LMFDB - Embedded newform 1224.2.bq.e.865.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(145,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 0, 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("1224.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1224 = 2^{3} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1224.bq (of order $8$ , degree $4$ , 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:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64$ x^16 + 36x^14 + 466x^12 + 2956x^10 + 10049x^8 + 18032x^6 + 14800x^4 + 3200*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			865.4
Root			$-1.71472i$ of defining polynomial
Character	$\chi$	$=$	1224.865
Dual form			1224.2.bq.e.433.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50807 - 1.03888i) q^{5} +(2.22010 + 0.919595i) q^{7} +(1.87280 - 4.52134i) q^{11} -1.07051i q^{13} +(-3.25570 - 2.52991i) q^{17} +(1.61272 + 1.61272i) q^{19} +(2.41400 - 5.82790i) q^{23} +(1.67562 - 1.67562i) q^{25} +(-3.13555 + 1.29879i) q^{29} +(2.67656 + 6.46178i) q^{31} +6.52351 q^{35} +(-1.63047 - 3.93630i) q^{37} +(-7.08028 - 2.93275i) q^{41} +(-4.37197 + 4.37197i) q^{43} +8.19868i q^{47} +(-0.866566 - 0.866566i) q^{49} +(7.97888 + 7.97888i) q^{53} -13.2855i q^{55} +(8.89888 - 8.89888i) q^{59} +(-2.52414 - 1.04553i) q^{61} +(-1.11213 - 2.68491i) q^{65} +12.8012 q^{67} +(1.62627 + 3.92615i) q^{71} +(-9.47623 + 3.92518i) q^{73} +(8.31560 - 8.31560i) q^{77} +(0.464750 - 1.12200i) q^{79} +(-2.92403 - 2.92403i) q^{83} +(-10.7938 - 2.96293i) q^{85} -2.96634i q^{89} +(0.984434 - 2.37663i) q^{91} +(5.72025 + 2.36941i) q^{95} +(16.8658 - 6.98605i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 8 q^{11} - 8 q^{19} + 8 q^{23} - 8 q^{25} - 32 q^{29} + 8 q^{31} - 16 q^{35} + 8 q^{37} - 40 q^{41} - 24 q^{43} + 24 q^{49} + 24 q^{53} + 16 q^{59} + 64 q^{65} + 16 q^{71} + 24 q^{73} - 96 q^{79}+ \cdots + 56 q^{97}+O(q^{100})$ 16 * q + 8 * q^11 - 8 * q^19 + 8 * q^23 - 8 * q^25 - 32 * q^29 + 8 * q^31 - 16 * q^35 + 8 * q^37 - 40 * q^41 - 24 * q^43 + 24 * q^49 + 24 * q^53 + 16 * q^59 + 64 * q^65 + 16 * q^71 + 24 * q^73 - 96 * q^79 - 32 * q^85 + 8 * q^95 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$ .

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{8}\right)$	$1$

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
$5$	2.50807	−	1.03888i	1.12164	−	0.464600i	0.256711	−	0.966488i	$-0.417361\pi$
$5$	2.50807	−	1.03888i	1.12164	−	0.464600i	0.864933	+	0.501888i	$0.167361\pi$
$6$		0			0
$7$	2.22010	+	0.919595i	0.839118	+	0.347574i	0.760506	−	0.649331i	$-0.224951\pi$
$7$	2.22010	+	0.919595i	0.839118	+	0.347574i	0.0786124	+	0.996905i	$0.474951\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.87280	−	4.52134i	0.564671	−	1.36324i	−0.341324	−	0.939946i	$-0.610875\pi$
$11$	1.87280	−	4.52134i	0.564671	−	1.36324i	0.905994	−	0.423290i	$-0.139125\pi$
$12$		0			0
$13$		−	1.07051i		−	0.296906i	−0.988920	−	0.148453i	$-0.952571\pi$
$13$		−	1.07051i		−	0.296906i	0.988920	−	0.148453i	$-0.0474293\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.25570	−	2.52991i	−0.789622	−	0.613593i
$17$	−3.25570	−	2.52991i	−0.789622	−	0.613593i
$18$		0			0
$19$	1.61272	+	1.61272i	0.369984	+	0.369984i	0.867471	−	0.497487i	$-0.165744\pi$
