LMFDB - Embedded newform 1440.2.bi.c.1423.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(847,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))

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

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

chi := DirichletCharacter("1440.847");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bi (of order $4$, 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1423.1
Root			$-0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1440.1423
Dual form			1440.2.bi.c.847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.90211 + 1.17557i) q^{5} +(-1.17557 - 1.17557i) q^{7} +O(q^{10})$	$q+(-1.90211 + 1.17557i) q^{5} +(-1.17557 - 1.17557i) q^{7} +1.23607 q^{11} +(-3.07768 + 3.07768i) q^{13} +(1.00000 - 1.00000i) q^{17} -2.00000i q^{19} +(2.62866 - 2.62866i) q^{23} +(2.23607 - 4.47214i) q^{25} +1.45309 q^{29} -5.25731i q^{31} +(3.61803 + 0.854102i) q^{35} +(3.07768 + 3.07768i) q^{37} +7.70820 q^{41} +(-2.38197 - 2.38197i) q^{43} +(-7.33094 - 7.33094i) q^{47} -4.23607i q^{49} +(0.726543 - 0.726543i) q^{53} +(-2.35114 + 1.45309i) q^{55} -8.47214i q^{59} -9.95959i q^{61} +(2.23607 - 9.47214i) q^{65} +(2.38197 - 2.38197i) q^{67} +7.05342i q^{71} +(8.70820 + 8.70820i) q^{73} +(-1.45309 - 1.45309i) q^{77} +12.3107 q^{79} +(-4.38197 - 4.38197i) q^{83} +(-0.726543 + 3.07768i) q^{85} -6.47214i q^{89} +7.23607 q^{91} +(2.35114 + 3.80423i) q^{95} +(-0.236068 + 0.236068i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} + 8 q^{17} + 20 q^{35} + 8 q^{41} - 28 q^{43} + 28 q^{67} + 16 q^{73} - 44 q^{83} + 40 q^{91} + 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^11 + 8 * q^17 + 20 * q^35 + 8 * q^41 - 28 * q^43 + 28 * q^67 + 16 * q^73 - 44 * q^83 + 40 * q^91 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$-1$

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
$5$	−1.90211	+	1.17557i	−0.850651	+	0.525731i
$5$	−1.90211	+	1.17557i	−0.850651	+	0.525731i
$6$		0			0
$7$	−1.17557	−	1.17557i	−0.444324	−	0.444324i	0.449138	−	0.893462i	$-0.351731\pi$
$7$	−1.17557	−	1.17557i	−0.444324	−	0.444324i	−0.893462	+	0.449138i	$0.851731\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.23607			0.372689			0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607			0.372689			0.186344	+	0.982485i	$0.440336\pi$
$12$		0			0
$13$	−3.07768	+	3.07768i	−0.853596	+	0.853596i	−0.990574	−	0.136978i	$-0.956261\pi$
$13$	−3.07768	+	3.07768i	−0.853596	+	0.853596i	0.136978	+	0.990574i	$0.456261\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	−	1.00000i	0.242536	−	0.242536i	−0.575363	−	0.817898i	$-0.695139\pi$
$17$	1.00000	−	1.00000i	0.242536	−	0.242536i	0.817898	+	0.575363i	$0.195139\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.62866	−	2.62866i	0.548113	−	0.548113i	−0.377782	−	0.925895i	$-0.623313\pi$
$23$	2.62866	−	2.62866i	0.548113	−	0.548113i	0.925895	+	0.377782i	$0.123313\pi$
$24$		0			0
$25$	2.23607	−	4.47214i	0.447214	−	0.894427i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.45309			0.269831			0.134916	−	0.990857i	$-0.456924\pi$
$29$	1.45309			0.269831			0.134916	+	0.990857i	$0.456924\pi$
$30$		0			0
$31$		−	5.25731i		−	0.944241i	−0.881534	−	0.472120i	$-0.843489\pi$
$31$		−	5.25731i		−	0.944241i	0.881534	−	0.472120i	$-0.156511\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.61803	+	0.854102i	0.611559	+	0.144370i
$36$		0			0
$37$	3.07768	+	3.07768i	0.505968	+	0.505968i	0.913286	−	0.407318i	$-0.133536\pi$
$37$	3.07768	+	3.07768i	0.505968	+	0.505968i	−0.407318	+	0.913286i	$0.633536\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.70820			1.20382			0.601910	−	0.798564i	$-0.294407\pi$
$41$	7.70820			1.20382			0.601910	+	0.798564i	$0.294407\pi$
$42$		0			0
$43$	−2.38197	−	2.38197i	−0.363246	−	0.363246i	0.501760	−	0.865007i	$-0.332686\pi$
$43$	−2.38197	−	2.38197i	−0.363246	−	0.363246i	−0.865007	+	0.501760i	$0.832686\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.33094	−	7.33094i	−1.06933	−	1.06933i	−0.997411	−	0.0719165i	$-0.977088\pi$
$47$	−7.33094	−	7.33094i	−1.06933	−	1.06933i	−0.0719165	−	0.997411i	$-0.522912\pi$
$48$		0			0
$49$		−	4.23607i		−	0.605153i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.726543	−	0.726543i	0.0997983	−	0.0997983i	−0.655445	−	0.755243i	$-0.727519\pi$
$53$	0.726543	−	0.726543i	0.0997983	−	0.0997983i	0.755243	+	0.655445i	$0.227519\pi$
$54$		0			0
$55$	−2.35114	+	1.45309i	−0.317028	+	0.195934i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	8.47214i		−	1.10298i	−0.834182	−	0.551489i	$-0.814060\pi$
$59$		−	8.47214i		−	1.10298i	0.834182	−	0.551489i	$-0.185940\pi$
$60$		0			0
$61$		−	9.95959i		−	1.27520i	−0.770370	−	0.637598i	$-0.779928\pi$
$61$		−	9.95959i		−	1.27520i	0.770370	−	0.637598i	$-0.220072\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.23607	−	9.47214i	0.277350	−	1.17487i
$66$		0			0
$67$	2.38197	−	2.38197i	0.291003	−	0.291003i	−0.546473	−	0.837477i	$-0.684030\pi$
$67$	2.38197	−	2.38197i	0.291003	−	0.291003i	0.837477	+	0.546473i	$0.184030\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.05342i			0.837087i	0.908197	+	0.418544i	$0.137459\pi$
$71$			7.05342i			0.837087i	−0.908197	+	0.418544i	$0.862541\pi$
$72$		0			0
$73$	8.70820	+	8.70820i	1.01922	+	1.01922i	0.999812	+	0.0194065i	$0.00617767\pi$
$73$	8.70820	+	8.70820i	1.01922	+	1.01922i	0.0194065	+	0.999812i	$0.493822\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.45309	−	1.45309i	−0.165594	−	0.165594i
$78$		0			0
$79$	12.3107			1.38507			0.692533	−	0.721386i	$-0.256495\pi$
$79$	12.3107			1.38507			0.692533	+	0.721386i	$0.256495\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.38197	−	4.38197i	−0.480983	−	0.480983i	0.424462	−	0.905446i	$-0.360463\pi$
$83$	−4.38197	−	4.38197i	−0.480983	−	0.480983i	−0.905446	+	0.424462i	$0.860463\pi$
$84$		0			0
