LMFDB - Embedded newform 1380.2.n.a.689.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(689,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1380.689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.n (of order $2$, degree $1$, 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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			689.4
Character	$\chi$	$=$	1380.689
Dual form			1380.2.n.a.689.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.71333 + 0.253996i) q^{3} +(0.923440 - 2.03648i) q^{5} +2.63922 q^{7} +(2.87097 - 0.870355i) q^{9} +O(q^{10})$	\(q+(-1.71333 + 0.253996i) q^{3} +(0.923440 - 2.03648i) q^{5} +2.63922 q^{7} +(2.87097 - 0.870355i) q^{9} -4.99653 q^{11} -4.22915i q^{13} +(-1.06490 + 3.72371i) q^{15} -4.76572i q^{17} +1.53868i q^{19} +(-4.52184 + 0.670350i) q^{21} +(4.57258 + 1.44620i) q^{23} +(-3.29452 - 3.76114i) q^{25} +(-4.69784 + 2.22042i) q^{27} +0.927712i q^{29} +1.66883 q^{31} +(8.56068 - 1.26910i) q^{33} +(2.43716 - 5.37472i) q^{35} -7.06223 q^{37} +(1.07419 + 7.24591i) q^{39} +11.2571i q^{41} +0.983560 q^{43} +(0.878707 - 6.65040i) q^{45} -8.19428 q^{47} -0.0345317 q^{49} +(1.21047 + 8.16523i) q^{51} -10.8330i q^{53} +(-4.61399 + 10.1753i) q^{55} +(-0.390819 - 2.63626i) q^{57} -4.23688i q^{59} -12.0027i q^{61} +(7.57712 - 2.29706i) q^{63} +(-8.61258 - 3.90536i) q^{65} +5.05357 q^{67} +(-8.20165 - 1.31639i) q^{69} -10.6750i q^{71} +11.4209i q^{73} +(6.59990 + 5.60726i) q^{75} -13.1869 q^{77} +0.925445i q^{79} +(7.48496 - 4.99753i) q^{81} -12.6693i q^{83} +(-9.70530 - 4.40085i) q^{85} +(-0.235635 - 1.58947i) q^{87} -2.70682 q^{89} -11.1616i q^{91} +(-2.85926 + 0.423877i) q^{93} +(3.13350 + 1.42088i) q^{95} -4.80363 q^{97} +(-14.3449 + 4.34875i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) $ 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	−1.71333	+	0.253996i	−0.989189	+	0.146645i
$3$	−1.71333	+	0.253996i	−0.989189	+	0.146645i
$4$		0			0
$5$	0.923440	−	2.03648i	0.412975	−	0.910742i
$5$	0.923440	−	2.03648i	0.412975	−	0.910742i
$6$		0			0
$7$	2.63922			0.997530			0.498765	−	0.866737i	$-0.333787\pi$
$7$	2.63922			0.997530			0.498765	+	0.866737i	$0.333787\pi$
$8$		0			0
$9$	2.87097	−	0.870355i	0.956991	−	0.290118i
$10$		0			0
$11$	−4.99653			−1.50651			−0.753255	−	0.657729i	$-0.771517\pi$
$11$	−4.99653			−1.50651			−0.753255	+	0.657729i	$0.771517\pi$
$12$		0			0
$13$		−	4.22915i		−	1.17295i	−0.809966	−	0.586477i	$-0.800514\pi$
$13$		−	4.22915i		−	1.17295i	0.809966	−	0.586477i	$-0.199486\pi$
$14$		0			0
$15$	−1.06490	+	3.72371i	−0.274955	+	0.961457i
$16$		0			0
$17$		−	4.76572i		−	1.15586i	−0.816088	−	0.577928i	$-0.803861\pi$
$17$		−	4.76572i		−	1.15586i	0.816088	−	0.577928i	$-0.196139\pi$
$18$		0			0
$19$			1.53868i			0.352998i	0.984301	+	0.176499i	$0.0564772\pi$
$19$			1.53868i			0.352998i	−0.984301	+	0.176499i	$0.943523\pi$
$20$		0			0
$21$	−4.52184	+	0.670350i	−0.986746	+	0.146282i
$22$		0			0
$23$	4.57258	+	1.44620i	0.953449	+	0.301553i
$23$	4.57258	+	1.44620i	0.953449	+	0.301553i
$24$		0			0
$25$	−3.29452	−	3.76114i	−0.658904	−	0.752227i
$26$		0			0
$27$	−4.69784	+	2.22042i	−0.904101	+	0.427319i
$28$		0			0
$29$			0.927712i			0.172272i	0.996283	+	0.0861359i	$0.0274519\pi$
$29$			0.927712i			0.172272i	−0.996283	+	0.0861359i	$0.972548\pi$
$30$		0			0
$31$	1.66883			0.299731			0.149866	−	0.988706i	$-0.452116\pi$
$31$	1.66883			0.299731			0.149866	+	0.988706i	$0.452116\pi$
$32$		0			0
$33$	8.56068	−	1.26910i	1.49022	−	0.220921i
$34$		0			0
$35$	2.43716	−	5.37472i	0.411955	−	0.908493i
$36$		0			0
$37$	−7.06223			−1.16102			−0.580512	−	0.814252i	$-0.697147\pi$
$37$	−7.06223			−1.16102			−0.580512	+	0.814252i	$0.697147\pi$
$38$		0			0
$39$	1.07419	+	7.24591i	0.172007	+	1.16027i
$40$		0			0
$41$			11.2571i			1.75806i	0.476763	+	0.879032i	$0.341810\pi$
$41$			11.2571i			1.75806i	−0.476763	+	0.879032i	$0.658190\pi$
$42$		0			0
$43$	0.983560			0.149992			0.0749958	−	0.997184i	$-0.476106\pi$
$43$	0.983560			0.149992			0.0749958	+	0.997184i	$0.476106\pi$
$44$		0			0
$45$	0.878707	−	6.65040i	0.130990	−	0.991384i
$46$		0			0
$47$	−8.19428			−1.19526			−0.597629	−	0.801773i	$-0.703890\pi$
$47$	−8.19428			−1.19526			−0.597629	+	0.801773i	$0.703890\pi$
$48$		0			0
$49$	−0.0345317			−0.00493310
$50$		0			0
$51$	1.21047	+	8.16523i	0.169500	+	1.14336i
$52$		0			0
$53$		−	10.8330i		−	1.48803i	−0.668164	−	0.744014i	$-0.732919\pi$
$53$		−	10.8330i		−	1.48803i	0.668164	−	0.744014i	$-0.267081\pi$
$54$		0			0
$55$	−4.61399	+	10.1753i	−0.622150	+	1.37204i
$56$		0			0
$57$	−0.390819	−	2.63626i	−0.0517652	−	0.349182i
$58$		0			0
$59$		−	4.23688i		−	0.551595i	−0.961216	−	0.275797i	$-0.911058\pi$
$59$		−	4.23688i		−	0.551595i	0.961216	−	0.275797i	$-0.0889419\pi$
$60$		0			0
$61$		−	12.0027i		−	1.53679i	−0.639977	−	0.768394i	$-0.721056\pi$
$61$		−	12.0027i		−	1.53679i	0.639977	−	0.768394i	$-0.278944\pi$
$62$		0			0
$63$	7.57712	−	2.29706i	0.954627	−	0.289402i
$64$		0			0
$65$	−8.61258	−	3.90536i	−1.06826	−	0.484400i
$66$		0			0
$67$	5.05357			0.617392			0.308696	−	0.951161i	$-0.400107\pi$
$67$	5.05357			0.617392			0.308696	+	0.951161i	$0.400107\pi$
$68$		0			0
$69$	−8.20165	−	1.31639i	−0.987363	−	0.158475i
$70$		0			0
$71$		−	10.6750i		−	1.26689i	−0.773789	−	0.633443i	$-0.781641\pi$
$71$		−	10.6750i		−	1.26689i	0.773789	−	0.633443i	$-0.218359\pi$
$72$		0			0
$73$			11.4209i			1.33672i	0.743840	+	0.668358i	$0.233002\pi$
$73$			11.4209i			1.33672i	−0.743840	+	0.668358i	$0.766998\pi$
$74$		0			0
$75$	6.59990	+	5.60726i	0.762090	+	0.647471i
$76$		0			0
$77$	−13.1869			−1.50279
$78$		0			0
$79$			0.925445i			0.104121i	0.998644	+	0.0520604i	$0.0165788\pi$
$79$			0.925445i			0.104121i	−0.998644	+	0.0520604i	$0.983421\pi$
$80$		0			0
$81$	7.48496	−	4.99753i	0.831663	−	0.555281i
$82$		0			0