$19$	1.61272	+	1.61272i	0.369984	+	0.369984i	−0.497487	+	0.867471i	$0.665744\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.41400	−	5.82790i	0.503353	−	1.21520i	−0.444294	−	0.895881i	$-0.646545\pi$
$23$	2.41400	−	5.82790i	0.503353	−	1.21520i	0.947647	−	0.319321i	$-0.103455\pi$
$24$		0			0
$25$	1.67562	−	1.67562i	0.335125	−	0.335125i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.13555	+	1.29879i	−0.582257	+	0.241179i	−0.654315	−	0.756222i	$-0.727043\pi$
$29$	−3.13555	+	1.29879i	−0.582257	+	0.241179i	0.0720587	+	0.997400i	$0.477043\pi$
$30$		0			0
$31$	2.67656	+	6.46178i	0.480724	+	1.16057i	0.959266	+	0.282506i	$0.0911656\pi$
$31$	2.67656	+	6.46178i	0.480724	+	1.16057i	−0.478542	+	0.878065i	$0.658834\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.52351			1.10267
$36$		0			0
$37$	−1.63047	−	3.93630i	−0.268047	−	0.647123i	0.731344	−	0.682009i	$-0.238893\pi$
$37$	−1.63047	−	3.93630i	−0.268047	−	0.647123i	−0.999391	+	0.0348855i	$0.988893\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.08028	−	2.93275i	−1.10575	−	0.458018i	−0.246280	−	0.969199i	$-0.579208\pi$
$41$	−7.08028	−	2.93275i	−1.10575	−	0.458018i	−0.859473	+	0.511180i	$0.829208\pi$
$42$		0			0
$43$	−4.37197	+	4.37197i	−0.666719	+	0.666719i	−0.956955	−	0.290236i	$-0.906266\pi$
$43$	−4.37197	+	4.37197i	−0.666719	+	0.666719i	0.290236	+	0.956955i	$0.406266\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.19868i			1.19590i	0.801533	+	0.597950i	$0.204018\pi$
$47$			8.19868i			1.19590i	−0.801533	+	0.597950i	$0.795982\pi$
$48$		0			0
$49$	−0.866566	−	0.866566i	−0.123795	−	0.123795i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.97888	+	7.97888i	1.09598	+	1.09598i	0.994875	+	0.101108i	$0.0322389\pi$
$53$	7.97888	+	7.97888i	1.09598	+	1.09598i	0.101108	+	0.994875i	$0.467761\pi$
$54$		0			0
$55$		−	13.2855i		−	1.79141i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.89888	−	8.89888i	1.15854	−	1.15854i	0.173745	−	0.984791i	$-0.444413\pi$
$59$	8.89888	−	8.89888i	1.15854	−	1.15854i	0.984791	−	0.173745i	$-0.0555869\pi$
$60$		0			0
$61$	−2.52414	−	1.04553i	−0.323183	−	0.133867i	0.215193	−	0.976572i	$-0.430962\pi$
$61$	−2.52414	−	1.04553i	−0.323183	−	0.133867i	−0.538376	+	0.842705i	$0.680962\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.11213	−	2.68491i	−0.137942	−	0.333022i
$66$		0			0
$67$	12.8012			1.56392			0.781958	−	0.623331i	$-0.214221\pi$
$67$	12.8012			1.56392			0.781958	+	0.623331i	$0.214221\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.62627	+	3.92615i	0.193002	+	0.465948i	0.990524	−	0.137342i	$-0.0438559\pi$
$71$	1.62627	+	3.92615i	0.193002	+	0.465948i	−0.797521	+	0.603291i	$0.793856\pi$
$72$		0			0
$73$	−9.47623	+	3.92518i	−1.10911	+	0.459408i	−0.860631	−	0.509229i	$-0.829931\pi$
$73$	−9.47623	+	3.92518i	−1.10911	+	0.459408i	−0.248478	+	0.968637i	$0.579931\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.31560	−	8.31560i	0.947651	−	0.947651i
$78$		0			0
$79$	0.464750	−	1.12200i	0.0522884	−	0.126235i	−0.895577	−	0.444907i	$-0.853237\pi$
$79$	0.464750	−	1.12200i	0.0522884	−	0.126235i	0.947865	+	0.318672i	$0.103237\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.92403	−	2.92403i	−0.320954	−	0.320954i	0.528179	−	0.849133i	$-0.322875\pi$
