LMFDB - Embedded newform 2268.2.l.n.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(109,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 4]))

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

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

chi := DirichletCharacter("2268.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.l (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:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 9x^{14} + 31x^{12} - 282x^{10} + 1695x^{8} - 3318x^{6} + 4606x^{4} - 4116x^{2} + 2401 $ x^16 - 9x^14 + 31x^12 - 282x^10 + 1695x^8 - 3318x^6 + 4606x^4 - 4116*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.4
Root			$-2.40332 - 0.123797i$ of defining polynomial
Character	$\chi$	$=$	2268.109
Dual form			2268.2.l.n.541.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.343737 q^{5} +(2.14324 + 1.55130i) q^{7} +O(q^{10})$	$q-0.343737 q^{5} +(2.14324 + 1.55130i) q^{7} -4.91976 q^{11} +(0.974162 - 1.68730i) q^{13} +(-2.07359 + 3.59156i) q^{17} +(0.202315 + 0.350420i) q^{19} -2.75886 q^{23} -4.88184 q^{25} +(-1.63861 - 2.83815i) q^{29} +(1.64324 + 2.84617i) q^{31} +(-0.736710 - 0.533240i) q^{35} +(-3.38925 - 5.87035i) q^{37} +(2.81250 - 4.87139i) q^{41} +(-4.96295 - 8.59608i) q^{43} +(-6.60038 + 11.4322i) q^{47} +(2.18693 + 6.64961i) q^{49} +(-1.38163 + 2.39306i) q^{53} +1.69110 q^{55} +(-4.22560 - 7.31896i) q^{59} +(-5.37804 + 9.31503i) q^{61} +(-0.334856 + 0.579987i) q^{65} +(4.21277 + 7.29673i) q^{67} +5.08888 q^{71} +(4.58416 - 7.94000i) q^{73} +(-10.5442 - 7.63203i) q^{77} +(-2.24601 + 3.89020i) q^{79} +(-4.62966 - 8.01880i) q^{83} +(0.712770 - 1.23455i) q^{85} +(-3.54253 - 6.13584i) q^{89} +(4.70537 - 2.10506i) q^{91} +(-0.0695431 - 0.120452i) q^{95} +(-1.31231 - 2.27299i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 10 q^{13} + 8 q^{19} - 8 q^{31} - 4 q^{37} - 10 q^{43} - 20 q^{49} - 32 q^{55} + 28 q^{61} + 18 q^{67} - 20 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97}+O(q^{100}) $ 16 * q + 10 * q^13 + 8 * q^19 - 8 * q^31 - 4 * q^37 - 10 * q^43 - 20 * q^49 - 32 * q^55 + 28 * q^61 + 18 * q^67 - 20 * q^79 - 38 * q^85 - 2 * q^91 + 42 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.343737			−0.153724			−0.0768620	−	0.997042i	$-0.524490\pi$
$5$	−0.343737			−0.153724			−0.0768620	+	0.997042i	$0.524490\pi$
$6$		0			0
$7$	2.14324	+	1.55130i	0.810068	+	0.586337i
$7$	2.14324	+	1.55130i	0.810068	+	0.586337i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.91976			−1.48336			−0.741682	−	0.670752i	$-0.765971\pi$
$11$	−4.91976			−1.48336			−0.741682	+	0.670752i	$0.765971\pi$
$12$		0			0
$13$	0.974162	−	1.68730i	0.270184	−	0.467972i	−0.698725	−	0.715391i	$-0.746249\pi$
$13$	0.974162	−	1.68730i	0.270184	−	0.467972i	0.968909	+	0.247418i	$0.0795821\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.07359	+	3.59156i	−0.502919	+	0.871082i	0.497075	+	0.867708i	$0.334407\pi$
$17$	−2.07359	+	3.59156i	−0.502919	+	0.871082i	−0.999994	+	0.00337409i	$0.998926\pi$
$18$		0			0
$19$	0.202315	+	0.350420i	0.0464142	+	0.0803918i	0.888299	−	0.459265i	$-0.151887\pi$
$19$	0.202315	+	0.350420i	0.0464142	+	0.0803918i	−0.841885	+	0.539657i	$0.818554\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.75886			−0.575263			−0.287631	−	0.957741i	$-0.592868\pi$
$23$	−2.75886			−0.575263			−0.287631	+	0.957741i	$0.592868\pi$
$24$		0			0
$25$	−4.88184			−0.976369
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.63861	−	2.83815i	−0.304282	−	0.527031i	0.672819	−	0.739807i	$-0.265083\pi$
$29$	−1.63861	−	2.83815i	−0.304282	−	0.527031i	−0.977101	+	0.212775i	$0.931750\pi$
$30$		0			0
$31$	1.64324	+	2.84617i	0.295134	+	0.511187i	0.975016	−	0.222134i	$-0.0713023\pi$
$31$	1.64324	+	2.84617i	0.295134	+	0.511187i	−0.679882	+	0.733322i	$0.737969\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.736710	−	0.533240i	−0.124527	−	0.0901340i
$36$		0			0
$37$	−3.38925	−	5.87035i	−0.557189	−	0.965079i	−0.997730	−	0.0673466i	$-0.978547\pi$
$37$	−3.38925	−	5.87035i	−0.557189	−	0.965079i	0.440541	−	0.897733i	$-0.354787\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.81250	−	4.87139i	0.439238	−	0.760783i	−0.558392	−	0.829577i	$-0.688582\pi$
$41$	2.81250	−	4.87139i	0.439238	−	0.760783i	0.997631	+	0.0687936i	$0.0219150\pi$
$42$		0			0
$43$	−4.96295	−	8.59608i	−0.756843	−	1.31089i	−0.944453	−	0.328647i	$-0.893407\pi$
$43$	−4.96295	−	8.59608i	−0.756843	−	1.31089i	0.187610	−	0.982244i	$-0.439926\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.60038	+	11.4322i	−0.962764	+	1.66756i	−0.247257	+	0.968950i	$0.579529\pi$
$47$	−6.60038	+	11.4322i	−0.962764	+	1.66756i	−0.715507	+	0.698606i	$0.753804\pi$
$48$		0			0
$49$	2.18693	+	6.64961i	0.312419	+	0.949944i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.38163	+	2.39306i	−0.189782	+	0.328711i	−0.945177	−	0.326557i	$-0.894111\pi$
$53$	−1.38163	+	2.39306i	−0.189782	+	0.328711i	0.755396	+	0.655269i	$0.227445\pi$
$54$		0			0
$55$	1.69110			0.228028
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.22560	−	7.31896i	−0.550127	−	0.952847i	−0.998265	−	0.0588830i	$-0.981246\pi$
$59$	−4.22560	−	7.31896i	−0.550127	−	0.952847i	0.448138	−	0.893964i	$-0.352087\pi$
$60$		0			0
$61$	−5.37804	+	9.31503i	−0.688587	+	1.19267i	0.283708	+	0.958911i	$0.408435\pi$
$61$	−5.37804	+	9.31503i	−0.688587	+	1.19267i	−0.972295	+	0.233757i	$0.924898\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.334856	+	0.579987i	−0.0415337	+	0.0719386i
$66$		0			0
$67$	4.21277	+	7.29673i	0.514672	+	0.891438i	0.999855	+	0.0170251i	$0.00541953\pi$
$67$	4.21277	+	7.29673i	0.514672	+	0.891438i	−0.485183	+	0.874412i	$0.661247\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.08888			0.603939			0.301970	−	0.953318i	$-0.402356\pi$
$71$	5.08888			0.603939			0.301970	+	0.953318i	$0.402356\pi$
$72$		0			0