$85$	−0.726543	+	3.07768i	−0.0788046	+	0.333822i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.47214i		−	0.686045i	−0.939327	−	0.343023i	$-0.888549\pi$
$89$		−	6.47214i		−	0.686045i	0.939327	−	0.343023i	$-0.111451\pi$
$90$		0			0
$91$	7.23607			0.758546
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.35114	+	3.80423i	0.241222	+	0.390305i
$96$		0			0
$97$	−0.236068	+	0.236068i	−0.0239691	+	0.0239691i	−0.718990	−	0.695021i	$-0.755395\pi$
$97$	−0.236068	+	0.236068i	−0.0239691	+	0.0239691i	0.695021	+	0.718990i	$0.255395\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	12.3107i		−	1.22496i	−0.790485	−	0.612482i	$-0.790171\pi$
$101$		−	12.3107i		−	1.22496i	0.790485	−	0.612482i	$-0.209829\pi$
$102$		0			0
$103$	−7.33094	+	7.33094i	−0.722339	+	0.722339i	−0.969081	−	0.246742i	$-0.920640\pi$
$103$	−7.33094	+	7.33094i	−0.722339	+	0.722339i	0.246742	+	0.969081i	$0.420640\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0902	−	12.0902i	1.16880	−	1.16880i	0.186310	−	0.982491i	$-0.440347\pi$
$107$	12.0902	−	12.0902i	1.16880	−	1.16880i	0.982491	−	0.186310i	$-0.0596528\pi$
$108$		0			0
$109$	6.71040			0.642739			0.321370	−	0.946954i	$-0.395857\pi$
$109$	6.71040			0.642739			0.321370	+	0.946954i	$0.395857\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.70820	+	4.70820i	0.442911	+	0.442911i	0.892989	−	0.450078i	$-0.148604\pi$
$113$	4.70820	+	4.70820i	0.442911	+	0.442911i	−0.450078	+	0.892989i	$0.648604\pi$
$114$		0			0
$115$	−1.90983	+	8.09017i	−0.178093	+	0.754412i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.35114			−0.215529
$120$		0			0
$121$	−9.47214			−0.861103
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00406	+	11.1352i	0.0898056	+	0.995959i
$126$		0			0
$127$	−8.78402	−	8.78402i	−0.779456	−	0.779456i	0.200282	−	0.979738i	$-0.435814\pi$
$127$	−8.78402	−	8.78402i	−0.779456	−	0.779456i	−0.979738	+	0.200282i	$0.935814\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.291796			0.0254943			0.0127472	−	0.999919i	$-0.495942\pi$
$131$	0.291796			0.0254943			0.0127472	+	0.999919i	$0.495942\pi$
$132$		0			0
$133$	−2.35114	+	2.35114i	−0.203870	+	0.203870i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.47214	+	3.47214i	−0.296645	+	0.296645i	−0.839698	−	0.543054i	$-0.817268\pi$
$137$	−3.47214	+	3.47214i	−0.296645	+	0.296645i	0.543054	+	0.839698i	$0.317268\pi$
$138$		0			0
$139$		−	5.41641i		−	0.459414i	−0.973260	−	0.229707i	$-0.926223\pi$
$139$		−	5.41641i		−	0.459414i	0.973260	−	0.229707i	$-0.0737767\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.80423	+	3.80423i	−0.318125	+	0.318125i
$144$		0			0
$145$	−2.76393	+	1.70820i	−0.229532	+	0.141859i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.2088			−1.08211			−0.541053	−	0.840988i	$-0.681974\pi$
$149$	−13.2088			−1.08211			−0.541053	+	0.840988i	$0.681974\pi$
$150$		0			0
$151$			14.6619i			1.19317i	0.802551	+	0.596583i	$0.203475\pi$
$151$			14.6619i			1.19317i	−0.802551	+	0.596583i	$0.796525\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.18034	+	10.0000i	0.496417	+	0.803219i
$156$		0			0
$157$	−9.78808	−	9.78808i	−0.781174	−	0.781174i	0.198855	−	0.980029i	$-0.436278\pi$
$157$	−9.78808	−	9.78808i	−0.781174	−	0.781174i	−0.980029	+	0.198855i	$0.936278\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.18034			−0.487079
$162$		0			0
$163$	7.14590	+	7.14590i	0.559710	+	0.559710i	0.929225	−	0.369515i	$-0.120476\pi$
$163$	7.14590	+	7.14590i	0.559710	+	0.559710i	−0.369515	+	0.929225i	$0.620476\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.277515	+	0.277515i	0.0214747	+	0.0214747i	0.717763	−	0.696288i	$-0.245166\pi$
$167$	0.277515	+	0.277515i	0.0214747	+	0.0214747i	−0.696288	+	0.717763i	$0.745166\pi$
$168$		0			0
$169$		−	5.94427i		−	0.457252i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.32688	−	6.32688i	0.481024	−	0.481024i	−0.424435	−	0.905459i	$-0.639527\pi$
$173$	6.32688	−	6.32688i	0.481024	−	0.481024i	0.905459	+	0.424435i	$0.139527\pi$
$174$		0			0
$175$	−7.88597	+	2.62866i	−0.596123	+	0.198708i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.4721i			1.23119i	0.788065	+	0.615593i	$0.211083\pi$
$179$			16.4721i			1.23119i	−0.788065	+	0.615593i	$0.788917\pi$
$180$		0			0
$181$		−	9.40456i		−	0.699036i	−0.936930	−	0.349518i	$-0.886345\pi$
$181$		−	9.40456i		−	0.699036i	0.936930	−	0.349518i	$-0.113655\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.47214	−	2.23607i	−0.696405	−	0.164399i
$186$		0			0
$187$	1.23607	−	1.23607i	0.0903902	−	0.0903902i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.8658i		−	0.930934i	−0.885065	−	0.465467i	$-0.845886\pi$
$191$		−	12.8658i		−	0.930934i	0.885065	−	0.465467i	$-0.154114\pi$
$192$		0			0
$193$	−7.47214	−	7.47214i	−0.537856	−	0.537856i	0.385043	−	0.922899i	$-0.374187\pi$
$193$	−7.47214	−	7.47214i	−0.537856	−	0.537856i	−0.922899	+	0.385043i	$0.874187\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.17963	−	2.17963i	−0.155292	−	0.155292i	0.625185	−	0.780477i	$-0.285024\pi$
$197$	−2.17963	−	2.17963i	−0.155292	−	0.155292i	−0.780477	+	0.625185i	$0.785024\pi$
$198$		0			0
$199$	18.1231			1.28471			0.642355	−	0.766407i	$-0.277957\pi$
$199$	18.1231			1.28471			0.642355	+	0.766407i	$0.277957\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.70820	−	1.70820i	−0.119892	−	0.119892i
$204$		0			0
$205$	−14.6619	+	9.06154i	−1.02403	+	0.632885i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.47214i		−	0.171001i
$210$		0			0
$211$	−15.7082			−1.08140			−0.540699	−	0.841216i	$-0.681840\pi$
$211$	−15.7082			−1.08140			−0.540699	+	0.841216i	$0.681840\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.33094	+	1.73060i	0.499966	+	0.118026i
$216$		0			0
$217$	−6.18034	+	6.18034i	−0.419549	+	0.419549i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			6.15537i			0.414055i