$83$	−2.92403	−	2.92403i	−0.320954	−	0.320954i	−0.849133	+	0.528179i	$0.822875\pi$
$84$		0			0
$85$	−10.7938	−	2.96293i	−1.17075	−	0.321374i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.96634i		−	0.314431i	−0.987564	−	0.157216i	$-0.949748\pi$
$89$		−	2.96634i		−	0.314431i	0.987564	−	0.157216i	$-0.0502518\pi$
$90$		0			0
$91$	0.984434	−	2.37663i	0.103197	−	0.249139i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.72025	+	2.36941i	0.586886	+	0.243096i
$96$		0			0
$97$	16.8658	−	6.98605i	1.71247	−	0.709326i	0.712495	−	0.701678i	$-0.247565\pi$
$97$	16.8658	−	6.98605i	1.71247	−	0.709326i	0.999971	−	0.00764885i	$-0.00243473\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.04503			0.900014			0.450007	−	0.893025i	$-0.351421\pi$
$101$	9.04503			0.900014			0.450007	+	0.893025i	$0.351421\pi$
$102$		0			0
$103$	−2.23568			−0.220288			−0.110144	−	0.993916i	$-0.535131\pi$
$103$	−2.23568			−0.220288			−0.110144	+	0.993916i	$0.535131\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.04118	+	2.08813i	−0.487350	+	0.201867i	−0.612808	−	0.790232i	$-0.709960\pi$
$107$	−5.04118	+	2.08813i	−0.487350	+	0.201867i	0.125458	+	0.992099i	$0.459960\pi$
$108$		0			0
$109$	−2.44258	−	1.01175i	−0.233956	−	0.0969079i	0.262625	−	0.964898i	$-0.415412\pi$
$109$	−2.44258	−	1.01175i	−0.233956	−	0.0969079i	−0.496582	+	0.867990i	$0.665412\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.35434	+	12.9265i	−0.503694	+	1.21602i	0.443764	+	0.896144i	$0.353643\pi$
$113$	−5.35434	+	12.9265i	−0.503694	+	1.21602i	−0.947458	+	0.319880i	$0.896357\pi$
$114$		0			0
$115$		−	17.1246i		−	1.59688i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.90147	−	8.61057i	−0.449317	−	0.789330i
$120$		0			0
$121$	−9.15696	−	9.15696i	−0.832451	−	0.832451i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.73257	+	6.59701i	−0.244409	+	0.590054i
$126$		0			0
$127$	10.2828	−	10.2828i	0.912449	−	0.912449i	−0.0840159	−	0.996464i	$-0.526775\pi$
$127$	10.2828	−	10.2828i	0.912449	−	0.912449i	0.996464	+	0.0840159i	$0.0267746\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.23079	−	0.924024i	0.194905	−	0.0807323i	−0.283096	−	0.959092i	$-0.591361\pi$
$131$	2.23079	−	0.924024i	0.194905	−	0.0807323i	0.478001	+	0.878359i	$0.341361\pi$
$132$		0			0
$133$	2.09735	+	5.06346i	0.181864	+	0.439058i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.42184			0.121476			0.0607382	−	0.998154i	$-0.480655\pi$
$137$	1.42184			0.121476			0.0607382	+	0.998154i	$0.480655\pi$
$138$		0			0
$139$	5.94425	+	14.3507i	0.504184	+	1.21721i	0.947185	+	0.320687i	$0.103914\pi$
$139$	5.94425	+	14.3507i	0.504184	+	1.21721i	−0.443001	+	0.896521i	$0.646086\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.84013	−	2.00485i	−0.404752	−	0.167654i
$144$		0			0
$145$	−6.51490	+	6.51490i	−0.541033	+	0.541033i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			16.2390i			1.33035i	0.746687	+	0.665176i	$0.231643\pi$
$149$			16.2390i			1.33035i	−0.746687	+	0.665176i	$0.768357\pi$
$150$		0			0
$151$	2.48539	+	2.48539i	0.202258	+	0.202258i	0.800967	−	0.598709i	$-0.204319\pi$
$151$	2.48539	+	2.48539i	0.202258	+	0.202258i	−0.598709	+	0.800967i	$0.704319\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	13.4260	+	13.4260i	1.07840	+	1.07840i