$73$	4.58416	−	7.94000i	0.536535	−	0.929306i	−0.462552	−	0.886592i	$-0.653066\pi$
$73$	4.58416	−	7.94000i	0.536535	−	0.929306i	0.999087	−	0.0427143i	$-0.0136005\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.5442	−	7.63203i	−1.20162	−	0.869750i
$78$		0			0
$79$	−2.24601	+	3.89020i	−0.252696	+	0.437682i	−0.964267	−	0.264932i	$-0.914651\pi$
$79$	−2.24601	+	3.89020i	−0.252696	+	0.437682i	0.711571	+	0.702614i	$0.247984\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.62966	−	8.01880i	−0.508171	−	0.880178i	−0.999955	−	0.00946064i	$-0.996989\pi$
$83$	−4.62966	−	8.01880i	−0.508171	−	0.880178i	0.491784	−	0.870717i	$-0.336345\pi$
$84$		0			0
$85$	0.712770	−	1.23455i	0.0773107	−	0.133906i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.54253	−	6.13584i	−0.375507	−	0.650397i	0.614896	−	0.788608i	$-0.289198\pi$
$89$	−3.54253	−	6.13584i	−0.375507	−	0.650397i	−0.990403	+	0.138211i	$0.955865\pi$
$90$		0			0
$91$	4.70537	−	2.10506i	0.493257	−	0.220670i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.0695431	−	0.120452i	−0.00713498	−	0.0123581i
$96$		0			0
$97$	−1.31231	−	2.27299i	−0.133245	−	0.230787i	0.791681	−	0.610935i	$-0.209206\pi$
$97$	−1.31231	−	2.27299i	−0.133245	−	0.230787i	−0.924926	+	0.380148i	$0.875873\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−15.7054			−1.56274			−0.781372	−	0.624066i	$-0.785480\pi$
$101$	−15.7054			−1.56274			−0.781372	+	0.624066i	$0.785480\pi$
$102$		0			0
$103$	−7.74278			−0.762919			−0.381459	−	0.924386i	$-0.624578\pi$
$103$	−7.74278			−0.762919			−0.381459	+	0.924386i	$0.624578\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.12952	+	10.6166i	0.592563	+	1.02635i	0.993886	+	0.110413i	$0.0352173\pi$
$107$	6.12952	+	10.6166i	0.592563	+	1.02635i	−0.401323	+	0.915937i	$0.631449\pi$
$108$		0			0
$109$	4.89342	−	8.47565i	0.468705	−	0.811820i	−0.530656	−	0.847588i	$-0.678054\pi$
$109$	4.89342	−	8.47565i	0.468705	−	0.811820i	0.999360	+	0.0357675i	$0.0113876\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.54664	+	4.41091i	−0.239568	+	0.414944i	−0.960590	−	0.277968i	$-0.910339\pi$
$113$	−2.54664	+	4.41091i	−0.239568	+	0.414944i	0.721022	+	0.692912i	$0.243672\pi$
$114$		0			0
$115$	0.948324			0.0884317
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.0158	+	4.48081i	−0.918146	+	0.410755i
$120$		0			0
$121$	13.2040			1.20037
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.39676			0.303815
$126$		0			0
$127$	2.24242			0.198982			0.0994911	−	0.995038i	$-0.468279\pi$
$127$	2.24242			0.198982			0.0994911	+	0.995038i	$0.468279\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.45443			−0.651297			−0.325648	−	0.945491i	$-0.605583\pi$
$131$	−7.45443			−0.651297			−0.325648	+	0.945491i	$0.605583\pi$
$132$		0			0
$133$	−0.109997	+	1.06488i	−0.00953799	+	0.0923371i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.44681			−0.721659			−0.360830	−	0.932632i	$-0.617506\pi$
$137$	−8.44681			−0.721659			−0.360830	+	0.932632i	$0.617506\pi$
$138$		0			0
$139$	1.82352	−	3.15843i	0.154669	−	0.267895i	−0.778269	−	0.627931i	$-0.783902\pi$
$139$	1.82352	−	3.15843i	0.154669	−	0.267895i	0.932939	+	0.360036i	$0.117235\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.79264	+	8.30110i	−0.400781	+	0.694173i
$144$		0			0
$145$	0.563250	+	0.975578i	0.0467754	+	0.0810173i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.294156			0.0240982			0.0120491	−	0.999927i	$-0.496165\pi$
$149$	0.294156			0.0240982			0.0120491	+	0.999927i	$0.496165\pi$
$150$		0			0
$151$	−19.2040			−1.56280			−0.781401	−	0.624029i	$-0.785495\pi$
$151$	−19.2040			−1.56280			−0.781401	+	0.624029i	$0.785495\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.564842	−	0.978335i	−0.0453692	−	0.0785817i
$156$		0			0
$157$	6.55832	+	11.3593i	0.523411	+	0.906575i	0.999629	+	0.0272471i	$0.00867410\pi$
$157$	6.55832	+	11.3593i	0.523411	+	0.906575i	−0.476218	+	0.879327i	$0.657993\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.91290	−	4.27983i	−0.466002	−	0.337298i
$162$		0			0
$163$	0.496191	+	0.859428i	0.0388647	+	0.0673156i	0.884803	−	0.465964i	$-0.154293\pi$
$163$	0.496191	+	0.859428i	0.0388647	+	0.0673156i	−0.845939	+	0.533280i	$0.820959\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.93040	−	12.0038i	0.536291	−	0.928883i	−0.462809	−	0.886458i	$-0.653158\pi$
$167$	6.93040	−	12.0038i	0.536291	−	0.928883i	0.999100	−	0.0424246i	$-0.0135082\pi$
$168$		0			0
$169$	4.60202	+	7.97093i	0.354001	+	0.613148i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.50243	−	4.33434i	0.190256	−	0.329534i	−0.755079	−	0.655634i	$-0.772401\pi$
$173$	2.50243	−	4.33434i	0.190256	−	0.329534i	0.945335	+	0.326100i	$0.105735\pi$
$174$		0			0
$175$	−10.4630	−	7.57321i	−0.790925	−	0.572481i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.8101	+	18.7236i	−0.807982	+	1.39947i	0.106277	+	0.994337i	$0.466107\pi$
$179$	−10.8101	+	18.7236i	−0.807982	+	1.39947i	−0.914259	+	0.405130i	$0.867226\pi$
$180$		0			0
$181$	−26.0342			−1.93511			−0.967553	−	0.252666i	$-0.918693\pi$
$181$	−26.0342			−1.93511			−0.967553	+	0.252666i	$0.918693\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.16501	+	2.01786i	0.0856532	+	0.148356i
$186$		0			0
$187$	10.2016	−	17.6696i	0.746012	−	1.29213i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.85252	+	15.3330i	−0.640546	+	1.10946i	0.344766	+	0.938689i	$0.387958\pi$
$191$	−8.85252	+	15.3330i	−0.640546	+	1.10946i	−0.985311	+	0.170769i	$0.945375\pi$
$192$		0			0
$193$	1.48214	+	2.56715i	0.106687	+	0.184787i	0.914426	−	0.404753i	$-0.132642\pi$
$193$	1.48214	+	2.56715i	0.106687	+	0.184787i	−0.807739	+	0.589540i	$0.799309\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.2357			−1.29924			−0.649618	−	0.760260i	$-0.725071\pi$
$197$	−18.2357			−1.29924			−0.649618	+	0.760260i	$0.725071\pi$
$198$		0			0
$199$	1.61323	−	2.79419i	0.114359	−	0.198075i	−0.803165	−	0.595757i	$-0.796852\pi$