$222$		0			0
$223$	−1.73060	+	1.73060i	−0.115890	+	0.115890i	−0.762673	−	0.646784i	$-0.776114\pi$
$223$	−1.73060	+	1.73060i	−0.115890	+	0.115890i	0.646784	+	0.762673i	$0.276114\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.6180	+	11.6180i	−0.771116	+	0.771116i	−0.978302	−	0.207186i	$-0.933570\pi$
$227$	−11.6180	+	11.6180i	−0.771116	+	0.771116i	0.207186	+	0.978302i	$0.433570\pi$
$228$		0			0
$229$	21.3723			1.41232			0.706160	−	0.708053i	$-0.250426\pi$
$229$	21.3723			1.41232			0.706160	+	0.708053i	$0.250426\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.47214	+	3.47214i	0.227467	+	0.227467i	0.811634	−	0.584167i	$-0.198579\pi$
$233$	3.47214	+	3.47214i	0.227467	+	0.227467i	−0.584167	+	0.811634i	$0.698579\pi$
$234$		0			0
$235$	22.5623	+	5.32624i	1.47180	+	0.347445i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	29.3238			1.89680			0.948398	−	0.317083i	$-0.102703\pi$
$239$	29.3238			1.89680			0.948398	+	0.317083i	$0.102703\pi$
$240$		0			0
$241$	6.76393			0.435703			0.217852	−	0.975982i	$-0.430095\pi$
$241$	6.76393			0.435703			0.217852	+	0.975982i	$0.430095\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.97980	+	8.05748i	0.318148	+	0.514774i
$246$		0			0
$247$	6.15537	+	6.15537i	0.391657	+	0.391657i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−22.1803			−1.40001			−0.700005	−	0.714138i	$-0.746819\pi$
$251$	−22.1803			−1.40001			−0.700005	+	0.714138i	$0.746819\pi$
$252$		0			0
$253$	3.24920	−	3.24920i	0.204275	−	0.204275i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.7082	+	18.7082i	−1.16699	+	1.16699i	−0.184073	+	0.982913i	$0.558928\pi$
$257$	−18.7082	+	18.7082i	−1.16699	+	1.16699i	−0.982913	+	0.184073i	$0.941072\pi$
$258$		0			0
$259$		−	7.23607i		−	0.449627i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.3925	+	16.3925i	−1.01080	+	1.01080i	−0.0108623	+	0.999941i	$0.503458\pi$
$263$	−16.3925	+	16.3925i	−1.01080	+	1.01080i	−0.999941	+	0.0108623i	$0.996542\pi$
$264$		0			0
$265$	−0.527864	+	2.23607i	−0.0324264	+	0.137361i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.9111			1.09206			0.546029	−	0.837766i	$-0.316139\pi$
$269$	17.9111			1.09206			0.546029	+	0.837766i	$0.316139\pi$
$270$		0			0
$271$		−	31.6749i		−	1.92411i	−0.272851	−	0.962056i	$-0.587967\pi$
$271$		−	31.6749i		−	1.92411i	0.272851	−	0.962056i	$-0.412033\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.76393	−	5.52786i	0.166671	−	0.333343i
$276$		0			0
$277$	−2.17963	−	2.17963i	−0.130961	−	0.130961i	0.638588	−	0.769549i	$-0.279519\pi$
$277$	−2.17963	−	2.17963i	−0.130961	−	0.130961i	−0.769549	+	0.638588i	$0.779519\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.70820			−0.221213			−0.110606	−	0.993864i	$-0.535279\pi$
$281$	−3.70820			−0.221213			−0.110606	+	0.993864i	$0.535279\pi$
$282$		0			0
$283$	−15.6180	−	15.6180i	−0.928396	−	0.928396i	0.0692066	−	0.997602i	$-0.477953\pi$
$283$	−15.6180	−	15.6180i	−0.928396	−	0.928396i	−0.997602	+	0.0692066i	$0.977953\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.06154	−	9.06154i	−0.534886	−	0.534886i
$288$		0			0
$289$			15.0000i			0.882353i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.726543	+	0.726543i	−0.0424451	+	0.0424451i	−0.728011	−	0.685566i	$-0.759555\pi$
$293$	−0.726543	+	0.726543i	−0.0424451	+	0.0424451i	0.685566	+	0.728011i	$0.259555\pi$
$294$		0			0
$295$	9.95959	+	16.1150i	0.579870	+	0.938249i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			16.1803i			0.935733i
$300$		0			0
$301$			5.60034i			0.322798i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.7082	+	18.9443i	0.670410	+	1.08475i
$306$		0			0
$307$	−6.56231	+	6.56231i	−0.374531	+	0.374531i	−0.869124	−	0.494594i	$-0.835317\pi$
$307$	−6.56231	+	6.56231i	−0.374531	+	0.374531i	0.494594	+	0.869124i	$0.335317\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			8.16348i			0.462909i	0.972846	+	0.231454i	$0.0743484\pi$
$311$			8.16348i			0.462909i	−0.972846	+	0.231454i	$0.925652\pi$
$312$		0			0
$313$	−6.23607	−	6.23607i	−0.352483	−	0.352483i	0.508549	−	0.861033i	$-0.330182\pi$
$313$	−6.23607	−	6.23607i	−0.352483	−	0.352483i	−0.861033	+	0.508549i	$0.830182\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.6861	+	10.6861i	0.600193	+	0.600193i	0.940364	−	0.340171i	$-0.110485\pi$
$317$	10.6861	+	10.6861i	0.600193	+	0.600193i	−0.340171	+	0.940364i	$0.610485\pi$
$318$		0			0
$319$	1.79611			0.100563
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.00000	−	2.00000i	−0.111283	−	0.111283i
$324$		0			0
$325$	6.88191	+	20.6457i	0.381740	+	1.14522i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			17.2361i			0.950255i
$330$		0			0
$331$	−28.0689			−1.54281			−0.771403	−	0.636347i	$-0.780445\pi$
$331$	−28.0689			−1.54281			−0.771403	+	0.636347i	$0.780445\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.73060	+	7.33094i	−0.0945528	+	0.400532i
$336$		0			0
$337$	−2.05573	+	2.05573i	−0.111983	+	0.111983i	−0.760878	−	0.648895i	$-0.775231\pi$
$337$	−2.05573	+	2.05573i	−0.111983	+	0.111983i	0.648895	+	0.760878i	$0.275231\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.49839i		−	0.351908i
$342$		0			0
$343$	−13.2088	+	13.2088i	−0.713208	+	0.713208i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.03444	+	3.03444i	−0.162897	+	0.162897i	−0.783849	−	0.620952i	$-0.786746\pi$
$347$	−3.03444	+	3.03444i	−0.162897	+	0.162897i	0.620952	+	0.783849i	$0.286746\pi$
$348$		0			0
$349$	15.5599			0.832904			0.416452	−	0.909158i	$-0.363273\pi$
$349$	15.5599			0.832904			0.416452	+	0.909158i	$0.363273\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.41641	−	4.41641i	−0.235062	−	0.235062i	0.579740	−	0.814802i	$-0.303154\pi$
$353$	−4.41641	−	4.41641i	−0.235062	−	0.235062i	−0.814802	+	0.579740i	$0.803154\pi$
$354$		0			0