$156$		0			0
$157$			21.3904i			1.70714i	0.520975	+	0.853572i	$0.325568\pi$
$157$			21.3904i			1.70714i	−0.520975	+	0.853572i	$0.674432\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.7186	−	10.7186i	0.844746	−	0.844746i
$162$		0			0
$163$	18.3602	+	7.60506i	1.43809	+	0.595675i	0.959333	−	0.282275i	$-0.0910892\pi$
$163$	18.3602	+	7.60506i	1.43809	+	0.595675i	0.478752	+	0.877950i	$0.341089\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.03768	−	7.33360i	−0.235063	−	0.567491i	0.761697	−	0.647934i	$-0.224367\pi$
$167$	−3.03768	−	7.33360i	−0.235063	−	0.567491i	−0.996759	+	0.0804426i	$0.974367\pi$
$168$		0			0
$169$	11.8540			0.911847
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.18012	−	22.1628i	−0.697951	−	1.68500i	−0.728112	−	0.685458i	$-0.759602\pi$
$173$	−9.18012	−	22.1628i	−0.697951	−	1.68500i	0.0301604	−	0.999545i	$-0.490398\pi$
$174$		0			0
$175$	5.26094	−	2.17915i	0.397690	−	0.164728i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.0828	+	17.0828i	−1.27683	+	1.27683i	−0.334392	+	0.942434i	$0.608531\pi$
$179$	−17.0828	+	17.0828i	−1.27683	+	1.27683i	−0.942434	+	0.334392i	$0.891469\pi$
$180$		0			0
$181$	4.65785	−	11.2451i	0.346216	−	0.835838i	−0.650844	−	0.759211i	$-0.725585\pi$
$181$	4.65785	−	11.2451i	0.346216	−	0.835838i	0.997060	−	0.0766270i	$-0.0244151\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.17866	−	8.17866i	−0.601307	−	0.601307i
$186$		0			0
$187$	−17.5359	+	9.98209i	−1.28235	+	0.729963i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3.27804i		−	0.237191i	−0.992943	−	0.118595i	$-0.962161\pi$
$191$		−	3.27804i		−	0.237191i	0.992943	−	0.118595i	$-0.0378392\pi$
$192$		0			0
$193$	−7.94091	+	19.1711i	−0.571599	+	1.37996i	0.328593	+	0.944472i	$0.393425\pi$
$193$	−7.94091	+	19.1711i	−0.571599	+	1.37996i	−0.900193	+	0.435492i	$0.856575\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.2135	−	7.54429i	−1.29766	−	0.537508i	−0.376401	−	0.926457i	$-0.622838\pi$
$197$	−18.2135	−	7.54429i	−1.29766	−	0.537508i	−0.921259	+	0.388949i	$0.872838\pi$
$198$		0			0
$199$	−14.0934	+	5.83768i	−0.999055	+	0.413822i	−0.821451	−	0.570280i	$-0.806835\pi$
$199$	−14.0934	+	5.83768i	−0.999055	+	0.413822i	−0.177605	+	0.984102i	$0.556835\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.15558			−0.572410
$204$		0			0
$205$	−20.8046			−1.45306
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.3120	−	4.27137i	0.713295	−	0.295457i
$210$		0			0
$211$	−11.5631	−	4.78961i	−0.796039	−	0.329730i	−0.0526704	−	0.998612i	$-0.516773\pi$
$211$	−11.5631	−	4.78961i	−0.796039	−	0.329730i	−0.743369	+	0.668882i	$0.766773\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.42327	+	15.5071i	−0.438063	+	1.05758i
$216$		0			0
$217$			16.8071i			1.14094i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.70829	+	3.48525i	−0.182179	+	0.234443i
$222$		0			0
$223$	−5.32288	−	5.32288i	−0.356446	−	0.356446i	0.506055	−	0.862501i	$-0.331103\pi$
$223$	−5.32288	−	5.32288i	−0.356446	−	0.356446i	−0.862501	+	0.506055i	$0.831103\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.83963	+	4.44126i	−0.122101	+	0.294777i	−0.973097	−	0.230394i	$-0.925999\pi$
$227$	−1.83963	+	4.44126i	−0.122101	+	0.294777i	0.850997	+	0.525171i	$0.175999\pi$
