114 - 7.93477i) q^{43} +4.83772 q^{47} +(-1.50000 - 2.59808i) q^{49} +12.4868 q^{53} +(-2.41886 - 4.18959i) q^{55} +(1.16228 - 2.01312i) q^{59} +(-0.0811388 + 0.140537i) q^{61} +(-0.337722 - 15.0035i) q^{65} +(-1.41886 - 2.45754i) q^{67} +(3.58114 - 6.20271i) q^{71} -1.00000 q^{73} +3.67544 q^{77} +4.00000 q^{79} -3.48683 q^{83} +(6.24342 - 10.8139i) q^{85} +(6.00000 + 10.3923i) q^{89} +(10.0000 + 5.47723i) q^{91} +(10.7434 - 18.6081i) q^{95} +(2.00000 - 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} - 4 q^{11} - 6 q^{13} - 6 q^{17} - 4 q^{19} - 8 q^{23} + 24 q^{25} - 10 q^{29} - 20 q^{35} - 14 q^{37} + 6 q^{41} + 12 q^{43} + 32 q^{47} - 6 q^{49} + 12 q^{53} - 16 q^{55} - 8 q^{59} + 6 q^{61} - 14 q^{65} - 12 q^{67} + 8 q^{71} - 4 q^{73} + 40 q^{77} + 16 q^{79} + 24 q^{83} + 6 q^{85} + 24 q^{89} + 40 q^{91} + 24 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^11 - 6 * q^13 - 6 * q^17 - 4 * q^19 - 8 * q^23 + 24 * q^25 - 10 * q^29 - 20 * q^35 - 14 * q^37 + 6 * q^41 + 12 * q^43 + 32 * q^47 - 6 * q^49 + 12 * q^53 - 16 * q^55 - 8 * q^59 + 6 * q^61 - 14 * q^65 - 12 * q^67 + 8 * q^71 - 4 * q^73 + 40 * q^77 + 16 * q^79 + 24 * q^83 + 6 * q^85 + 24 * q^89 + 40 * q^91 + 24 * q^95 + 8 * q^97

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