$83$		−	12.6693i		−	1.39063i	−0.718703	−	0.695317i	$-0.755264\pi$
$83$		−	12.6693i		−	1.39063i	0.718703	−	0.695317i	$-0.244736\pi$
$84$		0			0
$85$	−9.70530	−	4.40085i	−1.05269	−	0.477339i
$86$		0			0
$87$	−0.235635	−	1.58947i	−0.0252627	−	0.170409i
$88$		0			0
$89$	−2.70682			−0.286923			−0.143461	−	0.989656i	$-0.545823\pi$
$89$	−2.70682			−0.286923			−0.143461	+	0.989656i	$0.545823\pi$
$90$		0			0
$91$		−	11.1616i		−	1.17006i
$92$		0			0
$93$	−2.85926	+	0.423877i	−0.296491	+	0.0439540i
$94$		0			0
$95$	3.13350	+	1.42088i	0.321490	+	0.145779i
$96$		0			0
$97$	−4.80363			−0.487735			−0.243867	−	0.969809i	$-0.578416\pi$
$97$	−4.80363			−0.487735			−0.243867	+	0.969809i	$0.578416\pi$
$98$		0			0
$99$	−14.3449	+	4.34875i	−1.44172	+	0.437066i
$100$		0			0
$101$		−	3.23471i		−	0.321866i	−0.986965	−	0.160933i	$-0.948550\pi$
$101$		−	3.23471i		−	0.321866i	0.986965	−	0.160933i	$-0.0514502\pi$
$102$		0			0
$103$	−16.1378			−1.59010			−0.795052	−	0.606541i	$-0.792557\pi$
$103$	−16.1378			−1.59010			−0.795052	+	0.606541i	$0.792557\pi$
$104$		0			0
$105$	−2.81049	+	9.82767i	−0.274276	+	0.959083i
$106$		0			0
$107$		−	1.32946i		−	0.128523i	−0.997933	−	0.0642617i	$-0.979531\pi$
$107$		−	1.32946i		−	0.128523i	0.997933	−	0.0642617i	$-0.0204692\pi$
$108$		0			0
$109$		−	14.3765i		−	1.37702i	−0.725227	−	0.688510i	$-0.758265\pi$
$109$		−	14.3765i		−	1.37702i	0.725227	−	0.688510i	$-0.241735\pi$
$110$		0			0
$111$	12.0999	−	1.79378i	1.14847	−	0.170258i
$112$		0			0
$113$		−	4.62228i		−	0.434828i	−0.976079	−	0.217414i	$-0.930238\pi$
$113$		−	4.62228i		−	0.434828i	0.976079	−	0.217414i	$-0.0697622\pi$
$114$		0			0
$115$	7.16766	−	7.97650i	0.668388	−	0.743813i
$116$		0			0
$117$	−3.68086	−	12.1418i	−0.340296	−	1.12251i
$118$		0			0
$119$		−	12.5778i		−	1.15300i
$120$		0			0
$121$	13.9653			1.26957
$122$		0			0
$123$	−2.85926	−	19.2871i	−0.257810	−	1.73906i
$124$		0			0
$125$	−10.7018	+	3.23605i	−0.957196	+	0.289441i
$126$		0			0
$127$		−	20.9728i		−	1.86104i	−0.366242	−	0.930520i	$-0.619356\pi$
$127$		−	20.9728i		−	1.86104i	0.366242	−	0.930520i	$-0.380644\pi$
$128$		0			0
$129$	−1.68516	+	0.249820i	−0.148370	+	0.0219954i
$130$		0			0
$131$			18.5554i			1.62119i	0.585608	+	0.810594i	$0.300856\pi$
$131$			18.5554i			1.62119i	−0.585608	+	0.810594i	$0.699144\pi$
$132$		0			0
$133$			4.06092i			0.352126i
$134$		0			0
$135$	0.183664	+	11.6175i	0.0158072	+	0.999875i
$136$		0			0
$137$			1.99433i			0.170387i	0.996364	+	0.0851936i	$0.0271509\pi$
$137$			1.99433i			0.170387i	−0.996364	+	0.0851936i	$0.972849\pi$
$138$		0			0
$139$	−20.5244			−1.74086			−0.870430	−	0.492293i	$-0.836159\pi$
$139$	−20.5244			−1.74086			−0.870430	+	0.492293i	$0.836159\pi$
$140$		0			0
$141$	14.0395	−	2.08131i	1.18234	−	0.175278i
$142$		0			0
$143$			21.1310i			1.76707i
$144$		0			0
$145$	1.88927	+	0.856686i	0.156895	+	0.0711439i
$146$		0			0
$147$	0.0591640	−	0.00877090i	0.00487977	−	0.000723412i
$148$		0			0
$149$	18.3080			1.49985			0.749926	−	0.661522i	$-0.230089\pi$
$149$	18.3080			1.49985			0.749926	+	0.661522i	$0.230089\pi$
$150$		0			0
$151$	12.8860			1.04865			0.524323	−	0.851519i	$-0.324318\pi$
$151$	12.8860			1.04865			0.524323	+	0.851519i	$0.324318\pi$
$152$		0			0
$153$	−4.14787	−	13.6822i	−0.335335	−	1.10614i
$154$		0			0
$155$	1.54107	−	3.39855i	0.123781	−	0.272978i
$156$		0			0
$157$	−3.14798			−0.251236			−0.125618	−	0.992079i	$-0.540091\pi$
$157$	−3.14798			−0.251236			−0.125618	+	0.992079i	$0.540091\pi$
$158$		0			0
$159$	2.75154	+	18.5605i	0.218211	+	1.47194i
$160$		0			0
$161$	12.0680	+	3.81683i	0.951095	+	0.300809i
$162$		0			0
$163$			15.0621i			1.17975i	0.807493	+	0.589877i	$0.200824\pi$
$163$			15.0621i			1.17975i	−0.807493	+	0.589877i	$0.799176\pi$
$164$		0			0
$165$	5.32078	−	18.6056i	0.414222	−	1.44844i
$166$		0			0
$167$	9.28900			0.718804			0.359402	−	0.933183i	$-0.382981\pi$
$167$	9.28900			0.718804			0.359402	+	0.933183i	$0.382981\pi$
$168$		0			0
$169$	−4.88567			−0.375821
$170$		0			0
$171$	1.33920	+	4.41751i	0.102411	+	0.337816i
$172$		0			0
$173$	14.0195			1.06588			0.532941	−	0.846153i	$-0.321087\pi$
$173$	14.0195			1.06588			0.532941	+	0.846153i	$0.321087\pi$
$174$		0			0
$175$	−8.69495	−	9.92646i	−0.657276	−	0.750370i
$176$		0			0
$177$	1.07615	+	7.25916i	0.0808884	+	0.545632i
$178$		0			0
$179$			18.7560i			1.40189i	0.713216	+	0.700944i	$0.247238\pi$
$179$			18.7560i			1.40189i	−0.713216	+	0.700944i	$0.752762\pi$
$180$		0			0
$181$		−	9.40105i		−	0.698774i	−0.936978	−	0.349387i	$-0.886390\pi$
$181$		−	9.40105i		−	0.698774i	0.936978	−	0.349387i	$-0.113610\pi$
$182$		0			0
$183$	3.04863	+	20.5645i	0.225362	+	1.52017i
$184$		0			0
$185$	−6.52154	+	14.3821i	−0.479473	+	1.05739i
$186$		0			0
$187$			23.8120i			1.74131i
$188$		0			0
$189$	−12.3986	+	5.86016i	−0.901868	+	0.426264i
$190$		0			0
$191$	13.0886			0.947056			0.473528	−	0.880779i	$-0.342980\pi$
$191$	13.0886			0.947056			0.473528	+	0.880779i	$0.342980\pi$
$192$		0			0
$193$		−	3.58943i		−	0.258373i	−0.991620	−	0.129186i	$-0.958763\pi$
$193$		−	3.58943i		−	0.258373i	0.991620	−	0.129186i	$-0.0412366\pi$
$194$		0			0
$195$	15.7481	+	4.50360i	1.12774	+	0.322509i
$196$		0			0
$197$	−19.9885			−1.42412			−0.712061	−	0.702117i	$-0.752238\pi$
$197$	−19.9885			−1.42412			−0.712061	+	0.702117i	$0.752238\pi$
$198$		0			0
$199$		−	18.0864i		−	1.28211i	−0.767494	−	0.641056i	$-0.778497\pi$
$199$		−	18.0864i		−	1.28211i	0.767494	−	0.641056i	$-0.221503\pi$
$200$		0			0
$201$	−8.65842	+	1.28359i	−0.610718	+	0.0905372i
$202$		0			0
$203$			2.44843i			0.171846i
$204$		0			0
$205$	22.9249	+	10.3953i	1.60114	+	0.726036i
$206$		0			0
$207$	14.3865	+	0.172225i	0.999928	+	0.0119705i
$208$		0			0
$209$		−	7.68807i		−	0.531795i
$210$		0			0
$211$	12.4857			0.859554			0.429777	−	0.902935i	$-0.358592\pi$
$211$	12.4857			0.859554			0.429777	+	0.902935i	$0.358592\pi$