$228$		0			0
$229$	−9.47732	+	9.47732i	−0.626279	+	0.626279i	−0.947130	−	0.320851i	$-0.896031\pi$
$229$	−9.47732	+	9.47732i	−0.626279	+	0.626279i	0.320851	+	0.947130i	$0.396031\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.7306	+	5.27318i	−0.834008	+	0.345457i	−0.758488	−	0.651687i	$-0.774062\pi$
$233$	−12.7306	+	5.27318i	−0.834008	+	0.345457i	−0.0755198	+	0.997144i	$0.524062\pi$
$234$		0			0
$235$	8.51742	+	20.5629i	0.555616	+	1.34137i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.0060			1.10003			0.550014	−	0.835155i	$-0.314622\pi$
$239$	17.0060			1.10003			0.550014	+	0.835155i	$0.314622\pi$
$240$		0			0
$241$	−7.42077	−	17.9153i	−0.478014	−	1.15403i	−0.960539	−	0.278145i	$-0.910280\pi$
$241$	−7.42077	−	17.9153i	−0.478014	−	1.15403i	0.482525	−	0.875882i	$-0.339720\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.07366	−	1.27315i	−0.196369	−	0.0813388i
$246$		0			0
$247$	1.72644	−	1.72644i	0.109850	−	0.109850i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			21.5316i			1.35906i	0.733646	+	0.679532i	$0.237817\pi$
$251$			21.5316i			1.35906i	−0.733646	+	0.679532i	$0.762183\pi$
$252$		0			0
$253$	−21.8290	−	21.8290i	−1.37238	−	1.37238i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.5195	−	13.5195i	−0.843324	−	0.843324i	0.145966	−	0.989290i	$-0.453371\pi$
$257$	−13.5195	−	13.5195i	−0.843324	−	0.843324i	−0.989290	+	0.145966i	$0.953371\pi$
$258$		0			0
$259$		−	10.2383i		−	0.636179i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.83952	+	7.83952i	−0.483406	+	0.483406i	−0.906218	−	0.422812i	$-0.861043\pi$
$263$	−7.83952	+	7.83952i	−0.483406	+	0.483406i	0.422812	+	0.906218i	$0.361043\pi$
$264$		0			0
$265$	28.3007	+	11.7225i	1.73850	+	0.720109i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.69352	−	4.08852i	−0.103256	−	0.249282i	0.863805	−	0.503826i	$-0.168075\pi$
$269$	−1.69352	−	4.08852i	−0.103256	−	0.249282i	−0.967061	+	0.254544i	$0.918075\pi$
$270$		0			0
$271$	16.2476			0.986973			0.493486	−	0.869753i	$-0.335722\pi$
$271$	16.2476			0.986973			0.493486	+	0.869753i	$0.335722\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.43795	−	10.7142i	−0.267619	−	0.646089i
$276$		0			0
$277$	−0.591916	+	0.245180i	−0.0355648	+	0.0147314i	−0.400395	−	0.916343i	$-0.631127\pi$
$277$	−0.591916	+	0.245180i	−0.0355648	+	0.0147314i	0.364830	+	0.931074i	$0.381127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.0521155	+	0.0521155i	−0.00310895	+	0.00310895i	−0.708660	−	0.705551i	$-0.750700\pi$
$281$	−0.0521155	+	0.0521155i	−0.00310895	+	0.00310895i	0.705551	+	0.708660i	$0.250700\pi$
$282$		0			0
$283$	−7.82868	+	18.9001i	−0.465367	+	1.12350i	0.500797	+	0.865565i	$0.333040\pi$
$283$	−7.82868	+	18.9001i	−0.465367	+	1.12350i	−0.966164	+	0.257930i	$0.916960\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.0220	−	13.0220i	−0.768663	−	0.768663i
$288$		0			0
$289$	4.19911	+	16.4732i	0.247007	+	0.969014i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			7.88796i			0.460820i	0.973094	+	0.230410i	$0.0740067\pi$
$293$			7.88796i			0.460820i	−0.973094	+	0.230410i	$0.925993\pi$
$294$		0			0
$295$	13.0742	−	31.5639i	0.761209	−	1.83772i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.23882	−	2.58420i	−0.360800	−	0.149448i