$199$	1.61323	−	2.79419i	0.114359	−	0.198075i	0.917523	+	0.397682i	$0.130185\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.890902	−	8.62480i	0.0625290	−	0.605342i
$204$		0			0
$205$	−0.966760	+	1.67448i	−0.0675215	+	0.116951i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.995340	−	1.72398i	−0.0688491	−	0.119250i
$210$		0			0
$211$	−4.72740	+	8.18809i	−0.325447	+	0.563691i	−0.981603	−	0.190934i	$-0.938848\pi$
$211$	−4.72740	+	8.18809i	−0.325447	+	0.563691i	0.656155	+	0.754626i	$0.272182\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.70595	+	2.95479i	0.116345	+	0.201515i
$216$		0			0
$217$	−0.893419	+	8.64917i	−0.0606492	+	0.587144i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.04002	+	6.99753i	0.271761	+	0.470705i
$222$		0			0
$223$	9.63203	+	16.6832i	0.645008	+	1.11719i	0.984300	+	0.176506i	$0.0564794\pi$
$223$	9.63203	+	16.6832i	0.645008	+	1.11719i	−0.339291	+	0.940681i	$0.610187\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.17757			0.144530			0.0722652	−	0.997385i	$-0.476977\pi$
$227$	2.17757			0.144530			0.0722652	+	0.997385i	$0.476977\pi$
$228$		0			0
$229$	6.21238			0.410525			0.205263	−	0.978707i	$-0.434195\pi$
$229$	6.21238			0.410525			0.205263	+	0.978707i	$0.434195\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.76391	−	11.7154i	−0.443118	−	0.767503i	0.554801	−	0.831983i	$-0.312794\pi$
$233$	−6.76391	−	11.7154i	−0.443118	−	0.767503i	−0.997919	+	0.0644799i	$0.979461\pi$
$234$		0			0
$235$	2.26879	−	3.92967i	0.148000	−	0.256343i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.21398	+	3.83473i	−0.143211	+	0.248048i	−0.928704	−	0.370822i	$-0.879076\pi$
$239$	−2.21398	+	3.83473i	−0.143211	+	0.248048i	0.785493	+	0.618870i	$0.212409\pi$
$240$		0			0
$241$	−22.1090			−1.42417			−0.712084	−	0.702095i	$-0.752248\pi$
$241$	−22.1090			−1.42417			−0.712084	+	0.702095i	$0.752248\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.751730	−	2.28572i	−0.0480263	−	0.146029i
$246$		0			0
$247$	0.788350			0.0501615
$248$		0			0
$249$		0			0
$250$		0			0
$251$	29.4679			1.86000			0.929998	−	0.367564i	$-0.119808\pi$
$251$	29.4679			1.86000			0.929998	+	0.367564i	$0.119808\pi$
$252$		0			0
$253$	13.5729			0.853324
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−31.2054			−1.94654			−0.973270	−	0.229662i	$-0.926238\pi$
$257$	−31.2054			−1.94654			−0.973270	+	0.229662i	$0.926238\pi$
$258$		0			0
$259$	1.84271	−	17.8393i	0.114501	−	1.10848i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	22.8117			1.40663			0.703314	−	0.710879i	$-0.251703\pi$
$263$	22.8117			1.40663			0.703314	+	0.710879i	$0.251703\pi$
$264$		0			0
$265$	0.474918	−	0.822582i	0.0291740	−	0.0505308i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.7838	−	20.4101i	0.718471	−	1.24443i	−0.243135	−	0.969992i	$-0.578176\pi$
$269$	11.7838	−	20.4101i	0.718471	−	1.24443i	0.961606	−	0.274435i	$-0.0884908\pi$
$270$		0			0
$271$	0.0112106	+	0.0194174i	0.000680998	+	0.00117952i	0.866366	−	0.499410i	$-0.166450\pi$
$271$	0.0112106	+	0.0194174i	0.000680998	+	0.00117952i	−0.865685	+	0.500590i	$0.833117\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	24.0175			1.44831
$276$		0			0
$277$	11.0783			0.665628			0.332814	−	0.942992i	$-0.392002\pi$
$277$	11.0783			0.665628			0.332814	+	0.942992i	$0.392002\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.136542	+	0.236498i	0.00814541	+	0.0141083i	0.870069	−	0.492929i	$-0.164074\pi$
$281$	0.136542	+	0.236498i	0.00814541	+	0.0141083i	−0.861924	+	0.507038i	$0.830741\pi$
$282$		0			0
$283$	−5.57676	−	9.65923i	−0.331504	−	0.574181i	0.651303	−	0.758818i	$-0.274223\pi$
$283$	−5.57676	−	9.65923i	−0.331504	−	0.574181i	−0.982807	+	0.184636i	$0.940889\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.5848	−	6.07751i	0.801888	−	0.358744i
$288$		0			0
$289$	−0.0995427	−	0.172413i	−0.00585545	−	0.0101419i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.0597	−	24.3522i	0.821379	−	1.42267i	−0.0832760	−	0.996527i	$-0.526538\pi$
$293$	14.0597	−	24.3522i	0.821379	−	1.42267i	0.904655	−	0.426144i	$-0.140128\pi$
$294$		0			0
$295$	1.45250	+	2.51580i	0.0845676	+	0.146475i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.68758	+	4.65503i	−0.155427	+	0.269207i
$300$		0			0
$301$	2.69833	−	26.1225i	0.155529	−	1.50567i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.84863	−	3.20192i	0.105852	−	0.183342i
$306$		0			0
$307$	22.3340			1.27467			0.637333	−	0.770588i	$-0.280037\pi$
$307$	22.3340			1.27467			0.637333	+	0.770588i	$0.280037\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.9513	+	24.1643i	0.791105	+	1.37023i	0.925283	+	0.379277i	$0.123827\pi$
$311$	13.9513	+	24.1643i	0.791105	+	1.37023i	−0.134179	+	0.990957i	$0.542840\pi$
$312$		0			0
$313$	6.15786	−	10.6657i	0.348063	−	0.602863i	−0.637842	−	0.770167i	$-0.720173\pi$
$313$	6.15786	−	10.6657i	0.348063	−	0.602863i	0.985905	+	0.167304i	$0.0535062\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.7767	−	23.8619i	0.773775	−	1.34022i	−0.161705	−	0.986839i	$-0.551699\pi$
$317$	13.7767	−	23.8619i	0.773775	−	1.34022i	0.935480	−	0.353379i	$-0.114967\pi$
$318$		0			0
$319$	8.06155	+	13.9630i	0.451360	+	0.781779i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.67807			−0.0933704
$324$		0			0
$325$	−4.75571	+	8.23713i	−0.263799	+	0.456914i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−31.8809	+	14.2627i	−1.75765	+	0.786329i
$330$		0			0
$331$	−5.05585	+	8.75699i	−0.277895	+	0.481327i	−0.970861	−	0.239642i	$-0.922970\pi$
$331$	−5.05585	+	8.75699i	−0.277895	+	0.481327i	0.692967	+	0.720970i	$0.256303\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.44809	−	2.50816i	−0.0791174	−	0.137035i
$336$		0			0
$337$	−5.91508	+	10.2452i	−0.322215	+	0.558093i	−0.980945	−	0.194287i	$-0.937761\pi$