$355$	−8.29180	−	13.4164i	−0.440083	−	0.712069i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.79611			−0.0947952			−0.0473976	−	0.998876i	$-0.515093\pi$
$359$	−1.79611			−0.0947952			−0.0473976	+	0.998876i	$0.515093\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.8011	−	6.32688i	−1.40283	−	0.331164i
$366$		0			0
$367$	−24.0009	−	24.0009i	−1.25284	−	1.25284i	−0.954442	−	0.298396i	$-0.903548\pi$
$367$	−24.0009	−	24.0009i	−1.25284	−	1.25284i	−0.298396	−	0.954442i	$-0.596452\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.70820			−0.0886855
$372$		0			0
$373$	−15.0454	+	15.0454i	−0.779021	+	0.779021i	−0.979664	−	0.200644i	$-0.935697\pi$
$373$	−15.0454	+	15.0454i	−0.779021	+	0.779021i	0.200644	+	0.979664i	$0.435697\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.47214	+	4.47214i	−0.230327	+	0.230327i
$378$		0			0
$379$			35.8885i			1.84347i	0.387820	+	0.921735i	$0.373228\pi$
$379$			35.8885i			1.84347i	−0.387820	+	0.921735i	$0.626772\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.1352	−	11.1352i	0.568980	−	0.568980i	−0.362862	−	0.931843i	$-0.618201\pi$
$383$	11.1352	−	11.1352i	0.568980	−	0.568980i	0.931843	+	0.362862i	$0.118201\pi$
$384$		0			0
$385$	4.47214	+	1.05573i	0.227921	+	0.0538049i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	23.7234			1.20282			0.601412	−	0.798939i	$-0.294605\pi$
$389$	23.7234			1.20282			0.601412	+	0.798939i	$0.294605\pi$
$390$		0			0
$391$		−	5.25731i		−	0.265874i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−23.4164	+	14.4721i	−1.17821	+	0.728172i
$396$		0			0
$397$	−7.22494	−	7.22494i	−0.362609	−	0.362609i	0.502164	−	0.864773i	$-0.332538\pi$
$397$	−7.22494	−	7.22494i	−0.362609	−	0.362609i	−0.864773	+	0.502164i	$0.832538\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.88854			0.194185			0.0970923	−	0.995275i	$-0.469046\pi$
$401$	3.88854			0.194185			0.0970923	+	0.995275i	$0.469046\pi$
$402$		0			0
$403$	16.1803	+	16.1803i	0.806000	+	0.806000i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.80423	+	3.80423i	0.188568	+	0.188568i
$408$		0			0
$409$		−	27.5967i		−	1.36457i	−0.731086	−	0.682286i	$-0.760986\pi$
$409$		−	27.5967i		−	1.36457i	0.731086	−	0.682286i	$-0.239014\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.95959	+	9.95959i	−0.490080	+	0.490080i
$414$		0			0
$415$	13.4863	+	3.18368i	0.662017	+	0.156281i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	24.8328i		−	1.21316i	−0.795022	−	0.606581i	$-0.792540\pi$
$419$		−	24.8328i		−	1.21316i	0.795022	−	0.606581i	$-0.207460\pi$
$420$		0			0
$421$		−	3.46120i		−	0.168689i	−0.996437	−	0.0843443i	$-0.973120\pi$
$421$		−	3.46120i		−	0.168689i	0.996437	−	0.0843443i	$-0.0268795\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.23607	−	6.70820i	−0.108465	−	0.325396i
$426$		0			0
$427$	−11.7082	+	11.7082i	−0.566600	+	0.566600i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	11.7557i		−	0.566252i	−0.959083	−	0.283126i	$-0.908628\pi$
$431$		−	11.7557i		−	0.566252i	0.959083	−	0.283126i	$-0.0913715\pi$
$432$		0			0
$433$	23.1803	+	23.1803i	1.11398	+	1.11398i	0.992608	+	0.121369i	$0.0387283\pi$
$433$	23.1803	+	23.1803i	1.11398	+	1.11398i	0.121369	+	0.992608i	$0.461272\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.25731	−	5.25731i	−0.251491	−	0.251491i
$438$		0			0
$439$	−11.2007			−0.534579			−0.267290	−	0.963616i	$-0.586128\pi$
$439$	−11.2007			−0.534579			−0.267290	+	0.963616i	$0.586128\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.0902	+	10.0902i	0.479398	+	0.479398i	0.904939	−	0.425541i	$-0.139916\pi$
$443$	10.0902	+	10.0902i	0.479398	+	0.479398i	−0.425541	+	0.904939i	$0.639916\pi$
$444$		0			0
$445$	7.60845	+	12.3107i	0.360675	+	0.583585i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	31.5967i		−	1.49114i	−0.666426	−	0.745571i	$-0.732177\pi$
$449$		−	31.5967i		−	1.49114i	0.666426	−	0.745571i	$-0.267823\pi$
$450$		0			0
$451$	9.52786			0.448650
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−13.7638	+	8.50651i	−0.645258	+	0.398791i
$456$		0			0
$457$	−21.6525	+	21.6525i	−1.01286	+	1.01286i	−0.0129439	+	0.999916i	$0.504120\pi$
$457$	−21.6525	+	21.6525i	−1.01286	+	1.01286i	−0.999916	+	0.0129439i	$0.995880\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			6.49839i			0.302660i	0.988483	+	0.151330i	$0.0483557\pi$
$461$			6.49839i			0.302660i	−0.988483	+	0.151330i	$0.951644\pi$
$462$		0			0
$463$	17.2905	−	17.2905i	0.803559	−	0.803559i	−0.180091	−	0.983650i	$-0.557639\pi$
$463$	17.2905	−	17.2905i	0.803559	−	0.803559i	0.983650	+	0.180091i	$0.0576392\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.67376	+	2.67376i	−0.123727	+	0.123727i	−0.766259	−	0.642532i	$-0.777884\pi$
$467$	−2.67376	+	2.67376i	−0.123727	+	0.123727i	0.642532	+	0.766259i	$0.277884\pi$
$468$		0			0
$469$	−5.60034			−0.258600
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.94427	−	2.94427i	−0.135378	−	0.135378i
$474$		0			0
$475$	−8.94427	−	4.47214i	−0.410391	−	0.205196i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.60845			−0.347639			−0.173820	−	0.984778i	$-0.555611\pi$
$479$	−7.60845			−0.347639			−0.173820	+	0.984778i	$0.555611\pi$
$480$		0			0
$481$	−18.9443			−0.863784
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.171513	−	0.726543i	0.00778802	−	0.0329906i
$486$		0			0
$487$	9.33905	+	9.33905i	0.423193	+	0.423193i	0.886302	−	0.463109i	$-0.153266\pi$
$487$	9.33905	+	9.33905i	0.423193	+	0.423193i	−0.463109	+	0.886302i	$0.653266\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.7639			−0.485769			−0.242885	−	0.970055i	$-0.578094\pi$
$491$	−10.7639			−0.485769			−0.242885	+	0.970055i	$0.578094\pi$
$492$		0			0
$493$	1.45309	−	1.45309i	0.0654437	−	0.0654437i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.29180	−	8.29180i	0.371938	−	0.371938i
$498$		0			0
$499$		−	23.8885i		−	1.06940i	−0.845043	−	0.534699i	$-0.820425\pi$