$300$		0			0
$301$	−13.7266	+	5.68576i	−0.791190	+	0.327722i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.41691			−0.424691
$306$		0			0
$307$	−4.63954			−0.264793			−0.132396	−	0.991197i	$-0.542267\pi$
$307$	−4.63954			−0.264793			−0.132396	+	0.991197i	$0.542267\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.16892	−	2.14104i	0.293103	−	0.121407i	−0.231287	−	0.972886i	$-0.574294\pi$
$311$	5.16892	−	2.14104i	0.293103	−	0.121407i	0.524389	+	0.851479i	$0.324294\pi$
$312$		0			0
$313$	0.810266	+	0.335623i	0.0457990	+	0.0189705i	0.405465	−	0.914110i	$-0.367109\pi$
$313$	0.810266	+	0.335623i	0.0457990	+	0.0189705i	−0.359666	+	0.933081i	$0.617109\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.3854	+	27.4867i	−0.639467	+	1.54381i	0.187925	+	0.982183i	$0.439824\pi$
$317$	−11.3854	+	27.4867i	−0.639467	+	1.54381i	−0.827392	+	0.561625i	$0.810176\pi$
$318$		0			0
$319$			16.6092i			0.929939i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.17049	−	9.33059i	−0.0651280	−	0.519168i
$324$		0			0
$325$	−1.79377	−	1.79377i	−0.0995003	−	0.0995003i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.53947	+	18.2019i	−0.415664	+	1.00350i
$330$		0			0
$331$	−8.18438	+	8.18438i	−0.449854	+	0.449854i	−0.895306	−	0.445452i	$-0.853043\pi$
$331$	−8.18438	+	8.18438i	−0.449854	+	0.449854i	0.445452	+	0.895306i	$0.353043\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	32.1063	−	13.2989i	1.75416	−	0.726595i
$336$		0			0
$337$	6.47644	+	15.6355i	0.352794	+	0.851720i	0.996273	+	0.0862562i	$0.0274904\pi$
$337$	6.47644	+	15.6355i	0.352794	+	0.851720i	−0.643479	+	0.765464i	$0.722510\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	34.2286			1.85358
$342$		0			0
$343$	−7.56414	−	18.2614i	−0.408425	−	0.986025i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.02695	+	0.839592i	0.108813	+	0.0450716i	0.436426	−	0.899740i	$-0.356244\pi$
$347$	2.02695	+	0.839592i	0.108813	+	0.0450716i	−0.327613	+	0.944812i	$0.606244\pi$
$348$		0			0
$349$	18.7192	−	18.7192i	1.00201	−	1.00201i	0.00201610	−	0.999998i	$-0.499358\pi$
$349$	18.7192	−	18.7192i	1.00201	−	1.00201i	0.999998	−	0.00201610i	$-0.000641744\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			24.9345i			1.32713i	0.748120	+	0.663564i	$0.230957\pi$
$353$			24.9345i			1.32713i	−0.748120	+	0.663564i	$0.769043\pi$
$354$		0			0
$355$	8.15758	+	8.15758i	0.432959	+	0.432959i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.09699	−	2.09699i	−0.110675	−	0.110675i	0.649601	−	0.760276i	$-0.274936\pi$
$359$	−2.09699	−	2.09699i	−0.110675	−	0.110675i	−0.760276	+	0.649601i	$0.774936\pi$
$360$		0			0
$361$		−	13.7982i		−	0.726223i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−19.6893	+	19.6893i	−1.03058	+	1.03058i
$366$		0			0
$367$	−21.6341	−	8.96114i	−1.12929	−	0.467768i	−0.261751	−	0.965135i	$-0.584300\pi$
$367$	−21.6341	−	8.96114i	−1.12929	−	0.467768i	−0.867540	+	0.497368i	$0.834300\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.3766	+	25.0512i	0.538724	+	1.30060i
$372$		0			0
$373$	30.4720			1.57778			0.788890	−	0.614534i	$-0.210656\pi$
$373$	30.4720			1.57778			0.788890	+	0.614534i	$0.210656\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.39036	+	3.35663i	0.0716073	+	0.172875i