$337$	−5.91508	+	10.2452i	−0.322215	+	0.558093i	0.658730	+	0.752380i	$0.271094\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.08433	−	14.0025i	−0.437791	−	0.758276i
$342$		0			0
$343$	−5.62843	+	17.6443i	−0.303907	+	0.952702i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.2249	+	24.6383i	0.763634	+	1.32265i	0.940966	+	0.338502i	$0.109920\pi$
$347$	14.2249	+	24.6383i	0.763634	+	1.32265i	−0.177331	+	0.984151i	$0.556746\pi$
$348$		0			0
$349$	8.79028	+	15.2252i	0.470533	+	0.814987i	0.999432	−	0.0336976i	$-0.0107283\pi$
$349$	8.79028	+	15.2252i	0.470533	+	0.814987i	−0.528899	+	0.848685i	$0.677395\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	27.5145			1.46445			0.732225	−	0.681063i	$-0.238482\pi$
$353$	27.5145			1.46445			0.732225	+	0.681063i	$0.238482\pi$
$354$		0			0
$355$	−1.74924			−0.0928400
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.73182	+	16.8560i	0.513626	+	0.889626i	0.999875	+	0.0158059i	$0.00503137\pi$
$359$	9.73182	+	16.8560i	0.513626	+	0.889626i	−0.486249	+	0.873820i	$0.661635\pi$
$360$		0			0
$361$	9.41814	−	16.3127i	0.495691	−	0.858563i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.57575	+	2.72927i	−0.0824783	+	0.142857i
$366$		0			0
$367$	6.96429			0.363533			0.181766	−	0.983342i	$-0.441819\pi$
$367$	6.96429			0.363533			0.181766	+	0.983342i	$0.441819\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.67351	+	2.98556i	−0.346471	+	0.155003i
$372$		0			0
$373$	−13.8149			−0.715310			−0.357655	−	0.933854i	$-0.616424\pi$
$373$	−13.8149			−0.715310			−0.357655	+	0.933854i	$0.616424\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.38507			−0.328848
$378$		0			0
$379$	−13.3822			−0.687398			−0.343699	−	0.939080i	$-0.611680\pi$
$379$	−13.3822			−0.687398			−0.343699	+	0.939080i	$0.611680\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.46789			−0.228299			−0.114149	−	0.993464i	$-0.536414\pi$
$383$	−4.46789			−0.228299			−0.114149	+	0.993464i	$0.536414\pi$
$384$		0			0
$385$	3.62444	+	2.62341i	0.184718	+	0.133701i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	28.2463			1.43214			0.716072	−	0.698026i	$-0.245938\pi$
$389$	28.2463			1.43214			0.716072	+	0.698026i	$0.245938\pi$
$390$		0			0
$391$	5.72075	−	9.90863i	0.289311	−	0.501101i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.772037	−	1.33721i	0.0388454	−	0.0672822i
$396$		0			0
$397$	0.293513	+	0.508379i	0.0147310	+	0.0255148i	0.873297	−	0.487188i	$-0.161977\pi$
$397$	0.293513	+	0.508379i	0.0147310	+	0.0255148i	−0.858566	+	0.512703i	$0.828644\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	29.0301			1.44969			0.724846	−	0.688911i	$-0.241911\pi$
$401$	29.0301			1.44969			0.724846	+	0.688911i	$0.241911\pi$
$402$		0			0
$403$	6.40312			0.318962
$404$		0			0
$405$		0			0
$406$		0			0
$407$	16.6743	+	28.8807i	0.826513	+	1.43156i
$408$		0			0
$409$	−11.0021	−	19.0562i	−0.544018	−	0.942267i	−0.998668	−	0.0515965i	$-0.983569\pi$
$409$	−11.0021	−	19.0562i	−0.544018	−	0.942267i	0.454650	−	0.890670i	$-0.349764\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.29744	−	22.2414i	0.113049	−	1.09443i
$414$		0			0
$415$	1.59138	+	2.75636i	0.0781180	+	0.135304i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.5615	−	21.7572i	0.613671	−	1.06291i	−0.376945	−	0.926236i	$-0.623025\pi$
$419$	12.5615	−	21.7572i	0.613671	−	1.06291i	0.990616	−	0.136674i	$-0.0436413\pi$
$420$		0			0
$421$	−0.0961261	−	0.166495i	−0.00468490	−	0.00811449i	0.863673	−	0.504052i	$-0.168158\pi$
$421$	−0.0961261	−	0.166495i	−0.00468490	−	0.00811449i	−0.868358	+	0.495937i	$0.834825\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.1229	−	17.5334i	0.491035	−	0.850497i
$426$		0			0
$427$	−25.9768	+	11.6214i	−1.25711	+	0.562398i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.7344	+	18.5925i	−0.517057	+	0.895569i	0.482747	+	0.875760i	$0.339639\pi$
$431$	−10.7344	+	18.5925i	−0.517057	+	0.895569i	−0.999804	+	0.0198093i	$0.993694\pi$
$432$		0			0
$433$	−40.3807			−1.94057			−0.970286	−	0.241960i	$-0.922210\pi$
$433$	−40.3807			−1.94057			−0.970286	+	0.241960i	$0.922210\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.558159	−	0.966760i	−0.0267004	−	0.0462464i
$438$		0			0
$439$	16.8288	−	29.1484i	0.803196	−	1.39118i	−0.114306	−	0.993446i	$-0.536464\pi$
$439$	16.8288	−	29.1484i	0.803196	−	1.39118i	0.917502	−	0.397731i	$-0.130202\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.77416	−	10.0011i	0.274339	−	0.475168i	−0.695629	−	0.718401i	$-0.744874\pi$
$443$	5.77416	−	10.0011i	0.274339	−	0.475168i	0.969968	+	0.243232i	$0.0782078\pi$
$444$		0			0
$445$	1.21770	+	2.10911i	0.0577244	+	0.0999816i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−22.6025			−1.06668			−0.533338	−	0.845902i	$-0.679063\pi$
$449$	−22.6025			−1.06668			−0.533338	+	0.845902i	$0.679063\pi$
$450$		0			0
$451$	−13.8368	+	23.9661i	−0.651550	+	1.12852i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.61741	+	0.723588i	−0.0758253	+	0.0339223i
$456$		0			0
$457$	−1.04064	+	1.80244i	−0.0486792	+	0.0843148i	−0.889338	−	0.457250i	$-0.848835\pi$
$457$	−1.04064	+	1.80244i	−0.0486792	+	0.0843148i	0.840659	+	0.541565i	$0.182168\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.21838	−	7.30644i	−0.196469	−	0.340295i	0.750912	−	0.660402i	$-0.229614\pi$
$461$	−4.21838	−	7.30644i	−0.196469	−	0.340295i	−0.947381	+	0.320108i	$0.896281\pi$
$462$		0			0
$463$	2.92231	−	5.06159i	0.135811	−	0.235232i	−0.790096	−	0.612983i	$-0.789969\pi$
$463$	2.92231	−	5.06159i	0.135811	−	0.235232i	0.925907	+	0.377751i	$0.123303\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.04009	−	1.80149i	−0.0481298	−	0.0833632i	0.840957	−	0.541102i	$-0.181993\pi$
$467$	−1.04009	−	1.80149i	−0.0481298	−	0.0833632i	−0.889087	+	0.457739i	$0.848659\pi$
$468$		0			0
$469$	−2.29046	+	22.1739i	−0.105764	+	1.02390i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	24.4165	+	42.2907i	1.12267	+	1.94453i
$474$		0			0