$499$		−	23.8885i		−	1.06940i	0.845043	−	0.534699i	$-0.179575\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.53483	−	5.53483i	0.246786	−	0.246786i	−0.572864	−	0.819650i	$-0.694168\pi$
$503$	5.53483	−	5.53483i	0.246786	−	0.246786i	0.819650	+	0.572864i	$0.194168\pi$
$504$		0			0
$505$	14.4721	+	23.4164i	0.644002	+	1.04202i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.06154			0.401646			0.200823	−	0.979628i	$-0.435638\pi$
$509$	9.06154			0.401646			0.200823	+	0.979628i	$0.435638\pi$
$510$		0			0
$511$		−	20.4742i		−	0.905726i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.32624	−	22.5623i	0.234702	−	0.994214i
$516$		0			0
$517$	−9.06154	−	9.06154i	−0.398526	−	0.398526i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.47214			−0.371171			−0.185586	−	0.982628i	$-0.559418\pi$
$521$	−8.47214			−0.371171			−0.185586	+	0.982628i	$0.559418\pi$
$522$		0			0
$523$	16.7426	+	16.7426i	0.732105	+	0.732105i	0.971036	−	0.238932i	$-0.0767972\pi$
$523$	16.7426	+	16.7426i	0.732105	+	0.732105i	−0.238932	+	0.971036i	$0.576797\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.25731	−	5.25731i	−0.229012	−	0.229012i
$528$		0			0
$529$			9.18034i			0.399145i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−23.7234	+	23.7234i	−1.02758	+	1.02758i
$534$		0			0
$535$	−8.78402	+	37.2097i	−0.379766	+	1.60872i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	5.23607i		−	0.225533i
$540$		0			0
$541$		−	2.90617i		−	0.124946i	−0.998047	−	0.0624730i	$-0.980101\pi$
$541$		−	2.90617i		−	0.124946i	0.998047	−	0.0624730i	$-0.0198987\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.7639	+	7.88854i	−0.546747	+	0.337908i
$546$		0			0
$547$	8.56231	−	8.56231i	0.366098	−	0.366098i	−0.499954	−	0.866052i	$-0.666650\pi$
$547$	8.56231	−	8.56231i	0.366098	−	0.366098i	0.866052	+	0.499954i	$0.166650\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2.90617i		−	0.123807i
$552$		0			0
$553$	−14.4721	−	14.4721i	−0.615418	−	0.615418i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.1845	−	17.1845i	−0.728132	−	0.728132i	0.242116	−	0.970247i	$-0.422159\pi$
$557$	−17.1845	−	17.1845i	−0.728132	−	0.728132i	−0.970247	+	0.242116i	$0.922159\pi$
$558$		0			0
$559$	14.6619			0.620131
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.3262	+	11.3262i	0.477344	+	0.477344i	0.904281	−	0.426937i	$-0.140407\pi$
$563$	11.3262	+	11.3262i	0.477344	+	0.477344i	−0.426937	+	0.904281i	$0.640407\pi$
$564$		0			0
$565$	−14.4904	−	3.42071i	−0.609614	−	0.143910i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			13.1246i			0.550212i	0.961414	+	0.275106i	$0.0887130\pi$
$569$			13.1246i			0.550212i	−0.961414	+	0.275106i	$0.911287\pi$
$570$		0			0
$571$	8.65248			0.362095			0.181047	−	0.983474i	$-0.442051\pi$
$571$	8.65248			0.362095			0.181047	+	0.983474i	$0.442051\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.87785	−	17.6336i	−0.245123	−	0.735370i
$576$		0			0
$577$	21.7639	−	21.7639i	0.906044	−	0.906044i	−0.0899059	−	0.995950i	$-0.528657\pi$
$577$	21.7639	−	21.7639i	0.906044	−	0.906044i	0.995950	+	0.0899059i	$0.0286566\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			10.3026i			0.427425i
$582$		0			0
$583$	0.898056	−	0.898056i	0.0371937	−	0.0371937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.90983	−	5.90983i	0.243925	−	0.243925i	−0.574547	−	0.818472i	$-0.694822\pi$
$587$	5.90983	−	5.90983i	0.243925	−	0.243925i	0.818472	+	0.574547i	$0.194822\pi$
$588$		0			0
$589$	−10.5146			−0.433247
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.41641	+	6.41641i	0.263490	+	0.263490i	0.826470	−	0.562980i	$-0.190345\pi$
$593$	6.41641	+	6.41641i	0.263490	+	0.263490i	−0.562980	+	0.826470i	$0.690345\pi$
$594$		0			0
$595$	4.47214	−	2.76393i	0.183340	−	0.113310i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	27.5276			1.12475			0.562374	−	0.826883i	$-0.309888\pi$
$599$	27.5276			1.12475			0.562374	+	0.826883i	$0.309888\pi$
$600$		0			0
$601$	−4.29180			−0.175066			−0.0875330	−	0.996162i	$-0.527898\pi$
$601$	−4.29180			−0.175066			−0.0875330	+	0.996162i	$0.527898\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	18.0171	−	11.1352i	0.732498	−	0.452709i
$606$		0			0
$607$	−6.08985	−	6.08985i	−0.247180	−	0.247180i	0.572633	−	0.819812i	$-0.305922\pi$
$607$	−6.08985	−	6.08985i	−0.247180	−	0.247180i	−0.819812	+	0.572633i	$0.805922\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	45.1246			1.82555
$612$		0			0
$613$	27.3561	−	27.3561i	1.10490	−	1.10490i	0.111094	−	0.993810i	$-0.464565\pi$
$613$	27.3561	−	27.3561i	1.10490	−	1.10490i	0.993810	−	0.111094i	$-0.0354353\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.8885	−	30.8885i	1.24353	−	1.24353i	0.284998	−	0.958528i	$-0.408007\pi$
$617$	30.8885	−	30.8885i	1.24353	−	1.24353i	0.958528	−	0.284998i	$-0.0919929\pi$
$618$		0			0
$619$		−	27.3050i		−	1.09748i	−0.835994	−	0.548739i	$-0.815108\pi$
$619$		−	27.3050i		−	1.09748i	0.835994	−	0.548739i	$-0.184892\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.60845	+	7.60845i	−0.304826	+	0.304826i
$624$		0			0
$625$	−15.0000	−	20.0000i	−0.600000	−	0.800000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.15537			0.245431
$630$		0			0
$631$			28.0827i			1.11795i	0.829183	+	0.558977i	$0.188806\pi$
$631$			28.0827i			1.11795i	−0.829183	+	0.558977i	$0.811194\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	27.0344	+	6.38197i	1.07283	+	0.253261i
$636$		0			0
$637$	13.0373	+	13.0373i	0.516556	+	0.516556i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.34752			0.290210			0.145105	−	0.989416i	$-0.453648\pi$
$641$	7.34752			0.290210			0.145105	+	0.989416i	$0.453648\pi$
$642$		0			0
$643$	−8.56231	−	8.56231i	−0.337664	−	0.337664i	0.517823	−	0.855488i	$-0.326742\pi$