$212$		0			0
$213$	2.71140	+	18.2897i	0.185782	+	1.25319i
$214$		0			0
$215$	0.908258	−	2.00300i	0.0619427	−	0.136604i
$216$		0			0
$217$	4.40442			0.298991
$218$		0			0
$219$	−2.90086	−	19.5677i	−0.196022	−	1.32226i
$220$		0			0
$221$	−20.1549			−1.35577
$222$		0			0
$223$		−	17.3492i		−	1.16179i	−0.813979	−	0.580894i	$-0.802703\pi$
$223$		−	17.3492i		−	1.16179i	0.813979	−	0.580894i	$-0.197297\pi$
$224$		0			0
$225$	−12.7320	−	7.93072i	−0.848800	−	0.528714i
$226$		0			0
$227$			17.0179i			1.12952i	0.825255	+	0.564760i	$0.191031\pi$
$227$			17.0179i			1.12952i	−0.825255	+	0.564760i	$0.808969\pi$
$228$		0			0
$229$			14.9898i			0.990551i	0.868736	+	0.495275i	$0.164933\pi$
$229$			14.9898i			0.990551i	−0.868736	+	0.495275i	$0.835067\pi$
$230$		0			0
$231$	22.5935	−	3.34942i	1.48654	−	0.220376i
$232$		0			0
$233$	14.2774			0.935342			0.467671	−	0.883903i	$-0.345093\pi$
$233$	14.2774			0.935342			0.467671	+	0.883903i	$0.345093\pi$
$234$		0			0
$235$	−7.56692	+	16.6875i	−0.493611	+	1.08857i
$236$		0			0
$237$	−0.235059	−	1.58559i	−0.0152687	−	0.102995i
$238$		0			0
$239$			22.4108i			1.44963i	0.688943	+	0.724815i	$0.258075\pi$
$239$			22.4108i			1.44963i	−0.688943	+	0.724815i	$0.741925\pi$
$240$		0			0
$241$			3.82812i			0.246591i	0.992370	+	0.123295i	$0.0393463\pi$
$241$			3.82812i			0.246591i	−0.992370	+	0.123295i	$0.960654\pi$
$242$		0			0
$243$	−11.5548	+	10.4636i	−0.741243	+	0.671237i
$244$		0			0
$245$	−0.0318879	+	0.0703231i	−0.00203724	+	0.00449278i
$246$		0			0
$247$	6.50731			0.414050
$248$		0			0
$249$	3.21794	+	21.7066i	0.203929	+	1.37560i
$250$		0			0
$251$	−10.2528			−0.647149			−0.323574	−	0.946203i	$-0.604885\pi$
$251$	−10.2528			−0.647149			−0.323574	+	0.946203i	$0.604885\pi$
$252$		0			0
$253$	−22.8470	−	7.22597i	−1.43638	−	0.454293i
$254$		0			0
$255$	17.7461	+	5.07499i	1.11131	+	0.317808i
$256$		0			0
$257$	−4.15679			−0.259294			−0.129647	−	0.991560i	$-0.541384\pi$
$257$	−4.15679			−0.259294			−0.129647	+	0.991560i	$0.541384\pi$
$258$		0			0
$259$	−18.6388			−1.15816
$260$		0			0
$261$	0.807439	+	2.66344i	0.0499792	+	0.164862i
$262$		0			0
$263$			15.3343i			0.945553i	0.881183	+	0.472776i	$0.156748\pi$
$263$			15.3343i			0.945553i	−0.881183	+	0.472776i	$0.843252\pi$
$264$		0			0
$265$	−22.0612	−	10.0036i	−1.35521	−	0.614518i
$266$		0			0
$267$	4.63767	−	0.687522i	0.283821	−	0.0420757i
$268$		0			0
$269$		−	17.1602i		−	1.04628i	−0.852247	−	0.523139i	$-0.824761\pi$
$269$		−	17.1602i		−	1.04628i	0.852247	−	0.523139i	$-0.175239\pi$
$270$		0			0
$271$	3.07092			0.186545			0.0932726	−	0.995641i	$-0.470267\pi$
$271$	3.07092			0.186545			0.0932726	+	0.995641i	$0.470267\pi$
$272$		0			0
$273$	2.83501	+	19.1235i	0.171583	+	1.15741i
$274$		0			0
$275$	16.4612	+	18.7926i	0.992645	+	1.13324i
$276$		0			0
$277$		−	4.19493i		−	0.252049i	−0.992027	−	0.126024i	$-0.959778\pi$
$277$		−	4.19493i		−	0.252049i	0.992027	−	0.126024i	$-0.0402218\pi$
$278$		0			0
$279$	4.79118	−	1.45248i	0.286840	−	0.0869576i
$280$		0			0
$281$	14.4274			0.860668			0.430334	−	0.902670i	$-0.358396\pi$
$281$	14.4274			0.860668			0.430334	+	0.902670i	$0.358396\pi$
$282$		0			0
$283$	−12.6707			−0.753196			−0.376598	−	0.926377i	$-0.622906\pi$
$283$	−12.6707			−0.753196			−0.376598	+	0.926377i	$0.622906\pi$
$284$		0			0
$285$	−5.72960	−	1.63854i	−0.339392	−	0.0970584i
$286$		0			0
$287$			29.7099i			1.75372i
$288$		0			0
$289$	−5.71206			−0.336003
$290$		0			0
$291$	8.23019	−	1.22010i	0.482462	−	0.0715237i
$292$		0			0
$293$		−	12.2712i		−	0.716893i	−0.933550	−	0.358446i	$-0.883307\pi$
$293$		−	12.2712i		−	0.716893i	0.933550	−	0.358446i	$-0.116693\pi$
$294$		0			0
$295$	−8.62833	−	3.91250i	−0.502361	−	0.227795i
$296$		0			0
$297$	23.4729	−	11.0944i	1.36204	−	0.643761i
$298$		0			0
$299$	6.11618	−	19.3381i	0.353708	−	1.11835i
$300$		0			0
$301$	2.59583			0.149621
$302$		0			0
$303$	0.821603	+	5.54211i	0.0471998	+	0.318386i
$304$		0			0
$305$	−24.4433	−	11.0838i	−1.39962	−	0.634654i
$306$		0			0
$307$			14.2541i			0.813527i	0.913534	+	0.406763i	$0.133343\pi$
$307$			14.2541i			0.813527i	−0.913534	+	0.406763i	$0.866657\pi$
$308$		0			0
$309$	27.6493	−	4.09893i	1.57291	−	0.233180i
$310$		0			0
$311$		−	11.0779i		−	0.628173i	−0.949394	−	0.314086i	$-0.898302\pi$
$311$		−	11.0779i		−	0.628173i	0.949394	−	0.314086i	$-0.101698\pi$
$312$		0			0
$313$	24.9448			1.40997			0.704983	−	0.709225i	$-0.250955\pi$
$313$	24.9448			1.40997			0.704983	+	0.709225i	$0.250955\pi$
$314$		0			0
$315$	2.31910	−	17.5519i	0.130666	−	0.988935i
$316$		0			0
$317$	4.87859			0.274009			0.137004	−	0.990570i	$-0.456253\pi$
$317$	4.87859			0.274009			0.137004	+	0.990570i	$0.456253\pi$
$318$		0			0
$319$		−	4.63534i		−	0.259529i
$320$		0			0
$321$	0.337676	+	2.27779i	0.0188472	+	0.127134i
$322$		0			0
$323$	7.33292			0.408015
$324$		0			0
$325$	−15.9064	+	13.9330i	−0.882328	+	0.772864i
$326$		0			0
$327$	3.65157	+	24.6317i	0.201933	+	1.36213i
$328$		0			0
$329$	−21.6265			−1.19231
$330$		0			0
$331$	13.2533			0.728469			0.364235	−	0.931307i	$-0.381331\pi$
$331$	13.2533			0.728469			0.364235	+	0.931307i	$0.381331\pi$
$332$		0			0
$333$	−20.2755	+	6.14665i	−1.11109	+	0.336834i
$334$		0			0
$335$	4.66667	−	10.2915i	0.254967	−	0.562286i
$336$		0			0
$337$	−18.0638			−0.983998			−0.491999	−	0.870596i	$-0.663734\pi$
$337$	−18.0638			−0.983998			−0.491999	+	0.870596i	$0.663734\pi$
$338$		0			0
$339$	1.17404	+	7.91948i	0.0637651	+	0.430127i
$340$		0			0
$341$	−8.33837			−0.451548
$342$		0			0
$343$	−18.5657			−1.00245
$344$		0			0
$345$	−10.2545	+	15.4869i	−0.552086	+	0.833787i
$346$		0			0
$347$	15.4122			0.827368			0.413684	−	0.910421i	$-0.364242\pi$
$347$	15.4122			0.827368			0.413684	+	0.910421i	$0.364242\pi$
$348$		0			0
$349$	10.5977			0.567283			0.283642	−	0.958930i	$-0.408457\pi$