$475$	−0.987670	−	1.71069i	−0.0453174	−	0.0784920i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.0308			0.778155			0.389077	−	0.921205i	$-0.372794\pi$
$479$	17.0308			0.778155			0.389077	+	0.921205i	$0.372794\pi$
$480$		0			0
$481$	−13.2067			−0.602174
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.451091	+	0.781312i	0.0204830	+	0.0354775i
$486$		0			0
$487$	−11.0758	+	19.1838i	−0.501892	+	0.869302i	0.498106	+	0.867116i	$0.334029\pi$
$487$	−11.0758	+	19.1838i	−0.501892	+	0.869302i	−0.999998	+	0.00218582i	$0.999304\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.88390	+	4.99505i	−0.130148	+	0.225424i	−0.923734	−	0.383035i	$-0.874879\pi$
$491$	−2.88390	+	4.99505i	−0.130148	+	0.225424i	0.793585	+	0.608459i	$0.208212\pi$
$492$		0			0
$493$	13.5912			0.612116
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.9067	+	7.89439i	0.489232	+	0.354112i
$498$		0			0
$499$	−10.3207			−0.462017			−0.231008	−	0.972952i	$-0.574203\pi$
$499$	−10.3207			−0.462017			−0.231008	+	0.972952i	$0.574203\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	29.4644			1.31375			0.656876	−	0.753998i	$-0.271877\pi$
$503$	29.4644			1.31375			0.656876	+	0.753998i	$0.271877\pi$
$504$		0			0
$505$	5.39852			0.240231
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.74929			−0.166184			−0.0830922	−	0.996542i	$-0.526480\pi$
$509$	−3.74929			−0.166184			−0.0830922	+	0.996542i	$0.526480\pi$
$510$		0			0
$511$	22.1423	−	9.90589i	0.979516	−	0.438211i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.66148			0.117279
$516$		0			0
$517$	32.4723	−	56.2436i	1.42813	−	2.47359i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.9592	+	24.1780i	−0.611562	+	1.05926i	0.379415	+	0.925226i	$0.376125\pi$
$521$	−13.9592	+	24.1780i	−0.611562	+	1.05926i	−0.990977	+	0.134030i	$0.957208\pi$
$522$		0			0
$523$	21.0680	+	36.4909i	0.921240	+	1.59563i	0.797499	+	0.603320i	$0.206156\pi$
$523$	21.0680	+	36.4909i	0.921240	+	1.59563i	0.123741	+	0.992315i	$0.460511\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.6296			−0.593714
$528$		0			0
$529$	−15.3887			−0.669072
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.47966	−	9.49105i	−0.237350	−	0.411103i
$534$		0			0
$535$	−2.10694	−	3.64934i	−0.0910912	−	0.157775i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10.7592	−	32.7145i	−0.463431	−	1.40911i
$540$		0			0
$541$	−14.5992	−	25.2865i	−0.627667	−	1.08715i	−0.988019	−	0.154334i	$-0.950677\pi$
$541$	−14.5992	−	25.2865i	−0.627667	−	1.08715i	0.360352	−	0.932816i	$-0.382657\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.68205	+	2.91340i	−0.0720511	+	0.124796i
$546$		0			0
$547$	5.76122	+	9.97872i	0.246332	+	0.426659i	0.962505	−	0.271263i	$-0.0874414\pi$
$547$	5.76122	+	9.97872i	0.246332	+	0.426659i	−0.716173	+	0.697922i	$0.754108\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.663029	−	1.14840i	0.0282460	−	0.0489235i
$552$		0			0
$553$	−10.8486	+	4.85339i	−0.461330	+	0.206387i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.97631	−	10.3513i	0.253224	−	0.438597i	−0.711187	−	0.703002i	$-0.751842\pi$
$557$	5.97631	−	10.3513i	0.253224	−	0.438597i	0.964412	+	0.264405i	$0.0851756\pi$
$558$		0			0
$559$	−19.3389			−0.817947
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.8099	+	20.4553i	0.497728	+	0.862090i	0.999997	−	0.00262192i	$-0.000834585\pi$
$563$	11.8099	+	20.4553i	0.497728	+	0.862090i	−0.502269	+	0.864711i	$0.667501\pi$
$564$		0			0
$565$	0.875375	−	1.51619i	0.0368273	−	0.0637868i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.9745	−	20.7405i	0.501999	−	0.869487i	−0.497999	−	0.867178i	$-0.665932\pi$
$569$	11.9745	−	20.7405i	0.501999	−	0.869487i	0.999997	−	0.00230931i	$-0.000735078\pi$
$570$		0			0
$571$	−5.04958	−	8.74614i	−0.211319	−	0.366014i	0.740809	−	0.671716i	$-0.234442\pi$
$571$	−5.04958	−	8.74614i	−0.211319	−	0.366014i	−0.952127	+	0.305701i	$0.901109\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.4683			0.561669
$576$		0			0
$577$	16.1744	−	28.0149i	0.673348	−	1.16627i	−0.303600	−	0.952800i	$-0.598189\pi$
$577$	16.1744	−	28.0149i	0.673348	−	1.16627i	0.976949	−	0.213474i	$-0.0684780\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.51712	−	24.3682i	0.104428	−	1.01096i
$582$		0			0
$583$	6.79729	−	11.7733i	0.281515	−	0.487598i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.02746	+	8.70781i	0.207505	+	0.359410i	0.950928	−	0.309412i	$-0.100132\pi$
$587$	5.02746	+	8.70781i	0.207505	+	0.359410i	−0.743423	+	0.668822i	$0.766799\pi$
$588$		0			0
$589$	−0.664903	+	1.15165i	−0.0273968	+	0.0474527i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.0673	−	29.5613i	−0.700868	−	1.21394i	−0.968162	−	0.250324i	$-0.919463\pi$
$593$	−17.0673	−	29.5613i	−0.700868	−	1.21394i	0.267294	−	0.963615i	$-0.413870\pi$
$594$		0			0
$595$	3.44280	−	1.54022i	0.141141	−	0.0631429i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.2141	+	26.3517i	0.621633	+	1.07670i	0.989182	+	0.146695i	$0.0468637\pi$
$599$	15.2141	+	26.3517i	0.621633	+	1.07670i	−0.367549	+	0.930004i	$0.619803\pi$
$600$		0			0
$601$	23.6966	+	41.0438i	0.966606	+	1.67421i	0.705238	+	0.708971i	$0.250840\pi$
$601$	23.6966	+	41.0438i	0.966606	+	1.67421i	0.261368	+	0.965239i	$0.415826\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.53872			−0.184525
$606$		0			0
$607$	−43.8119			−1.77827			−0.889135	−	0.457644i	$-0.848693\pi$
$607$	−43.8119			−1.77827			−0.889135	+	0.457644i	$0.848693\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.8597	+	22.2736i	0.520247	+	0.901094i
$612$		0			0
$613$	0.997163	−	1.72714i	0.0402750	−	0.0697584i	−0.845185	−	0.534474i	$-0.820510\pi$
$613$	0.997163	−	1.72714i	0.0402750	−	0.0697584i	0.885460	+	0.464715i	$0.153843\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.0562	+	36.4704i	−0.847690	+	1.46824i	0.0355743	+	0.999367i	$0.488674\pi$
$617$	−21.0562	+	36.4704i	−0.847690	+	1.46824i	−0.883264	+	0.468875i	$0.844659\pi$