$643$	−8.56231	−	8.56231i	−0.337664	−	0.337664i	−0.855488	+	0.517823i	$0.826742\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	22.8909	+	22.8909i	0.899933	+	0.899933i	0.995430	−	0.0954968i	$-0.0304440\pi$
$647$	22.8909	+	22.8909i	0.899933	+	0.899933i	−0.0954968	+	0.995430i	$0.530444\pi$
$648$		0			0
$649$		−	10.4721i		−	0.411067i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.2412	−	11.2412i	0.439901	−	0.439901i	−0.452078	−	0.891979i	$-0.649317\pi$
$653$	11.2412	−	11.2412i	0.439901	−	0.439901i	0.891979	+	0.452078i	$0.149317\pi$
$654$		0			0
$655$	−0.555029	+	0.343027i	−0.0216868	+	0.0134032i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	18.0000i		−	0.701180i	−0.936529	−	0.350590i	$-0.885981\pi$
$659$		−	18.0000i		−	0.701180i	0.936529	−	0.350590i	$-0.114019\pi$
$660$		0			0
$661$		−	2.35114i		−	0.0914488i	−0.998954	−	0.0457244i	$-0.985440\pi$
$661$		−	2.35114i		−	0.0914488i	0.998954	−	0.0457244i	$-0.0145596\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.70820	−	7.23607i	0.0662413	−	0.280603i
$666$		0			0
$667$	3.81966	−	3.81966i	0.147898	−	0.147898i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	12.3107i		−	0.475251i
$672$		0			0
$673$	−30.7082	−	30.7082i	−1.18371	−	1.18371i	−0.978775	−	0.204940i	$-0.934300\pi$
$673$	−30.7082	−	30.7082i	−1.18371	−	1.18371i	−0.204940	−	0.978775i	$-0.565700\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.33499	+	8.33499i	0.320340	+	0.320340i	0.848897	−	0.528558i	$-0.177267\pi$
$677$	8.33499	+	8.33499i	0.320340	+	0.320340i	−0.528558	+	0.848897i	$0.677267\pi$
$678$		0			0
$679$	0.555029			0.0213001
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.79837	+	1.79837i	0.0688129	+	0.0688129i	0.740676	−	0.671863i	$-0.234506\pi$
$683$	1.79837	+	1.79837i	0.0688129	+	0.0688129i	−0.671863	+	0.740676i	$0.734506\pi$
$684$		0			0
$685$	2.52265	−	10.6861i	0.0963857	−	0.408296i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.47214i			0.170375i
$690$		0			0
$691$	31.1246			1.18404			0.592018	−	0.805925i	$-0.298331\pi$
$691$	31.1246			1.18404			0.592018	+	0.805925i	$0.298331\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.36737	+	10.3026i	0.241528	+	0.390801i
$696$		0			0
$697$	7.70820	−	7.70820i	0.291969	−	0.291969i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	40.3934i		−	1.52564i	−0.646612	−	0.762819i	$-0.723815\pi$
$701$		−	40.3934i		−	1.52564i	0.646612	−	0.762819i	$-0.276185\pi$
$702$		0			0
$703$	6.15537	−	6.15537i	0.232154	−	0.232154i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14.4721	+	14.4721i	−0.544281	+	0.544281i
$708$		0			0
$709$	13.7638			0.516911			0.258456	−	0.966023i	$-0.416786\pi$
$709$	13.7638			0.516911			0.258456	+	0.966023i	$0.416786\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.8197	−	13.8197i	−0.517550	−	0.517550i
$714$		0			0
$715$	2.76393	−	11.7082i	0.103365	−	0.437862i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−44.5407			−1.66109			−0.830543	−	0.556954i	$-0.811970\pi$
$719$	−44.5407			−1.66109			−0.830543	+	0.556954i	$0.811970\pi$
$720$		0			0
$721$	17.2361			0.641905
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.24920	−	6.49839i	0.120672	−	0.241344i
$726$		0			0
$727$	−5.87785	−	5.87785i	−0.217997	−	0.217997i	0.589657	−	0.807654i	$-0.299263\pi$
$727$	−5.87785	−	5.87785i	−0.217997	−	0.217997i	−0.807654	+	0.589657i	$0.799263\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.76393			−0.176200
$732$		0			0
$733$	1.28157	−	1.28157i	0.0473359	−	0.0473359i	−0.683043	−	0.730379i	$-0.739344\pi$
$733$	1.28157	−	1.28157i	0.0473359	−	0.0473359i	0.730379	+	0.683043i	$0.239344\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.94427	−	2.94427i	0.108454	−	0.108454i
$738$		0			0
$739$			17.4164i			0.640673i	0.947304	+	0.320336i	$0.103796\pi$
$739$			17.4164i			0.640673i	−0.947304	+	0.320336i	$0.896204\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.4863	+	13.4863i	−0.494765	+	0.494765i	−0.909804	−	0.415039i	$-0.863768\pi$
$743$	−13.4863	+	13.4863i	−0.494765	+	0.494765i	0.415039	+	0.909804i	$0.363768\pi$
$744$		0			0
$745$	25.1246	−	15.5279i	0.920495	−	0.568897i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−28.4257			−1.03865
$750$		0			0
$751$			7.05342i			0.257383i	0.991685	+	0.128692i	$0.0410777\pi$
$751$			7.05342i			0.257383i	−0.991685	+	0.128692i	$0.958922\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17.2361	−	27.8885i	−0.627285	−	1.01497i
$756$		0			0
$757$	−38.0018	−	38.0018i	−1.38120	−	1.38120i	−0.842497	−	0.538701i	$-0.818915\pi$
$757$	−38.0018	−	38.0018i	−1.38120	−	1.38120i	−0.538701	−	0.842497i	$-0.681085\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14.9443			0.541729			0.270865	−	0.962617i	$-0.412690\pi$
$761$	14.9443			0.541729			0.270865	+	0.962617i	$0.412690\pi$
$762$		0			0
$763$	−7.88854	−	7.88854i	−0.285584	−	0.285584i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	26.0746	+	26.0746i	0.941498	+	0.941498i
$768$		0			0
$769$		−	2.47214i		−	0.0891475i	−0.999006	−	0.0445738i	$-0.985807\pi$
$769$		−	2.47214i		−	0.0891475i	0.999006	−	0.0445738i	$-0.0141930\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.18774	−	4.18774i	0.150623	−	0.150623i	−0.627773	−	0.778396i	$-0.716034\pi$
$773$	4.18774	−	4.18774i	0.150623	−	0.150623i	0.778396	+	0.627773i	$0.216034\pi$
$774$		0			0
$775$	−23.5114	−	11.7557i	−0.844555	−	0.422277i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	15.4164i		−	0.552350i
$780$		0			0
$781$			8.71851i			0.311973i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	30.1246	+	7.11146i	1.07519	+	0.253819i
$786$		0			0
$787$	−11.1459	+	11.1459i	−0.397308	+	0.397308i	−0.877283	−	0.479974i	$-0.840646\pi$
$787$	−11.1459	+	11.1459i	−0.397308	+	0.397308i	0.479974	+	0.877283i	$0.340646\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	11.0697i		−	0.393591i
$792$		0			0