$349$	10.5977			0.567283			0.283642	+	0.958930i	$0.408457\pi$
$350$		0			0
$351$	9.39047	+	19.8679i	0.501226	+	1.06047i
$352$		0			0
$353$	17.1999			0.915457			0.457729	−	0.889092i	$-0.348663\pi$
$353$	17.1999			0.915457			0.457729	+	0.889092i	$0.348663\pi$
$354$		0			0
$355$	−21.7394	−	9.85770i	−1.15381	−	0.523192i
$356$		0			0
$357$	3.19470	+	21.5498i	0.169081	+	1.14054i
$358$		0			0
$359$	26.6103			1.40444			0.702220	−	0.711960i	$-0.252192\pi$
$359$	26.6103			1.40444			0.702220	+	0.711960i	$0.252192\pi$
$360$		0			0
$361$	16.6325			0.875393
$362$		0			0
$363$	−23.9271	+	3.54712i	−1.25585	+	0.186176i
$364$		0			0
$365$	23.2585	+	10.5465i	1.21740	+	0.552030i
$366$		0			0
$367$	−10.3116			−0.538262			−0.269131	−	0.963104i	$-0.586736\pi$
$367$	−10.3116			−0.538262			−0.269131	+	0.963104i	$0.586736\pi$
$368$		0			0
$369$	9.79768	+	32.3188i	0.510047	+	1.68245i
$370$		0			0
$371$		−	28.5907i		−	1.48435i
$372$		0			0
$373$	25.5252			1.32165			0.660824	−	0.750541i	$-0.270207\pi$
$373$	25.5252			1.32165			0.660824	+	0.750541i	$0.270207\pi$
$374$		0			0
$375$	17.5137	−	8.26261i	0.904403	−	0.426679i
$376$		0			0
$377$	3.92343			0.202067
$378$		0			0
$379$		−	14.0643i		−	0.722435i	−0.932482	−	0.361217i	$-0.882361\pi$
$379$		−	14.0643i		−	0.722435i	0.932482	−	0.361217i	$-0.117639\pi$
$380$		0			0
$381$	5.32702	+	35.9333i	0.272911	+	1.84092i
$382$		0			0
$383$			10.6158i			0.542443i	0.962517	+	0.271222i	$0.0874276\pi$
$383$			10.6158i			0.542443i	−0.962517	+	0.271222i	$0.912572\pi$
$384$		0			0
$385$	−12.1773	+	26.8549i	−0.620614	+	1.36865i
$386$		0			0
$387$	2.82377	−	0.856047i	0.143541	−	0.0435153i
$388$		0			0
$389$	−18.3989			−0.932863			−0.466432	−	0.884557i	$-0.654461\pi$
$389$	−18.3989			−0.932863			−0.466432	+	0.884557i	$0.654461\pi$
$390$		0			0
$391$	6.89217	−	21.7916i	0.348552	−	1.10205i
$392$		0			0
$393$	−4.71298	−	31.7914i	−0.237738	−	1.60366i
$394$		0			0
$395$	1.88465	+	0.854593i	0.0948272	+	0.0429992i
$396$		0			0
$397$			12.0606i			0.605305i	0.953101	+	0.302652i	$0.0978721\pi$
$397$			12.0606i			0.605305i	−0.953101	+	0.302652i	$0.902128\pi$
$398$		0			0
$399$	−1.03146	−	6.95767i	−0.0516374	−	0.348319i
$400$		0			0
$401$	−1.84688			−0.0922287			−0.0461144	−	0.998936i	$-0.514684\pi$
$401$	−1.84688			−0.0922287			−0.0461144	+	0.998936i	$0.514684\pi$
$402$		0			0
$403$		−	7.05774i		−	0.351571i
$404$		0			0
$405$	−3.26547	−	19.8579i	−0.162263	−	0.986748i
$406$		0			0
$407$	35.2866			1.74909
$408$		0			0
$409$	16.6668			0.824120			0.412060	−	0.911157i	$-0.364809\pi$
$409$	16.6668			0.824120			0.412060	+	0.911157i	$0.364809\pi$
$410$		0			0
$411$	−0.506551	−	3.41694i	−0.0249863	−	0.168545i
$412$		0			0
$413$		−	11.1821i		−	0.550233i
$414$		0			0
$415$	−25.8008	−	11.6993i	−1.26651	−	0.574297i
$416$		0			0
$417$	35.1650	−	5.21312i	1.72204	−	0.255288i
$418$		0			0
$419$	29.6399			1.44801			0.724003	−	0.689797i	$-0.242300\pi$
$419$	29.6399			1.44801			0.724003	+	0.689797i	$0.242300\pi$
$420$		0			0
$421$		−	21.3723i		−	1.04162i	−0.853672	−	0.520811i	$-0.825630\pi$
$421$		−	21.3723i		−	1.04162i	0.853672	−	0.520811i	$-0.174370\pi$
$422$		0			0
$423$	−23.5255	+	7.13193i	−1.14385	+	0.346766i
$424$		0			0
$425$	−17.9245	+	15.7007i	−0.869467	+	0.761598i
$426$		0			0
$427$		−	31.6777i		−	1.53299i
$428$		0			0
$429$	−5.36720	−	36.2044i	−0.259131	−	1.74796i
$430$		0			0
$431$	−0.798473			−0.0384611			−0.0192306	−	0.999815i	$-0.506122\pi$
$431$	−0.798473			−0.0384611			−0.0192306	+	0.999815i	$0.506122\pi$
$432$		0			0
$433$	34.8937			1.67688			0.838442	−	0.544991i	$-0.183467\pi$
$433$	34.8937			1.67688			0.838442	+	0.544991i	$0.183467\pi$
$434$		0			0
$435$	−3.45453	−	0.987916i	−0.165632	−	0.0473669i
$436$		0			0
$437$	−2.22524	+	7.03575i	−0.106448	+	0.336566i
$438$		0			0
$439$	−14.6301			−0.698255			−0.349128	−	0.937075i	$-0.613522\pi$
$439$	−14.6301			−0.698255			−0.349128	+	0.937075i	$0.613522\pi$
$440$		0			0
$441$	−0.0991395	+	0.0300548i	−0.00472093	+	0.00143118i
$442$		0			0
$443$	0.387490			0.0184102			0.00920510	−	0.999958i	$-0.497070\pi$
$443$	0.387490			0.0184102			0.00920510	+	0.999958i	$0.497070\pi$
$444$		0			0
$445$	−2.49959	+	5.51240i	−0.118492	+	0.261313i
$446$		0			0
$447$	−31.3676	+	4.65016i	−1.48364	+	0.219945i
$448$		0			0
$449$			33.4370i			1.57799i	0.614401	+	0.788994i	$0.289398\pi$
$449$			33.4370i			1.57799i	−0.614401	+	0.788994i	$0.710602\pi$
$450$		0			0
$451$		−	56.2464i		−	2.64854i
$452$		0			0
$453$	−22.0779	+	3.27298i	−1.03731	+	0.153778i
$454$		0			0
$455$	−22.7305	−	10.3071i	−1.06562	−	0.483204i
$456$		0			0
$457$	−35.0846			−1.64119			−0.820595	−	0.571510i	$-0.806358\pi$
$457$	−35.0846			−1.64119			−0.820595	+	0.571510i	$0.806358\pi$
$458$		0			0
$459$	10.5819	+	22.3886i	0.493920	+	1.04501i
$460$		0			0
$461$			0.608136i			0.0283237i	0.999900	+	0.0141619i	$0.00450801\pi$
$461$			0.608136i			0.0283237i	−0.999900	+	0.0141619i	$0.995492\pi$
$462$		0			0
$463$		−	6.68914i		−	0.310871i	−0.987846	−	0.155435i	$-0.950322\pi$
$463$		−	6.68914i		−	0.310871i	0.987846	−	0.155435i	$-0.0496780\pi$
$464$		0			0
$465$	−1.77713	+	6.21425i	−0.0824126	+	0.288179i
$466$		0			0
$467$		−	35.4003i		−	1.63813i	−0.573700	−	0.819066i	$-0.694492\pi$
$467$		−	35.4003i		−	1.63813i	0.573700	−	0.819066i	$-0.305508\pi$
$468$		0			0
$469$	13.3375			0.615868
$470$		0			0
$471$	5.39351	−	0.799572i	0.248520	−	0.0368424i
$472$		0			0
$473$	−4.91439			−0.225964
$474$		0			0
$475$	5.78719	−	5.06922i	0.265535	−	0.232592i
$476$		0			0
$477$	−9.42857	−	31.1013i	−0.431704	−	1.42403i
$478$		0			0
$479$	−23.9530			−1.09444			−0.547219	−	0.836989i	$-0.684314\pi$
$479$	−23.9530			−1.09444			−0.547219	+	0.836989i	$0.684314\pi$
$480$		0			0
$481$			29.8672i			1.36183i
$482$		0			0
$483$	−21.6459	−	3.47425i	−0.984925	−	0.158084i
$484$		0			0
$485$	−4.43586	+	9.78251i	−0.201422	+	0.444201i