$618$		0			0
$619$	11.7644			0.472852			0.236426	−	0.971650i	$-0.424024\pi$
$619$	11.7644			0.472852			0.236426	+	0.971650i	$0.424024\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.92605	−	18.6461i	0.0771656	−	0.747039i
$624$		0			0
$625$	23.2416			0.929665
$626$		0			0
$627$		0			0
$628$		0			0
$629$	28.1116			1.12088
$630$		0			0
$631$	37.3202			1.48570			0.742848	−	0.669460i	$-0.233475\pi$
$631$	37.3202			1.48570			0.742848	+	0.669460i	$0.233475\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.770802			−0.0305883
$636$		0			0
$637$	13.3503	+	2.78779i	0.528958	+	0.110456i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	40.4592			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$641$	40.4592			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$642$		0			0
$643$	1.43445	−	2.48455i	0.0565693	−	0.0979809i	−0.836354	−	0.548190i	$-0.815317\pi$
$643$	1.43445	−	2.48455i	0.0565693	−	0.0979809i	0.892923	+	0.450209i	$0.148650\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.71712	−	2.97413i	0.0675068	−	0.116925i	−0.830296	−	0.557322i	$-0.811829\pi$
$647$	1.71712	−	2.97413i	0.0675068	−	0.116925i	0.897803	+	0.440397i	$0.145162\pi$
$648$		0			0
$649$	20.7889	+	36.0075i	0.816038	+	1.41342i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24.4951			0.958566			0.479283	−	0.877660i	$-0.340897\pi$
$653$	24.4951			0.958566			0.479283	+	0.877660i	$0.340897\pi$
$654$		0			0
$655$	2.56236			0.100120
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.6375	+	20.1567i	0.453332	+	0.785194i	0.998591	−	0.0530740i	$-0.0169019\pi$
$659$	11.6375	+	20.1567i	0.453332	+	0.785194i	−0.545259	+	0.838268i	$0.683569\pi$
$660$		0			0
$661$	16.7899	+	29.0809i	0.653051	+	1.13112i	0.982379	+	0.186902i	$0.0598446\pi$
$661$	16.7899	+	29.0809i	0.653051	+	1.13112i	−0.329328	+	0.944216i	$0.606822\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.0378102	−	0.366040i	0.00146622	−	0.0141944i
$666$		0			0
$667$	4.52069	+	7.83007i	0.175042	+	0.303182i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	26.4586	−	45.8277i	1.02142	−	1.76916i
$672$		0			0
$673$	−23.3581	−	40.4574i	−0.900388	−	1.55952i	−0.826991	−	0.562215i	$-0.809949\pi$
$673$	−23.3581	−	40.4574i	−0.900388	−	1.55952i	−0.0733972	−	0.997303i	$-0.523384\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.73653	+	3.00777i	−0.0667404	+	0.115598i	−0.897465	−	0.441086i	$-0.854593\pi$
$677$	−1.73653	+	3.00777i	−0.0667404	+	0.115598i	0.830724	+	0.556684i	$0.187927\pi$
$678$		0			0
$679$	0.713497	−	6.90735i	0.0273815	−	0.265080i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.49075	+	6.04615i	−0.133570	+	0.231349i	−0.925050	−	0.379845i	$-0.875977\pi$
$683$	−3.49075	+	6.04615i	−0.133570	+	0.231349i	0.791480	+	0.611194i	$0.209311\pi$
$684$		0			0
$685$	2.90348			0.110936
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.69187	+	4.66245i	0.102552	+	0.177625i
$690$		0			0
$691$	−16.5416	+	28.6509i	−0.629272	+	1.08993i	0.358426	+	0.933558i	$0.383314\pi$
$691$	−16.5416	+	28.6509i	−0.629272	+	1.08993i	−0.987698	+	0.156373i	$0.950020\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.626813	+	1.08567i	−0.0237764	+	0.0411819i
$696$		0			0
$697$	11.6639	+	20.2025i	0.441803	+	0.765225i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	39.4243			1.48904			0.744518	−	0.667602i	$-0.232679\pi$
$701$	39.4243			1.48904			0.744518	+	0.667602i	$0.232679\pi$
$702$		0			0
$703$	1.37139	−	2.37532i	0.0517230	−	0.0895868i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−33.6604	−	24.3638i	−1.26593	−	0.916294i
$708$		0			0
$709$	−10.5771	+	18.3201i	−0.397232	+	0.688026i	−0.993383	−	0.114846i	$-0.963363\pi$
$709$	−10.5771	+	18.3201i	−0.397232	+	0.688026i	0.596151	+	0.802872i	$0.296696\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.53347	−	7.85220i	−0.169780	−	0.294067i
$714$		0			0
$715$	1.64741	−	2.85340i	0.0616096	−	0.106711i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.7586	−	23.8305i	−0.513108	−	0.888729i	−0.999884	−	0.0152024i	$-0.995161\pi$
$719$	−13.7586	−	23.8305i	−0.513108	−	0.888729i	0.486777	−	0.873527i	$-0.338173\pi$
$720$		0			0
$721$	−16.5946	−	12.0114i	−0.618016	−	0.447327i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.99942	+	13.8554i	0.297091	+	0.514577i
$726$		0			0
$727$	−10.5095	−	18.2030i	−0.389775	−	0.675110i	0.602644	−	0.798010i	$-0.294114\pi$
$727$	−10.5095	−	18.2030i	−0.389775	−	0.675110i	−0.992419	+	0.122900i	$0.960781\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	41.1645			1.52252
$732$		0			0
$733$	−6.42438			−0.237290			−0.118645	−	0.992937i	$-0.537855\pi$
$733$	−6.42438			−0.237290			−0.118645	+	0.992937i	$0.537855\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−20.7258	−	35.8982i	−0.763445	−	1.32233i
$738$		0			0
$739$	19.2219	−	33.2933i	0.707089	−	1.22471i	−0.258844	−	0.965919i	$-0.583342\pi$
$739$	19.2219	−	33.2933i	0.707089	−	1.22471i	0.965932	−	0.258794i	$-0.0833252\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.85728	+	17.0733i	−0.361629	+	0.626359i	−0.988229	−	0.152982i	$-0.951112\pi$
$743$	−9.85728	+	17.0733i	−0.361629	+	0.626359i	0.626600	+	0.779341i	$0.284446\pi$
$744$		0			0
$745$	−0.101112			−0.00370447
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.33259	+	32.2627i	−0.121770	+	1.17885i
$750$		0			0
$751$	22.9247			0.836536			0.418268	−	0.908324i	$-0.362637\pi$
$751$	22.9247			0.836536			0.418268	+	0.908324i	$0.362637\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.60114			0.240240
$756$		0			0
$757$	−52.3408			−1.90236			−0.951179	−	0.308639i	$-0.900127\pi$
$757$	−52.3408			−1.90236			−0.951179	+	0.308639i	$0.900127\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.87815			−0.0680830			−0.0340415	−	0.999420i	$-0.510838\pi$
$761$	−1.87815			−0.0680830			−0.0340415	+	0.999420i	$0.510838\pi$
$762$		0			0