$378$		0			0
$379$	−2.43151	+	1.00717i	−0.124898	+	0.0517346i	−0.444257	−	0.895899i	$-0.646532\pi$
$379$	−2.43151	+	1.00717i	−0.124898	+	0.0517346i	0.319359	+	0.947634i	$0.396532\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.8259	+	13.8259i	−0.706473	+	0.706473i	−0.965792	−	0.259319i	$-0.916502\pi$
$383$	−13.8259	+	13.8259i	−0.706473	+	0.706473i	0.259319	+	0.965792i	$0.416502\pi$
$384$		0			0
$385$	12.2172	−	29.4950i	0.622648	−	1.50321i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.3694	+	10.3694i	0.525748	+	0.525748i	0.919302	−	0.393553i	$-0.128754\pi$
$389$	10.3694	+	10.3694i	0.525748	+	0.525748i	−0.393553	+	0.919302i	$0.628754\pi$
$390$		0			0
$391$	−22.6033	+	12.8667i	−1.14310	+	0.650696i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	3.29689i		−	0.165884i
$396$		0			0
$397$	5.12132	−	12.3639i	0.257031	−	0.620529i	−0.741708	−	0.670723i	$-0.765984\pi$
$397$	5.12132	−	12.3639i	0.257031	−	0.620529i	0.998739	+	0.0501941i	$0.0159840\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.03313	+	2.49900i	0.301280	+	0.124794i	0.528202	−	0.849119i	$-0.322866\pi$
$401$	6.03313	+	2.49900i	0.301280	+	0.124794i	−0.226922	+	0.973913i	$0.572866\pi$
$402$		0			0
$403$	6.91739	−	2.86528i	0.344580	−	0.142730i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.8509			−1.03354
$408$		0			0
$409$	−6.45254			−0.319058			−0.159529	−	0.987193i	$-0.550997\pi$
$409$	−6.45254			−0.319058			−0.159529	+	0.987193i	$0.550997\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	27.9398	−	11.5730i	1.37483	−	0.569471i
$414$		0			0
$415$	−10.3714	−	4.29596i	−0.509111	−	0.210881i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.77323	+	21.1805i	−0.428600	+	1.03473i	0.551131	+	0.834419i	$0.314196\pi$
$419$	−8.77323	+	21.1805i	−0.428600	+	1.03473i	−0.979732	+	0.200314i	$0.935804\pi$
$420$		0			0
$421$		−	26.8644i		−	1.30929i	−0.755936	−	0.654645i	$-0.772818\pi$
$421$		−	26.8644i		−	1.30929i	0.755936	−	0.654645i	$-0.227182\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.69449	+	1.21614i	−0.470252	+	0.0589917i
$426$		0			0
$427$	−4.64238	−	4.64238i	−0.224660	−	0.224660i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.8232	−	35.7865i	0.714010	−	1.72377i	0.0242809	−	0.999705i	$-0.492270\pi$
$431$	14.8232	−	35.7865i	0.714010	−	1.72377i	0.689729	−	0.724068i	$-0.257730\pi$
$432$		0			0
$433$	−21.6205	+	21.6205i	−1.03901	+	1.03901i	−0.0398071	+	0.999207i	$0.512674\pi$
$433$	−21.6205	+	21.6205i	−1.03901	+	1.03901i	−0.999207	+	0.0398071i	$0.987326\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13.2919	−	5.50569i	0.635839	−	0.263373i
$438$		0			0
$439$	−0.602318	−	1.45412i	−0.0287471	−	0.0694016i	0.908854	−	0.417114i	$-0.136958\pi$
$439$	−0.602318	−	1.45412i	−0.0287471	−	0.0694016i	−0.937601	+	0.347713i	$0.886958\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.7788			−0.702164			−0.351082	−	0.936345i	$-0.614186\pi$
$443$	−14.7788			−0.702164			−0.351082	+	0.936345i	$0.614186\pi$
$444$		0			0
$445$	−3.08166	−	7.43979i	−0.146085	−	0.352680i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.04315	+	3.74579i	0.426773	+	0.176775i	0.585722	−	0.810512i	$-0.300811\pi$
$449$	9.04315	+	3.74579i	0.426773	+	0.176775i	−0.158950	+	0.987287i	$0.550811\pi$