$486$		0			0
$487$		−	1.57108i		−	0.0711926i	−0.999366	−	0.0355963i	$-0.988667\pi$
$487$		−	1.57108i		−	0.0711926i	0.999366	−	0.0355963i	$-0.0113331\pi$
$488$		0			0
$489$	−3.82571	−	25.8063i	−0.173004	−	1.16700i
$490$		0			0
$491$			11.1996i			0.505429i	0.967541	+	0.252715i	$0.0813234\pi$
$491$			11.1996i			0.505429i	−0.967541	+	0.252715i	$0.918677\pi$
$492$		0			0
$493$	4.42121			0.199121
$494$		0			0
$495$	−4.39048	+	33.2289i	−0.197337	+	1.49353i
$496$		0			0
$497$		−	28.1736i		−	1.26376i
$498$		0			0
$499$	−30.3308			−1.35779			−0.678896	−	0.734234i	$-0.737541\pi$
$499$	−30.3308			−1.35779			−0.678896	+	0.734234i	$0.737541\pi$
$500$		0			0
$501$	−15.9151	+	2.35937i	−0.711034	+	0.105409i
$502$		0			0
$503$			0.217832i			0.00971263i	0.999988	+	0.00485632i	$0.00154582\pi$
$503$			0.217832i			0.00971263i	−0.999988	+	0.00485632i	$0.998454\pi$
$504$		0			0
$505$	−6.58743	−	2.98706i	−0.293137	−	0.132922i
$506$		0			0
$507$	8.37075	−	1.24094i	0.371758	−	0.0551121i
$508$		0			0
$509$		−	15.3321i		−	0.679584i	−0.940501	−	0.339792i	$-0.889643\pi$
$509$		−	15.3321i		−	0.679584i	0.940501	−	0.339792i	$-0.110357\pi$
$510$		0			0
$511$			30.1422i			1.33341i
$512$		0			0
$513$	−3.41652	−	7.22849i	−0.150843	−	0.319146i
$514$		0			0
$515$	−14.9023	+	32.8643i	−0.656673	+	1.44818i
$516$		0			0
$517$	40.9429			1.80067
$518$		0			0
$519$	−24.0199	+	3.56089i	−1.05436	+	0.156306i
$520$		0			0
$521$	7.61115			0.333451			0.166725	−	0.986003i	$-0.446681\pi$
$521$	7.61115			0.333451			0.166725	+	0.986003i	$0.446681\pi$
$522$		0			0
$523$	40.4017			1.76664			0.883321	−	0.468769i	$-0.155302\pi$
$523$	40.4017			1.76664			0.883321	+	0.468769i	$0.155302\pi$
$524$		0			0
$525$	17.4186	+	14.7988i	0.760208	+	0.645872i
$526$		0			0
$527$		−	7.95319i		−	0.346446i
$528$		0			0
$529$	18.8170	+	13.2257i	0.818131	+	0.575031i
$530$		0			0
$531$	−3.68759	−	12.1640i	−0.160028	−	0.527871i
$532$		0			0
$533$	47.6079			2.06213
$534$		0			0
$535$	−2.70741	−	1.22767i	−0.117052	−	0.0530769i
$536$		0			0
$537$	−4.76394	−	32.1351i	−0.205579	−	1.38673i
$538$		0			0
$539$	0.172538			0.00743176
$540$		0			0
$541$	35.4425			1.52379			0.761896	−	0.647700i	$-0.224269\pi$
$541$	35.4425			1.52379			0.761896	+	0.647700i	$0.224269\pi$
$542$		0			0
$543$	2.38783	+	16.1071i	0.102471	+	0.691220i
$544$		0			0
$545$	−29.2775	−	13.2758i	−1.25411	−	0.568675i
$546$		0			0
$547$			21.2820i			0.909955i	0.890503	+	0.454977i	$0.150353\pi$
$547$			21.2820i			0.909955i	−0.890503	+	0.454977i	$0.849647\pi$
$548$		0			0
$549$	−10.4466	−	34.4594i	−0.445850	−	1.47069i
$550$		0			0
$551$	−1.42745			−0.0608116
$552$		0			0
$553$			2.44245i			0.103864i
$554$		0			0
$555$	7.52054	−	26.2977i	0.319229	−	1.11627i
$556$		0			0
$557$			15.5087i			0.657125i	0.944482	+	0.328562i	$0.106564\pi$
$557$			15.5087i			0.657125i	−0.944482	+	0.328562i	$0.893436\pi$
$558$		0			0
$559$		−	4.15962i		−	0.175933i
$560$		0			0
$561$	−6.04816	−	40.7978i	−0.255353	−	1.72248i
$562$		0			0
$563$		−	16.4495i		−	0.693266i	−0.938001	−	0.346633i	$-0.887325\pi$
$563$		−	16.4495i		−	0.693266i	0.938001	−	0.346633i	$-0.112675\pi$
$564$		0			0
$565$	−9.41320	−	4.26840i	−0.396016	−	0.179573i
$566$		0			0
$567$	19.7544	−	13.1896i	0.829609	−	0.553910i
$568$		0			0
$569$	−8.67991			−0.363881			−0.181940	−	0.983310i	$-0.558238\pi$
$569$	−8.67991			−0.363881			−0.181940	+	0.983310i	$0.558238\pi$
$570$		0			0
$571$			33.8700i			1.41742i	0.705502	+	0.708708i	$0.250722\pi$
$571$			33.8700i			1.41742i	−0.705502	+	0.708708i	$0.749278\pi$
$572$		0			0
$573$	−22.4250	+	3.32444i	−0.936818	+	0.138881i
$574$		0			0
$575$	−9.62511	−	21.9626i	−0.401395	−	0.915905i
$576$		0			0
$577$			8.89556i			0.370327i	0.982708	+	0.185164i	$0.0592815\pi$
$577$			8.89556i			0.370327i	−0.982708	+	0.185164i	$0.940719\pi$
$578$		0			0
$579$	0.911700	+	6.14986i	0.0378890	+	0.255580i
$580$		0			0
$581$		−	33.4370i		−	1.38720i
$582$		0			0
$583$			54.1274i			2.24173i
$584$		0			0
$585$	−28.1255	−	3.71618i	−1.16285	−	0.153645i
$586$		0			0
$587$	−12.4694			−0.514667			−0.257334	−	0.966323i	$-0.582844\pi$
$587$	−12.4694			−0.514667			−0.257334	+	0.966323i	$0.582844\pi$
$588$		0			0
$589$			2.56780i			0.105805i
$590$		0			0
$591$	34.2468	−	5.07700i	1.40873	−	0.208840i
$592$		0			0
$593$	−6.27127			−0.257530			−0.128765	−	0.991675i	$-0.541101\pi$
$593$	−6.27127			−0.257530			−0.128765	+	0.991675i	$0.541101\pi$
$594$		0			0
$595$	−25.6144	−	11.6148i	−1.05009	−	0.476161i
$596$		0			0
$597$	4.59387	+	30.9879i	0.188015	+	1.26825i
$598$		0			0
$599$		−	27.4178i		−	1.12026i	−0.828404	−	0.560131i	$-0.810751\pi$
$599$		−	27.4178i		−	1.12026i	0.828404	−	0.560131i	$-0.189249\pi$
$600$		0			0
$601$	6.86043			0.279843			0.139921	−	0.990163i	$-0.455315\pi$
$601$	6.86043			0.279843			0.139921	+	0.990163i	$0.455315\pi$
$602$		0			0
$603$	14.5087	−	4.39841i	0.590839	−	0.179117i
$604$		0			0
$605$	12.8961	−	28.4400i	0.524301	−	1.15625i
$606$		0			0
$607$			3.27489i			0.132924i	0.997789	+	0.0664619i	$0.0211711\pi$
$607$			3.27489i			0.132924i	−0.997789	+	0.0664619i	$0.978829\pi$
$608$		0			0
$609$	−0.621892	−	4.19496i	−0.0252003	−	0.169989i
$610$		0			0
$611$			34.6548i			1.40198i
$612$		0			0
$613$	−23.6108			−0.953630			−0.476815	−	0.879004i	$-0.658209\pi$
$613$	−23.6108			−0.953630			−0.476815	+	0.879004i	$0.658209\pi$
$614$		0			0
$615$	−41.9182	−	11.9876i	−1.69030	−	0.483388i
$616$		0			0
$617$			7.88855i			0.317581i	0.987312	+	0.158791i	$0.0507595\pi$
$617$			7.88855i			0.317581i	−0.987312	+	0.158791i	$0.949241\pi$
$618$		0			0
$619$			28.1910i			1.13309i	0.824030	+	0.566546i	$0.191721\pi$
$619$			28.1910i			1.13309i	−0.824030	+	0.566546i	$0.808279\pi$
$620$		0			0
$621$	−24.6924	+	3.35902i	−0.990874	+	0.134793i
$622$		0			0
$623$	−7.14390			−0.286214
$624$		0			0
$625$	−3.29230	+	24.7823i	−0.131692	+	0.991291i