$763$	23.6360	−	10.5742i	0.855682	−	0.382810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16.4657			−0.594542
$768$		0			0
$769$	−13.4060	+	23.2198i	−0.483431	+	0.837327i	−0.999819	−	0.0190276i	$-0.993943\pi$
$769$	−13.4060	+	23.2198i	−0.483431	+	0.837327i	0.516388	+	0.856355i	$0.327276\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.67835	+	8.10314i	−0.168268	+	0.291450i	−0.937811	−	0.347146i	$-0.887151\pi$
$773$	−4.67835	+	8.10314i	−0.168268	+	0.291450i	0.769543	+	0.638595i	$0.220484\pi$
$774$		0			0
$775$	−8.02203	−	13.8946i	−0.288160	−	0.499107i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.27604			0.0815476
$780$		0			0
$781$	−25.0361			−0.895862
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.25434	−	3.90463i	−0.0804608	−	0.139362i
$786$		0			0
$787$	−9.43675	−	16.3449i	−0.336384	−	0.582634i	0.647366	−	0.762179i	$-0.275871\pi$
$787$	−9.43675	−	16.3449i	−0.336384	−	0.582634i	−0.983750	+	0.179546i	$0.942537\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.3007	+	5.50302i	−0.437363	+	0.195665i
$792$		0			0
$793$	10.4782	+	18.1487i	0.372090	+	0.644479i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.9023	−	31.0078i	0.634134	−	1.09835i	−0.352564	−	0.935788i	$-0.614690\pi$
$797$	17.9023	−	31.0078i	0.634134	−	1.09835i	0.986698	−	0.162564i	$-0.0519765\pi$
$798$		0			0
$799$	−27.3729	−	47.4113i	−0.968385	−	1.67729i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22.5530	+	39.0629i	−0.795877	+	1.37850i
$804$		0			0
$805$	2.03248	+	1.47114i	0.0716356	+	0.0518507i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.8027	+	32.5672i	−0.661068	+	1.14500i	0.319268	+	0.947665i	$0.396563\pi$
$809$	−18.8027	+	32.5672i	−0.661068	+	1.14500i	−0.980335	+	0.197338i	$0.936770\pi$
$810$		0			0
$811$	−38.9673			−1.36833			−0.684163	−	0.729329i	$-0.739832\pi$
$811$	−38.9673			−1.36833			−0.684163	+	0.729329i	$0.739832\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.170559	−	0.295418i	−0.00597443	−	0.0103480i
$816$		0			0
$817$	2.00816	−	3.47823i	0.0702565	−	0.121688i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.5874	−	21.8020i	0.439304	−	0.760897i	−0.558332	−	0.829618i	$-0.688558\pi$
$821$	12.5874	−	21.8020i	0.439304	−	0.760897i	0.997636	+	0.0687210i	$0.0218918\pi$
$822$		0			0
$823$	−3.91792	−	6.78604i	−0.136570	−	0.236546i	0.789626	−	0.613588i	$-0.210275\pi$
$823$	−3.91792	−	6.78604i	−0.136570	−	0.236546i	−0.926196	+	0.377042i	$0.876941\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.4941			−0.643103			−0.321552	−	0.946892i	$-0.604204\pi$
$827$	−18.4941			−0.643103			−0.321552	+	0.946892i	$0.604204\pi$
$828$		0			0
$829$	−6.83264	+	11.8345i	−0.237307	+	0.411029i	−0.959941	−	0.280203i	$-0.909598\pi$
$829$	−6.83264	+	11.8345i	−0.237307	+	0.411029i	0.722633	+	0.691232i	$0.242932\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.4173	−	5.93406i	−0.984601	−	0.205603i
$834$		0			0
$835$	−2.38224	+	4.12616i	−0.0824407	+	0.142791i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.5170	−	44.1968i	−0.880945	−	1.52584i	−0.850291	−	0.526312i	$-0.823574\pi$
$839$	−25.5170	−	44.1968i	−0.880945	−	1.52584i	−0.0306540	−	0.999530i	$-0.509759\pi$
$840$		0			0
$841$	9.12994	−	15.8135i	0.314825	−	0.545294i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.58188	−	2.73990i	−0.0544185	−	0.0942556i
$846$		0			0
$847$	28.2994	+	20.4834i	0.972378	+	0.703819i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	9.35047	+	16.1955i	0.320530	+	0.555174i
$852$		0			0
$853$	7.92630	+	13.7287i	0.271391	+	0.470063i	0.969218	−	0.246203i	$-0.0791830\pi$
$853$	7.92630	+	13.7287i	0.271391	+	0.470063i	−0.697827	+	0.716266i	$0.745850\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−42.2001			−1.44153			−0.720764	−	0.693181i	$-0.756209\pi$
$857$	−42.2001			−1.44153			−0.720764	+	0.693181i	$0.756209\pi$
$858$		0			0
$859$	−28.3256			−0.966457			−0.483228	−	0.875494i	$-0.660536\pi$
$859$	−28.3256			−0.966457			−0.483228	+	0.875494i	$0.660536\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.6891	+	30.6385i	0.602145	+	1.04295i	0.992496	+	0.122279i	$0.0390203\pi$
$863$	17.6891	+	30.6385i	0.602145	+	1.04295i	−0.390351	+	0.920666i	$0.627646\pi$
$864$		0			0
$865$	−0.860179	+	1.48987i	−0.0292470	+	0.0506572i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.0498	−	19.1389i	0.374840	−	0.649241i
$870$		0			0
$871$	16.4157			0.556224
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.28006	+	5.26939i	0.246111	+	0.178138i
$876$		0			0
$877$	12.9533			0.437401			0.218700	−	0.975792i	$-0.429818\pi$
$877$	12.9533			0.437401			0.218700	+	0.975792i	$0.429818\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.1995			0.882683			0.441341	−	0.897339i	$-0.354503\pi$
$881$	26.1995			0.882683			0.441341	+	0.897339i	$0.354503\pi$
$882$		0			0
$883$	−6.36625			−0.214241			−0.107121	−	0.994246i	$-0.534163\pi$
$883$	−6.36625			−0.214241			−0.107121	+	0.994246i	$0.534163\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−48.1077			−1.61530			−0.807649	−	0.589664i	$-0.799260\pi$
$887$	−48.1077			−1.61530			−0.807649	+	0.589664i	$0.799260\pi$
$888$		0			0
$889$	4.80603	+	3.47866i	0.161189	+	0.116671i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.34142			−0.178744
$894$		0			0
$895$	3.71582	−	6.43599i	0.124206	−	0.215132i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.38524	−	9.32751i	0.179608	−	0.311090i
$900$		0			0
$901$	−5.72987	−	9.92443i	−0.190890	−	0.330631i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.94892			0.297472
$906$		0			0
$907$	−43.4081			−1.44134			−0.720671	−	0.693278i	$-0.756166\pi$
$907$	−43.4081			−1.44134			−0.720671	+	0.693278i	$0.756166\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11.5141	−	19.9430i	−0.381479	−	0.660740i	0.609795	−	0.792559i	$-0.291252\pi$
$911$	−11.5141	−	19.9430i	−0.381479	−	0.660740i	−0.991274	+	0.131819i	$0.957918\pi$