$793$	30.6525	+	30.6525i	1.08850	+	1.08850i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.7477	−	19.7477i	−0.699498	−	0.699498i	0.264804	−	0.964302i	$-0.414693\pi$
$797$	−19.7477	−	19.7477i	−0.699498	−	0.699498i	−0.964302	+	0.264804i	$0.914693\pi$
$798$		0			0
$799$	−14.6619			−0.518700
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.7639	+	10.7639i	0.379851	+	0.379851i
$804$		0			0
$805$	11.7557	−	7.26543i	0.414334	−	0.256073i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			12.9443i			0.455096i	0.973767	+	0.227548i	$0.0730709\pi$
$809$			12.9443i			0.455096i	−0.973767	+	0.227548i	$0.926929\pi$
$810$		0			0
$811$	32.0689			1.12609			0.563045	−	0.826426i	$-0.309630\pi$
$811$	32.0689			1.12609			0.563045	+	0.826426i	$0.309630\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21.9928	−	5.19180i	−0.770375	−	0.181861i
$816$		0			0
$817$	−4.76393	+	4.76393i	−0.166669	+	0.166669i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.4742i		−	0.714555i	−0.933998	−	0.357278i	$-0.883705\pi$
$821$		−	20.4742i		−	0.714555i	0.933998	−	0.357278i	$-0.116295\pi$
$822$		0			0
$823$	−24.3440	+	24.3440i	−0.848577	+	0.848577i	−0.989956	−	0.141379i	$-0.954846\pi$
$823$	−24.3440	+	24.3440i	−0.848577	+	0.848577i	0.141379	+	0.989956i	$0.454846\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.8541	−	14.8541i	0.516528	−	0.516528i	−0.399991	−	0.916519i	$-0.630987\pi$
$827$	14.8541	−	14.8541i	0.516528	−	0.516528i	0.916519	+	0.399991i	$0.130987\pi$
$828$		0			0
$829$	3.11817			0.108299			0.0541493	−	0.998533i	$-0.482755\pi$
$829$	3.11817			0.108299			0.0541493	+	0.998533i	$0.482755\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.23607	−	4.23607i	−0.146771	−	0.146771i
$834$		0			0
$835$	−0.854102	−	0.201626i	−0.0295574	−	0.00697756i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.40456			−0.324682			−0.162341	−	0.986735i	$-0.551904\pi$
$839$	−9.40456			−0.324682			−0.162341	+	0.986735i	$0.551904\pi$
$840$		0			0
$841$	−26.8885			−0.927191
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.98791	+	11.3067i	0.240391	+	0.388962i
$846$		0			0
$847$	11.1352	+	11.1352i	0.382609	+	0.382609i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	16.1803			0.554655
$852$		0			0
$853$	−24.7930	+	24.7930i	−0.848896	+	0.848896i	−0.989995	−	0.141100i	$-0.954936\pi$
$853$	−24.7930	+	24.7930i	−0.848896	+	0.848896i	0.141100	+	0.989995i	$0.454936\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.8328	−	17.8328i	0.609157	−	0.609157i	−0.333568	−	0.942726i	$-0.608253\pi$
$857$	17.8328	−	17.8328i	0.609157	−	0.609157i	0.942726	+	0.333568i	$0.108253\pi$
$858$		0			0
$859$		−	7.52786i		−	0.256847i	−0.991719	−	0.128424i	$-0.959008\pi$
$859$		−	7.52786i		−	0.256847i	0.991719	−	0.128424i	$-0.0409917\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.22088	−	6.22088i	0.211761	−	0.211761i	−0.593254	−	0.805015i	$-0.702157\pi$
$863$	6.22088	−	6.22088i	0.211761	−	0.211761i	0.805015	+	0.593254i	$0.202157\pi$
$864$		0			0
$865$	−4.59675	+	19.4721i	−0.156294	+	0.662072i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15.2169			0.516198
$870$		0			0
$871$			14.6619i			0.496799i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.9098	−	14.2705i	0.402626	−	0.482431i
$876$		0			0
$877$	−5.08580	−	5.08580i	−0.171735	−	0.171735i	0.616006	−	0.787741i	$-0.288750\pi$
$877$	−5.08580	−	5.08580i	−0.171735	−	0.171735i	−0.787741	+	0.616006i	$0.788750\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−47.1246			−1.58767			−0.793834	−	0.608134i	$-0.791918\pi$
$881$	−47.1246			−1.58767			−0.793834	+	0.608134i	$0.791918\pi$
$882$		0			0
$883$	−21.7984	−	21.7984i	−0.733574	−	0.733574i	0.237752	−	0.971326i	$-0.423589\pi$
$883$	−21.7984	−	21.7984i	−0.733574	−	0.733574i	−0.971326	+	0.237752i	$0.923589\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.1482	−	28.1482i	−0.945123	−	0.945123i	0.0534473	−	0.998571i	$-0.482979\pi$
$887$	−28.1482	−	28.1482i	−0.945123	−	0.945123i	−0.998571	+	0.0534473i	$0.982979\pi$
$888$		0			0
$889$			20.6525i			0.692662i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−14.6619	+	14.6619i	−0.490641	+	0.490641i
$894$		0			0
$895$	−19.3642	−	31.3319i	−0.647272	−	1.04731i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	7.63932i		−	0.254786i
$900$		0			0
$901$		−	1.45309i		−	0.0484093i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	11.0557	+	17.8885i	0.367505	+	0.594635i
$906$		0			0
$907$	23.3262	−	23.3262i	0.774535	−	0.774535i	−0.204361	−	0.978896i	$-0.565512\pi$
$907$	23.3262	−	23.3262i	0.774535	−	0.774535i	0.978896	+	0.204361i	$0.0655115\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	25.1765i		−	0.834135i	−0.908876	−	0.417067i	$-0.863058\pi$
$911$		−	25.1765i		−	0.834135i	0.908876	−	0.417067i	$-0.136942\pi$
$912$		0			0
$913$	−5.41641	−	5.41641i	−0.179257	−	0.179257i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.343027	−	0.343027i	−0.0113277	−	0.0113277i
$918$		0			0
$919$	21.7153			0.716322			0.358161	−	0.933660i	$-0.383404\pi$
$919$	21.7153			0.716322			0.358161	+	0.933660i	$0.383404\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−21.7082	−	21.7082i	−0.714534	−	0.714534i
$924$		0			0
$925$	20.6457	−	6.88191i	0.678827	−	0.226276i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	5.34752i		−	0.175447i	−0.996145	−	0.0877233i	$-0.972041\pi$
$929$		−	5.34752i		−	0.175447i	0.996145	−	0.0877233i	$-0.0279591\pi$
$930$		0			0
$931$	−8.47214			−0.277663
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.898056	+	3.80423i	−0.0293696	+	0.124411i
$936$		0			0
$937$	20.3050	−	20.3050i	0.663334	−	0.663334i	−0.292831	−	0.956164i	$-0.594597\pi$
$937$	20.3050	−	20.3050i	0.663334	−	0.663334i	0.956164	+	0.292831i	$0.0945972\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			60.8676i			1.98423i	0.125340	+	0.992114i	$0.459998\pi$