$626$		0			0
$627$	1.95274	+	13.1722i	0.0779848	+	0.526046i
$628$		0			0
$629$			33.6566i			1.34198i
$630$		0			0
$631$		−	32.8125i		−	1.30624i	−0.757252	−	0.653122i	$-0.773459\pi$
$631$		−	32.8125i		−	1.30624i	0.757252	−	0.653122i	$-0.226541\pi$
$632$		0			0
$633$	−21.3922	+	3.17133i	−0.850262	+	0.126049i
$634$		0			0
$635$	−42.7108	−	19.3672i	−1.69493	−	0.768562i
$636$		0			0
$637$			0.146039i			0.00578630i
$638$		0			0
$639$	−9.29102	−	30.6476i	−0.367547	−	1.21240i
$640$		0			0
$641$	46.9056			1.85266			0.926330	−	0.376713i	$-0.122946\pi$
$641$	46.9056			1.85266			0.926330	+	0.376713i	$0.122946\pi$
$642$		0			0
$643$	8.78913			0.346610			0.173305	−	0.984868i	$-0.444555\pi$
$643$	8.78913			0.346610			0.173305	+	0.984868i	$0.444555\pi$
$644$		0			0
$645$	−1.04739	+	3.66249i	−0.0412409	+	0.144210i
$646$		0			0
$647$	−1.09045			−0.0428699			−0.0214350	−	0.999770i	$-0.506823\pi$
$647$	−1.09045			−0.0428699			−0.0214350	+	0.999770i	$0.506823\pi$
$648$		0			0
$649$			21.1697i			0.830983i
$650$		0			0
$651$	−7.54620	+	1.11870i	−0.295759	+	0.0438454i
$652$		0			0
$653$	19.8447			0.776582			0.388291	−	0.921537i	$-0.373065\pi$
$653$	19.8447			0.776582			0.388291	+	0.921537i	$0.373065\pi$
$654$		0			0
$655$	37.7876	+	17.1347i	1.47649	+	0.669510i
$656$		0			0
$657$	9.94024	+	32.7891i	0.387806	+	1.27922i
$658$		0			0
$659$	−41.4315			−1.61394			−0.806971	−	0.590592i	$-0.798894\pi$
$659$	−41.4315			−1.61394			−0.806971	+	0.590592i	$0.798894\pi$
$660$		0			0
$661$			34.8741i			1.35645i	0.734856	+	0.678223i	$0.237250\pi$
$661$			34.8741i			1.35645i	−0.734856	+	0.678223i	$0.762750\pi$
$662$		0			0
$663$	34.5319	−	5.11926i	1.34111	−	0.198816i
$664$		0			0
$665$	8.26998	+	3.75001i	0.320696	+	0.145419i
$666$		0			0
$667$	−1.34166	+	4.24204i	−0.0519491	+	0.164252i
$668$		0			0
$669$	4.40662	+	29.7248i	0.170370	+	1.14923i
$670$		0			0
$671$			59.9718i			2.31519i
$672$		0			0
$673$		−	45.0347i		−	1.73596i	−0.496599	−	0.867980i	$-0.665418\pi$
$673$		−	45.0347i		−	1.73596i	0.496599	−	0.867980i	$-0.334582\pi$
$674$		0			0
$675$	23.8284	+	10.3540i	0.917157	+	0.398527i
$676$		0			0
$677$		−	27.7836i		−	1.06781i	−0.845544	−	0.533906i	$-0.820724\pi$
$677$		−	27.7836i		−	1.06781i	0.845544	−	0.533906i	$-0.179276\pi$
$678$		0			0
$679$	−12.6778			−0.486530
$680$		0			0
$681$	−4.32249	−	29.1573i	−0.165638	−	1.11731i
$682$		0			0
$683$	−17.5276			−0.670676			−0.335338	−	0.942098i	$-0.608850\pi$
$683$	−17.5276			−0.670676			−0.335338	+	0.942098i	$0.608850\pi$
$684$		0			0
$685$	4.06142	+	1.84164i	0.155179	+	0.0703656i
$686$		0			0
$687$	−3.80733	−	25.6823i	−0.145259	−	0.979842i
$688$		0			0
$689$	−45.8144			−1.74539
$690$		0			0
$691$	27.0517			1.02910			0.514548	−	0.857462i	$-0.327960\pi$
$691$	27.0517			1.02910			0.514548	+	0.857462i	$0.327960\pi$
$692$		0			0
$693$	−37.8593	+	11.4773i	−1.43816	+	0.435987i
$694$		0			0
$695$	−18.9531	+	41.7976i	−0.718931	+	1.58547i
$696$		0			0
$697$	53.6482			2.03207
$698$		0			0
$699$	−24.4618	+	3.62639i	−0.925230	+	0.137163i
$700$		0			0
$701$	−1.43549			−0.0542178			−0.0271089	−	0.999632i	$-0.508630\pi$
$701$	−1.43549			−0.0542178			−0.0271089	+	0.999632i	$0.508630\pi$
$702$		0			0
$703$		−	10.8665i		−	0.409839i
$704$		0			0
$705$	8.72605	−	30.5131i	0.328642	−	1.14919i
$706$		0			0
$707$		−	8.53710i		−	0.321071i
$708$		0			0
$709$			50.6146i			1.90087i	0.310921	+	0.950436i	$0.399363\pi$
$709$			50.6146i			1.90087i	−0.310921	+	0.950436i	$0.600637\pi$
$710$		0			0
$711$	0.805466	+	2.65693i	0.0302073	+	0.0996426i
$712$		0			0
$713$	7.63088	+	2.41346i	0.285779	+	0.0903850i
$714$		0			0
$715$	43.0330	+	19.5132i	1.60934	+	0.729754i
$716$		0			0
$717$	−5.69224	−	38.3969i	−0.212580	−	1.43396i
$718$		0			0
$719$			7.67412i			0.286196i	0.989708	+	0.143098i	$0.0457065\pi$
$719$			7.67412i			0.286196i	−0.989708	+	0.143098i	$0.954294\pi$
$720$		0			0
$721$	−42.5912			−1.58618
$722$		0			0
$723$	−0.972326	−	6.55882i	−0.0361612	−	0.243925i
$724$		0			0
$725$	3.48925	−	3.05636i	0.129588	−	0.113511i
$726$		0			0
$727$	−31.1521			−1.15537			−0.577685	−	0.816260i	$-0.696044\pi$
$727$	−31.1521			−1.15537			−0.577685	+	0.816260i	$0.696044\pi$
$728$		0			0
$729$	17.1395	−	20.8624i	0.634796	−	0.772680i
$730$		0			0
$731$		−	4.68737i		−	0.173369i
$732$		0			0
$733$	12.4805			0.460978			0.230489	−	0.973075i	$-0.425967\pi$
$733$	12.4805			0.460978			0.230489	+	0.973075i	$0.425967\pi$
$734$		0			0
$735$	0.0367726	−	0.128586i	0.00135638	−	0.00474296i
$736$		0			0
$737$	−25.2503			−0.930108
$738$		0			0
$739$	−24.5086			−0.901562			−0.450781	−	0.892635i	$-0.648854\pi$
$739$	−24.5086			−0.901562			−0.450781	+	0.892635i	$0.648854\pi$
$740$		0			0
$741$	−11.1491	+	1.65283i	−0.409574	+	0.0607182i
$742$		0			0
$743$		−	20.2346i		−	0.742334i	−0.928566	−	0.371167i	$-0.878958\pi$
$743$		−	20.2346i		−	0.742334i	0.928566	−	0.371167i	$-0.121042\pi$
$744$		0			0
$745$	16.9064	−	37.2840i	0.619401	−	1.36598i
$746$		0			0
$747$	−11.0268	−	36.3731i	−0.403448	−	1.33082i
$748$		0			0
$749$		−	3.50872i		−	0.128206i
$750$		0			0
$751$			5.76339i			0.210309i	0.994456	+	0.105155i	$0.0335338\pi$
$751$			5.76339i			0.210309i	−0.994456	+	0.105155i	$0.966466\pi$
$752$		0			0
$753$	17.5663	−	2.60416i	0.640153	−	0.0949008i
$754$		0			0
$755$	11.8994	−	26.2421i	0.433064	−	0.955046i
$756$		0			0
$757$	−4.51464			−0.164087			−0.0820436	−	0.996629i	$-0.526145\pi$
$757$	−4.51464			−0.164087			−0.0820436	+	0.996629i	$0.526145\pi$
$758$		0			0
$759$	40.9798	+	6.57739i	1.48747	+	0.238744i
$760$		0			0
$761$		−	27.6045i		−	1.00066i	−0.865834	−	0.500331i	$-0.833212\pi$
$761$		−	27.6045i		−	1.00066i	0.865834	−	0.500331i	$-0.166788\pi$
$762$		0			0
$763$		−	37.9428i		−	1.37362i
$764$		0			0
$765$	−31.6939	−	4.18767i	−1.14590	−	0.151405i
$766$		0			0
$767$	−17.9184			−0.646995
$768$		0			0