$912$		0			0
$913$	22.7768	+	39.4506i	0.753802	+	1.30562i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.9766	−	11.5641i	−0.527594	−	0.381879i
$918$		0			0
$919$	−17.2599	−	29.8950i	−0.569351	−	0.986145i	−0.996630	−	0.0820251i	$-0.973861\pi$
$919$	−17.2599	−	29.8950i	−0.569351	−	0.986145i	0.427279	−	0.904120i	$-0.359472\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.95740	−	8.58647i	0.163175	−	0.282627i
$924$		0			0
$925$	16.5458	+	28.6581i	0.544022	+	0.942273i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−14.3176	+	24.7988i	−0.469746	+	0.813623i	−0.999402	−	0.0345892i	$-0.988988\pi$
$929$	−14.3176	+	24.7988i	−0.469746	+	0.813623i	0.529656	+	0.848213i	$0.322321\pi$
$930$		0			0
$931$	−1.88771	+	2.11166i	−0.0618670	+	0.0692068i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.50666	+	6.07370i	−0.114680	+	0.198631i
$936$		0			0
$937$	13.9020			0.454158			0.227079	−	0.973876i	$-0.427083\pi$
$937$	13.9020			0.454158			0.227079	+	0.973876i	$0.427083\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−18.4340	−	31.9286i	−0.600930	−	1.04084i	−0.992681	−	0.120770i	$-0.961464\pi$
$941$	−18.4340	−	31.9286i	−0.600930	−	1.04084i	0.391750	−	0.920072i	$-0.371870\pi$
$942$		0			0
$943$	−7.75930	+	13.4395i	−0.252678	+	0.437650i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.8909	+	32.7200i	−0.613872	+	1.06326i	0.376710	+	0.926331i	$0.377055\pi$
$947$	−18.8909	+	32.7200i	−0.613872	+	1.06326i	−0.990581	+	0.136925i	$0.956278\pi$
$948$		0			0
$949$	−8.93143	−	15.4697i	−0.289926	−	0.502167i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−43.5184			−1.40970			−0.704850	−	0.709356i	$-0.748986\pi$
$953$	−43.5184			−1.40970			−0.704850	+	0.709356i	$0.748986\pi$
$954$		0			0
$955$	3.04294	−	5.27052i	0.0984672	−	0.170550i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.1035	−	13.1035i	−0.584593	−	0.423135i
$960$		0			0
$961$	10.0995	−	17.4929i	0.325792	−	0.564288i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.509468	−	0.882424i	−0.0164003	−	0.0284062i
$966$		0			0
$967$	−17.8032	+	30.8360i	−0.572512	+	0.991619i	0.423795	+	0.905758i	$0.360698\pi$
$967$	−17.8032	+	30.8360i	−0.572512	+	0.991619i	−0.996307	+	0.0858615i	$0.972636\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.2654	+	31.6366i	0.586164	+	1.01527i	0.994729	+	0.102537i	$0.0326960\pi$
$971$	18.2654	+	31.6366i	0.586164	+	1.01527i	−0.408565	+	0.912729i	$0.633971\pi$
$972$		0			0
$973$	8.80792	−	3.94044i	0.282369	−	0.126325i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.6369	+	20.1558i	0.372299	+	0.644840i	0.989919	−	0.141637i	$-0.0452364\pi$
$977$	11.6369	+	20.1558i	0.372299	+	0.644840i	−0.617620	+	0.786477i	$0.711903\pi$
$978$		0			0
$979$	17.4284	+	30.1868i	0.557013	+	0.964775i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.02341			−0.128327			−0.0641634	−	0.997939i	$-0.520438\pi$
$983$	−4.02341			−0.128327			−0.0641634	+	0.997939i	$0.520438\pi$
$984$		0			0
$985$	6.26827			0.199724
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.6921	+	23.7154i	0.435384	+	0.754107i
$990$		0			0
$991$	21.3820	−	37.0348i	0.679223	−	1.17645i	−0.295993	−	0.955190i	$-0.595650\pi$
$991$	21.3820	−	37.0348i	0.679223	−	1.17645i	0.975215	−	0.221258i	$-0.0710163\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.554526	+	0.960467i	−0.0175797	+	0.0304489i
$996$		0			0
$997$	−8.06648			−0.255468			−0.127734	−	0.991808i	$-0.540770\pi$
$997$	−8.06648			−0.255468			−0.127734	+	0.991808i	$0.540770\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.l.n.109.4		16
3.2	odd	2	inner	2268.2.l.n.109.5		16
7.2	even	3		2268.2.i.n.2053.5		16
9.2	odd	6		2268.2.i.n.865.4		16
9.4	even	3		2268.2.k.g.1621.5	yes	16
9.5	odd	6		2268.2.k.g.1621.4	yes	16
9.7	even	3		2268.2.i.n.865.5		16
21.2	odd	6		2268.2.i.n.2053.4		16
63.2	odd	6	inner	2268.2.l.n.541.5		16
63.16	even	3	inner	2268.2.l.n.541.4		16
63.23	odd	6		2268.2.k.g.1297.4	✓	16
63.58	even	3		2268.2.k.g.1297.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.i.n.865.4		16	9.2	odd	6
2268.2.i.n.865.5		16	9.7	even	3
2268.2.i.n.2053.4		16	21.2	odd	6
2268.2.i.n.2053.5		16	7.2	even	3
2268.2.k.g.1297.4	✓	16	63.23	odd	6
2268.2.k.g.1297.5	yes	16	63.58	even	3
2268.2.k.g.1621.4	yes	16	9.5	odd	6
2268.2.k.g.1621.5	yes	16	9.4	even	3
2268.2.l.n.109.4		16	1.1	even	1	trivial
2268.2.l.n.109.5		16	3.2	odd	2	inner
2268.2.l.n.541.4		16	63.16	even	3	inner
2268.2.l.n.541.5		16	63.2	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-0.343737 q^{5} +(2.14324 + 1.55130i) q^{7} +O(q^{10})\)	\(q-0.343737 q^{5} +(2.14324 + 1.55130i) q^{7} -4.91976 q^{11} +(0.974162 - 1.68730i) q^{13} +(-2.07359 + 3.59156i) q^{17} +(0.202315 + 0.350420i) q^{19} -2.75886 q^{23} -4.88184 q^{25} +(-1.63861 - 2.83815i) q^{29} +(1.64324 + 2.84617i) q^{31} +(-0.736710 - 0.533240i) q^{35} +(-3.38925 - 5.87035i) q^{37} +(2.81250 - 4.87139i) q^{41} +(-4.96295 - 8.59608i) q^{43} +(-6.60038 + 11.4322i) q^{47} +(2.18693 + 6.64961i) q^{49} +(-1.38163 + 2.39306i) q^{53} +1.69110 q^{55} +(-4.22560 - 7.31896i) q^{59} +(-5.37804 + 9.31503i) q^{61} +(-0.334856 + 0.579987i) q^{65} +(4.21277 + 7.29673i) q^{67} +5.08888 q^{71} +(4.58416 - 7.94000i) q^{73} +(-10.5442 - 7.63203i) q^{77} +(-2.24601 + 3.89020i) q^{79} +(-4.62966 - 8.01880i) q^{83} +(0.712770 - 1.23455i) q^{85} +(-3.54253 - 6.13584i) q^{89} +(4.70537 - 2.10506i) q^{91} +(-0.0695431 - 0.120452i) q^{95} +(-1.31231 - 2.27299i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 10 q^{13} + 8 q^{19} - 8 q^{31} - 4 q^{37} - 10 q^{43} - 20 q^{49} - 32 q^{55} + 28 q^{61} + 18 q^{67} - 20 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97}+O(q^{100}) \) 16 * q + 10 * q^13 + 8 * q^19 - 8 * q^31 - 4 * q^37 - 10 * q^43 - 20 * q^49 - 32 * q^55 + 28 * q^61 + 18 * q^67 - 20 * q^79 - 38 * q^85 - 2 * q^91 + 42 * q^97

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