$941$			60.8676i			1.98423i	−0.125340	+	0.992114i	$0.540002\pi$
$942$		0			0
$943$	20.2622	−	20.2622i	0.659828	−	0.659828i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.85410	+	8.85410i	−0.287720	+	0.287720i	−0.836178	−	0.548458i	$-0.815215\pi$
$947$	−8.85410	+	8.85410i	−0.287720	+	0.287720i	0.548458	+	0.836178i	$0.315215\pi$
$948$		0			0
$949$	−53.6022			−1.74000
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.81966	+	6.81966i	0.220910	+	0.220910i	0.808882	−	0.587971i	$-0.200073\pi$
$953$	6.81966	+	6.81966i	0.220910	+	0.220910i	−0.587971	+	0.808882i	$0.700073\pi$
$954$		0			0
$955$	15.1246	+	24.4721i	0.489421	+	0.791900i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.16348			0.263613
$960$		0			0
$961$	3.36068			0.108409
$962$		0			0
$963$		0			0
$964$		0			0
$965$	22.9969	+	5.42882i	0.740295	+	0.174760i
$966$		0			0
$967$	12.0332	+	12.0332i	0.386962	+	0.386962i	0.873602	−	0.486640i	$-0.161778\pi$
$967$	12.0332	+	12.0332i	0.386962	+	0.386962i	−0.486640	+	0.873602i	$0.661778\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33.5967			1.07817			0.539085	−	0.842251i	$-0.318770\pi$
$971$	33.5967			1.07817			0.539085	+	0.842251i	$0.318770\pi$
$972$		0			0
$973$	−6.36737	+	6.36737i	−0.204128	+	0.204128i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.2361	−	28.2361i	0.903352	−	0.903352i	−0.0923727	−	0.995725i	$-0.529445\pi$
$977$	28.2361	−	28.2361i	0.903352	−	0.903352i	0.995725	+	0.0923727i	$0.0294451\pi$
$978$		0			0
$979$		−	8.00000i		−	0.255681i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.620541	−	0.620541i	0.0197922	−	0.0197922i	−0.697141	−	0.716934i	$-0.745545\pi$
$983$	0.620541	−	0.620541i	0.0197922	−	0.0197922i	0.716934	+	0.697141i	$0.245545\pi$
$984$		0			0
$985$	6.70820	+	1.58359i	0.213741	+	0.0504574i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.5227			−0.398200
$990$		0			0
$991$		−	19.3642i		−	0.615123i	−0.951528	−	0.307561i	$-0.900487\pi$
$991$		−	19.3642i		−	0.615123i	0.951528	−	0.307561i	$-0.0995129\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−34.4721	+	21.3050i	−1.09284	+	0.675412i
$996$		0			0
$997$	35.6506	+	35.6506i	1.12907	+	1.12907i	0.990329	+	0.138738i	$0.0443045\pi$
$997$	35.6506	+	35.6506i	1.12907	+	1.12907i	0.138738	+	0.990329i	$0.455696\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bi.c.1423.1		8
3.2	odd	2		160.2.o.a.143.4		8
4.3	odd	2		360.2.w.c.163.3		8
5.2	odd	4	inner	1440.2.bi.c.847.4		8
8.3	odd	2	inner	1440.2.bi.c.1423.4		8
8.5	even	2		360.2.w.c.163.1		8
12.11	even	2		40.2.k.a.3.2	✓	8
15.2	even	4		160.2.o.a.47.3		8
15.8	even	4		800.2.o.g.207.1		8
15.14	odd	2		800.2.o.g.143.2		8
20.7	even	4		360.2.w.c.307.1		8
24.5	odd	2		40.2.k.a.3.4	yes	8
24.11	even	2		160.2.o.a.143.3		8
40.27	even	4	inner	1440.2.bi.c.847.1		8
40.37	odd	4		360.2.w.c.307.3		8
48.5	odd	4		1280.2.n.q.1023.4		8
48.11	even	4		1280.2.n.m.1023.2		8
48.29	odd	4		1280.2.n.m.1023.1		8
48.35	even	4		1280.2.n.q.1023.3		8
60.23	odd	4		200.2.k.h.107.1		8
60.47	odd	4		40.2.k.a.27.4	yes	8
60.59	even	2		200.2.k.h.43.3		8
120.29	odd	2		200.2.k.h.43.1		8
120.53	even	4		200.2.k.h.107.3		8
120.59	even	2		800.2.o.g.143.1		8
120.77	even	4		40.2.k.a.27.2	yes	8
120.83	odd	4		800.2.o.g.207.2		8
120.107	odd	4		160.2.o.a.47.4		8
240.77	even	4		1280.2.n.q.767.3		8
240.107	odd	4		1280.2.n.q.767.4		8
240.197	even	4		1280.2.n.m.767.2		8
240.227	odd	4		1280.2.n.m.767.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.k.a.3.2	✓	8	12.11	even	2
40.2.k.a.3.4	yes	8	24.5	odd	2
40.2.k.a.27.2	yes	8	120.77	even	4
40.2.k.a.27.4	yes	8	60.47	odd	4
160.2.o.a.47.3		8	15.2	even	4
160.2.o.a.47.4		8	120.107	odd	4
160.2.o.a.143.3		8	24.11	even	2
160.2.o.a.143.4		8	3.2	odd	2
200.2.k.h.43.1		8	120.29	odd	2
200.2.k.h.43.3		8	60.59	even	2
200.2.k.h.107.1		8	60.23	odd	4
200.2.k.h.107.3		8	120.53	even	4
360.2.w.c.163.1		8	8.5	even	2
360.2.w.c.163.3		8	4.3	odd	2
360.2.w.c.307.1		8	20.7	even	4
360.2.w.c.307.3		8	40.37	odd	4
800.2.o.g.143.1		8	120.59	even	2
800.2.o.g.143.2		8	15.14	odd	2
800.2.o.g.207.1		8	15.8	even	4
800.2.o.g.207.2		8	120.83	odd	4
1280.2.n.m.767.1		8	240.227	odd	4
1280.2.n.m.767.2		8	240.197	even	4
1280.2.n.m.1023.1		8	48.29	odd	4
1280.2.n.m.1023.2		8	48.11	even	4
1280.2.n.q.767.3		8	240.77	even	4
1280.2.n.q.767.4		8	240.107	odd	4
1280.2.n.q.1023.3		8	48.35	even	4
1280.2.n.q.1023.4		8	48.5	odd	4
1440.2.bi.c.847.1		8	40.27	even	4	inner
1440.2.bi.c.847.4		8	5.2	odd	4	inner
1440.2.bi.c.1423.1		8	1.1	even	1	trivial
1440.2.bi.c.1423.4		8	8.3	odd	2	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bi (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.90211 + 1.17557i) q^{5} +(-1.17557 - 1.17557i) q^{7} +O(q^{10})\)	\(q+(-1.90211 + 1.17557i) q^{5} +(-1.17557 - 1.17557i) q^{7} +1.23607 q^{11} +(-3.07768 + 3.07768i) q^{13} +(1.00000 - 1.00000i) q^{17} -2.00000i q^{19} +(2.62866 - 2.62866i) q^{23} +(2.23607 - 4.47214i) q^{25} +1.45309 q^{29} -5.25731i q^{31} +(3.61803 + 0.854102i) q^{35} +(3.07768 + 3.07768i) q^{37} +7.70820 q^{41} +(-2.38197 - 2.38197i) q^{43} +(-7.33094 - 7.33094i) q^{47} -4.23607i q^{49} +(0.726543 - 0.726543i) q^{53} +(-2.35114 + 1.45309i) q^{55} -8.47214i q^{59} -9.95959i q^{61} +(2.23607 - 9.47214i) q^{65} +(2.38197 - 2.38197i) q^{67} +7.05342i q^{71} +(8.70820 + 8.70820i) q^{73} +(-1.45309 - 1.45309i) q^{77} +12.3107 q^{79} +(-4.38197 - 4.38197i) q^{83} +(-0.726543 + 3.07768i) q^{85} -6.47214i q^{89} +7.23607 q^{91} +(2.35114 + 3.80423i) q^{95} +(-0.236068 + 0.236068i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} + 8 q^{17} + 20 q^{35} + 8 q^{41} - 28 q^{43} + 28 q^{67} + 16 q^{73} - 44 q^{83} + 40 q^{91} + 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^11 + 8 * q^17 + 20 * q^35 + 8 * q^41 - 28 * q^43 + 28 * q^67 + 16 * q^73 - 44 * q^83 + 40 * q^91 + 16 * q^97

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