$769$		−	15.9862i		−	0.576478i	−0.957559	−	0.288239i	$-0.906930\pi$
$769$		−	15.9862i		−	0.576478i	0.957559	−	0.288239i	$-0.0930697\pi$
$770$		0			0
$771$	7.12194	−	1.05581i	0.256491	−	0.0380240i
$772$		0			0
$773$		−	22.1131i		−	0.795352i	−0.917526	−	0.397676i	$-0.869817\pi$
$773$		−	22.1131i		−	0.795352i	0.917526	−	0.397676i	$-0.130183\pi$
$774$		0			0
$775$	−5.49800	−	6.27671i	−0.197494	−	0.225466i
$776$		0			0
$777$	31.9343	−	4.73417i	1.14564	−	0.169837i
$778$		0			0
$779$	−17.3211			−0.620593
$780$		0			0
$781$			53.3378i			1.90858i
$782$		0			0
$783$	−2.05991	−	4.35825i	−0.0736151	−	0.155751i
$784$		0			0
$785$	−2.90696	+	6.41079i	−0.103754	+	0.228811i
$786$		0			0
$787$	15.1177			0.538889			0.269445	−	0.963016i	$-0.413160\pi$
$787$	15.1177			0.538889			0.269445	+	0.963016i	$0.413160\pi$
$788$		0			0
$789$	−3.89484	−	26.2726i	−0.138660	−	0.935330i
$790$		0			0
$791$		−	12.1992i		−	0.433754i
$792$		0			0
$793$	−50.7611			−1.80258
$794$		0			0
$795$	40.3390	+	11.5360i	1.43068	+	0.409141i
$796$		0			0
$797$		−	12.0843i		−	0.428048i	−0.976828	−	0.214024i	$-0.931343\pi$
$797$		−	12.0843i		−	0.428048i	0.976828	−	0.214024i	$-0.0686570\pi$
$798$		0			0
$799$			39.0516i			1.38155i
$800$		0			0
$801$	−7.77122	+	2.35590i	−0.274583	+	0.0832416i
$802$		0			0
$803$		−	57.0648i		−	2.01377i
$804$		0			0
$805$	18.9170	−	21.0517i	0.666737	−	0.741976i
$806$		0			0
$807$	4.35863	+	29.4011i	0.153431	+	1.03497i
$808$		0			0
$809$			11.3618i			0.399460i	0.979851	+	0.199730i	$0.0640064\pi$
$809$			11.3618i			0.399460i	−0.979851	+	0.199730i	$0.935994\pi$
$810$		0			0
$811$	4.18744			0.147041			0.0735204	−	0.997294i	$-0.476577\pi$
$811$	4.18744			0.147041			0.0735204	+	0.997294i	$0.476577\pi$
$812$		0			0
$813$	−5.26149	+	0.780001i	−0.184528	+	0.0273558i
$814$		0			0
$815$	30.6737	+	13.9089i	1.07445	+	0.487209i
$816$		0			0
$817$			1.51339i			0.0529467i
$818$		0			0
$819$	−9.71459	−	32.0447i	−0.339455	−	1.11973i
$820$		0			0
$821$			29.7192i			1.03721i	0.855015	+	0.518603i	$0.173548\pi$
$821$			29.7192i			1.03721i	−0.855015	+	0.518603i	$0.826452\pi$
$822$		0			0
$823$		−	32.8181i		−	1.14397i	−0.820265	−	0.571984i	$-0.806174\pi$
$823$		−	32.8181i		−	1.14397i	0.820265	−	0.571984i	$-0.193826\pi$
$824$		0			0
$825$	−32.9766	−	28.0168i	−1.14810	−	0.975421i
$826$		0			0
$827$		−	22.1052i		−	0.768672i	−0.923193	−	0.384336i	$-0.874430\pi$
$827$		−	22.1052i		−	0.768672i	0.923193	−	0.384336i	$-0.125570\pi$
$828$		0			0
$829$	17.9308			0.622761			0.311380	−	0.950285i	$-0.399209\pi$
$829$	17.9308			0.622761			0.311380	+	0.950285i	$0.399209\pi$
$830$		0			0
$831$	1.06549	+	7.18728i	0.0369616	+	0.249324i
$832$		0			0
$833$			0.164568i			0.00570195i
$834$		0			0
$835$	8.57783	−	18.9169i	0.296848	−	0.654646i
$836$		0			0
$837$	−7.83992	+	3.70551i	−0.270987	+	0.128081i
$838$		0			0
$839$	31.2542			1.07902			0.539508	−	0.841980i	$-0.318610\pi$
$839$	31.2542			1.07902			0.539508	+	0.841980i	$0.318610\pi$
$840$		0			0
$841$	28.1394			0.970322
$842$		0			0
$843$	−24.7189	+	3.66450i	−0.851363	+	0.126212i
$844$		0			0
$845$	−4.51162	+	9.94958i	−0.155205	+	0.342276i
$846$		0			0
$847$	36.8574			1.26644
$848$		0			0
$849$	21.7091	−	3.21831i	0.745054	−	0.110452i
$850$		0			0
$851$	−32.2926	−	10.2134i	−1.10698	−	0.350110i
$852$		0			0
$853$			12.2913i			0.420847i	0.977610	+	0.210424i	$0.0674843\pi$
$853$			12.2913i			0.420847i	−0.977610	+	0.210424i	$0.932516\pi$
$854$		0			0
$855$	10.2329	+	1.35205i	0.349956	+	0.0462391i
$856$		0			0
$857$	1.69819			0.0580090			0.0290045	−	0.999579i	$-0.490766\pi$
$857$	1.69819			0.0580090			0.0290045	+	0.999579i	$0.490766\pi$
$858$		0			0
$859$	25.9892			0.886739			0.443369	−	0.896339i	$-0.353783\pi$
$859$	25.9892			0.886739			0.443369	+	0.896339i	$0.353783\pi$
$860$		0			0
$861$	−7.54620	−	50.9028i	−0.257174	−	1.73476i
$862$		0			0
$863$	46.7754			1.59225			0.796126	−	0.605131i	$-0.206879\pi$
$863$	46.7754			1.59225			0.796126	+	0.605131i	$0.206879\pi$
$864$		0			0
$865$	12.9461	−	28.5504i	0.440182	−	0.970743i
$866$		0			0
$867$	9.78661	−	1.45084i	0.332371	−	0.0492731i
$868$		0			0
$869$		−	4.62401i		−	0.156859i
$870$		0			0
$871$		−	21.3723i		−	0.724173i
$872$		0			0
$873$	−13.7911	+	4.18087i	−0.466758	+	0.141501i
$874$		0			0
$875$	−28.2443	+	8.54063i	−0.954832	+	0.288726i
$876$		0			0
$877$		−	25.1000i		−	0.847566i	−0.905764	−	0.423783i	$-0.860702\pi$
$877$		−	25.1000i		−	0.847566i	0.905764	−	0.423783i	$-0.139298\pi$
$878$		0			0
$879$	3.11684	+	21.0246i	0.105128	+	0.709142i
$880$		0			0
$881$	−39.9000			−1.34427			−0.672133	−	0.740430i	$-0.734622\pi$
$881$	−39.9000			−1.34427			−0.672133	+	0.740430i	$0.734622\pi$
$882$		0			0
$883$		−	25.5727i		−	0.860589i	−0.902689	−	0.430294i	$-0.858410\pi$
$883$		−	25.5727i		−	0.860589i	0.902689	−	0.430294i	$-0.141590\pi$
$884$		0			0
$885$	15.7769	+	4.51183i	0.530335	+	0.151664i
$886$		0			0
$887$	−15.0289			−0.504623			−0.252311	−	0.967646i	$-0.581191\pi$
$887$	−15.0289			−0.504623			−0.252311	+	0.967646i	$0.581191\pi$
$888$		0			0
$889$		−	55.3519i		−	1.85644i
$890$		0			0
$891$	−37.3988	+	24.9703i	−1.25291	+	0.836537i
$892$		0			0
$893$		−	12.6084i		−	0.421924i
$894$		0			0
$895$	38.1962	+	17.3200i	1.27676	+	0.578945i
$896$		0			0
$897$	−5.56722	+	34.6860i	−0.185884	+	1.15813i
$898$		0			0
$899$			1.54820i			0.0516353i
$900$		0			0
$901$	−51.6271			−1.71995
$902$		0			0
$903$	−4.44750	+	0.659330i	−0.148004	+	0.0219411i
$904$		0			0
$905$	−19.1451	−	8.68130i	−0.636404	−	0.288576i
$906$		0			0
$907$	−10.9325			−0.363007			−0.181503	−	0.983390i	$-0.558096\pi$
$907$	−10.9325			−0.363007			−0.181503	+	0.983390i	$0.558096\pi$
$908$		0			0
$909$	−2.81535	−	9.28676i	−0.0933791	−	0.308022i
$910$		0			0
$911$	17.5163			0.580341			0.290171	−	0.956975i	$-0.406288\pi$
$911$	17.5163			0.580341			0.290171	+	0.956975i	$0.406288\pi$
$912$		0			0
$913$			63.3024i			2.09500i
$914$		0			0
$915$	44.6945	+	12.7816i	1.47756	+	0.422547i
$916$		0			0
$917$			48.9716i			1.61719i
$918$		0			0
$919$			29.3722i			0.968899i	0.874819	+	0.484450i	$0.160980\pi$
$919$			29.3722i			0.968899i	−0.874819	+	0.484450i	$0.839020\pi$
$920$		0			0
$921$	−3.62049	−	24.4220i	−0.119299	−	0.804732i
$922$		0			0
$923$	−45.1460			−1.48600
$924$		0			0
$925$	23.2666	+	26.5620i	0.765003	+	0.873354i
$926$		0			0
$927$	−46.3312	+	14.0456i	−1.52172	+	0.461319i
$928$		0			0
$929$		−	30.5320i		−	1.00172i	−0.865527	−	0.500862i	$-0.833017\pi$
$929$		−	30.5320i		−	1.00172i	0.865527	−	0.500862i	$-0.166983\pi$
$930$		0			0
$931$		−	0.0531333i		−	0.00174137i
$932$		0			0
$933$	2.81375	+	18.9801i	0.0921181	+	0.621382i
$934$		0			0
$935$	48.4928	+	21.9890i	1.58588	+	0.719116i
$936$		0			0
$937$	7.85359			0.256566			0.128283	−	0.991738i	$-0.459053\pi$
$937$	7.85359			0.256566			0.128283	+	0.991738i	$0.459053\pi$
$938$		0			0
$939$	−42.7386	+	6.33588i	−1.39472	+	0.206764i
$940$		0			0
$941$	11.7490			0.383007			0.191504	−	0.981492i	$-0.438664\pi$
$941$	11.7490			0.383007			0.191504	+	0.981492i	$0.438664\pi$
$942$		0			0
$943$	−16.2800	+	51.4740i	−0.530150	+	1.67622i
$944$		0			0
$945$	0.484728	+	30.6611i	0.0157682	+	0.997406i
$946$		0			0
$947$	−46.3624			−1.50657			−0.753287	−	0.657692i	$-0.771533\pi$
$947$	−46.3624			−1.50657			−0.753287	+	0.657692i	$0.771533\pi$
$948$		0			0
$949$	48.3006			1.56791
$950$		0			0
$951$	−8.35862	+	1.23914i	−0.271047	+	0.0401819i
$952$		0			0
$953$		−	1.03342i		−	0.0334756i	−0.999860	−	0.0167378i	$-0.994672\pi$
$953$		−	1.03342i		−	0.0334756i	0.999860	−	0.0167378i	$-0.00532806\pi$
$954$		0			0
$955$	12.0865	−	26.6547i	0.391110	−	0.862524i
$956$		0			0
$957$	1.17736	+	7.94184i	0.0380585	+	0.256723i
$958$		0			0
$959$			5.26347i			0.169966i
$960$		0			0
$961$	−28.2150			−0.910161
$962$		0			0
$963$	−1.15710	−	3.81683i	−0.0372870	−	0.122996i
$964$		0			0
$965$	−7.30981	−	3.31462i	−0.235311	−	0.106701i
$966$		0			0
$967$			0.920038i			0.0295864i	0.999891	+	0.0147932i	$0.00470900\pi$
$967$			0.920038i			0.0295864i	−0.999891	+	0.0147932i	$0.995291\pi$
$968$		0			0
$969$	−12.5637	+	1.86253i	−0.403604	+	0.0598331i
$970$		0			0
$971$	−33.0102			−1.05935			−0.529674	−	0.848202i	$-0.677686\pi$
$971$	−33.0102			−1.05935			−0.529674	+	0.848202i	$0.677686\pi$
$972$		0			0
$973$	−54.1684			−1.73656
$974$		0			0
$975$	23.7139	−	27.9119i	0.759453	−	0.893897i
$976$		0			0
$977$			45.9689i			1.47067i	0.677701	+	0.735337i	$0.262976\pi$
$977$			45.9689i			1.47067i	−0.677701	+	0.735337i	$0.737024\pi$
$978$		0			0
$979$	13.5247			0.432252
$980$		0			0
$981$	−12.5127	−	41.2746i	−0.399499	−	1.31780i
$982$		0			0
$983$			18.0700i			0.576342i	0.957579	+	0.288171i	$0.0930472\pi$
$983$			18.0700i			0.576342i	−0.957579	+	0.288171i	$0.906953\pi$
$984$		0			0
$985$	−18.4582	+	40.7063i	−0.588127	+	1.29701i
$986$		0			0
$987$	37.0532	−	5.49303i	1.17942	−	0.174845i
$988$		0			0
$989$	4.49741	+	1.42242i	0.143009	+	0.0452304i
$990$		0			0
$991$	−36.1251			−1.14755			−0.573777	−	0.819012i	$-0.694522\pi$
$991$	−36.1251			−1.14755			−0.573777	+	0.819012i	$0.694522\pi$
$992$		0			0
$993$	−22.7073	+	3.36629i	−0.720594	+	0.106826i
$994$		0			0
$995$	−36.8326	−	16.7017i	−1.16767	−	0.529480i
$996$		0			0
$997$		−	32.8486i		−	1.04032i	−0.854068	−	0.520162i	$-0.825872\pi$
$997$		−	32.8486i		−	1.04032i	0.854068	−	0.520162i	$-0.174128\pi$
$998$		0			0
$999$	33.1773	−	15.6811i	1.04968	−	0.496128i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.n.a.689.4	yes	48
3.2	odd	2	inner	1380.2.n.a.689.47	yes	48
5.4	even	2	inner	1380.2.n.a.689.46	yes	48
15.14	odd	2	inner	1380.2.n.a.689.1	✓	48
23.22	odd	2	inner	1380.2.n.a.689.3	yes	48
69.68	even	2	inner	1380.2.n.a.689.48	yes	48
115.114	odd	2	inner	1380.2.n.a.689.45	yes	48
345.344	even	2	inner	1380.2.n.a.689.2	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.n.a.689.1	✓	48	15.14	odd	2	inner
1380.2.n.a.689.2	yes	48	345.344	even	2	inner
1380.2.n.a.689.3	yes	48	23.22	odd	2	inner
1380.2.n.a.689.4	yes	48	1.1	even	1	trivial
1380.2.n.a.689.45	yes	48	115.114	odd	2	inner
1380.2.n.a.689.46	yes	48	5.4	even	2	inner
1380.2.n.a.689.47	yes	48	3.2	odd	2	inner
1380.2.n.a.689.48	yes	48	69.68	even	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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.71333 + 0.253996i) q^{3} +(0.923440 - 2.03648i) q^{5} +2.63922 q^{7} +(2.87097 - 0.870355i) q^{9} +O(q^{10})\)	\(q+(-1.71333 + 0.253996i) q^{3} +(0.923440 - 2.03648i) q^{5} +2.63922 q^{7} +(2.87097 - 0.870355i) q^{9} -4.99653 q^{11} -4.22915i q^{13} +(-1.06490 + 3.72371i) q^{15} -4.76572i q^{17} +1.53868i q^{19} +(-4.52184 + 0.670350i) q^{21} +(4.57258 + 1.44620i) q^{23} +(-3.29452 - 3.76114i) q^{25} +(-4.69784 + 2.22042i) q^{27} +0.927712i q^{29} +1.66883 q^{31} +(8.56068 - 1.26910i) q^{33} +(2.43716 - 5.37472i) q^{35} -7.06223 q^{37} +(1.07419 + 7.24591i) q^{39} +11.2571i q^{41} +0.983560 q^{43} +(0.878707 - 6.65040i) q^{45} -8.19428 q^{47} -0.0345317 q^{49} +(1.21047 + 8.16523i) q^{51} -10.8330i q^{53} +(-4.61399 + 10.1753i) q^{55} +(-0.390819 - 2.63626i) q^{57} -4.23688i q^{59} -12.0027i q^{61} +(7.57712 - 2.29706i) q^{63} +(-8.61258 - 3.90536i) q^{65} +5.05357 q^{67} +(-8.20165 - 1.31639i) q^{69} -10.6750i q^{71} +11.4209i q^{73} +(6.59990 + 5.60726i) q^{75} -13.1869 q^{77} +0.925445i q^{79} +(7.48496 - 4.99753i) q^{81} -12.6693i q^{83} +(-9.70530 - 4.40085i) q^{85} +(-0.235635 - 1.58947i) q^{87} -2.70682 q^{89} -11.1616i q^{91} +(-2.85926 + 0.423877i) q^{93} +(3.13350 + 1.42088i) q^{95} -4.80363 q^{97} +(-14.3449 + 4.34875i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) \) 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

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