$450$		0			0
$451$	−26.5199	+	26.5199i	−1.24877	+	1.24877i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	6.98347i		−	0.327390i
$456$		0			0
$457$	−17.3054	−	17.3054i	−0.809513	−	0.809513i	0.175047	−	0.984560i	$-0.443992\pi$
$457$	−17.3054	−	17.3054i	−0.809513	−	0.809513i	−0.984560	+	0.175047i	$0.943992\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.62092	−	9.62092i	−0.448091	−	0.448091i	0.446629	−	0.894719i	$-0.352625\pi$
$461$	−9.62092	−	9.62092i	−0.448091	−	0.448091i	−0.894719	+	0.446629i	$0.852625\pi$
$462$		0			0
$463$		−	10.8498i		−	0.504233i	−0.967697	−	0.252117i	$-0.918873\pi$
$463$		−	10.8498i		−	0.504233i	0.967697	−	0.252117i	$-0.0811267\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.9692	+	10.9692i	−0.507594	+	0.507594i	−0.913787	−	0.406193i	$-0.866856\pi$
$467$	−10.9692	+	10.9692i	−0.507594	+	0.507594i	0.406193	+	0.913787i	$0.366856\pi$
$468$		0			0
$469$	28.4199	+	11.7719i	1.31231	+	0.543577i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11.5793	+	27.9550i	0.532418	+	1.28537i
$474$		0			0
$475$	5.40464			0.247982
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.4855	−	39.7996i	−0.753244	−	1.81849i	−0.540407	−	0.841403i	$-0.681730\pi$
$479$	−16.4855	−	39.7996i	−0.753244	−	1.81849i	−0.212836	−	0.977088i	$-0.568270\pi$
$480$		0			0
$481$	−4.21384	+	1.74543i	−0.192134	+	0.0795847i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	35.0430	−	35.0430i	1.59122	−	1.59122i
$486$		0			0
$487$	16.2097	−	39.1337i	0.734531	−	1.77331i	0.107666	−	0.994187i	$-0.465662\pi$
$487$	16.2097	−	39.1337i	0.734531	−	1.77331i	0.626865	−	0.779128i	$-0.284338\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.65738	−	3.65738i	−0.165055	−	0.165055i	0.619747	−	0.784802i	$-0.287235\pi$
$491$	−3.65738	−	3.65738i	−0.165055	−	0.165055i	−0.784802	+	0.619747i	$0.787235\pi$
$492$		0			0
$493$	13.4942	+	3.70420i	0.607748	+	0.166829i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			10.2119i			0.458068i
$498$		0			0
$499$	−8.48908	+	20.4945i	−0.380023	+	0.917458i	0.611937	+	0.790907i	$0.290391\pi$
$499$	−8.48908	+	20.4945i	−0.380023	+	0.917458i	−0.991960	+	0.126551i	$0.959609\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	25.0563	+	10.3786i	1.11720	+	0.462761i	0.863413	−	0.504498i	$-0.168322\pi$
$503$	25.0563	+	10.3786i	1.11720	+	0.462761i	0.253791	+	0.967259i	$0.418322\pi$
$504$		0			0
$505$	22.6856	−	9.39667i	1.00950	−	0.418147i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−25.9336			−1.14949			−0.574743	−	0.818334i	$-0.694898\pi$
$509$	−25.9336			−1.14949			−0.574743	+	0.818334i	$0.694898\pi$
$510$		0			0
$511$	−24.6478			−1.09035
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.60724	+	2.32259i	−0.247084	+	0.102346i
$516$		0			0
$517$	37.0690	+	15.3545i	1.63029	+	0.675290i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14.8942	−	35.9578i	0.652527	−	1.57534i	−0.156573	−	0.987666i	$-0.550045\pi$
$521$	14.8942	−	35.9578i	0.652527	−	1.57534i	0.809099	−	0.587672i	$-0.199955\pi$
$522$		0			0
$523$			29.2087i			1.27721i	0.769536	+	0.638603i	$0.220488\pi$
$523$			29.2087i			1.27721i	−0.769536	+	0.638603i	$0.779512\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.63367	−	27.8091i	0.332528	−	1.21138i
$528$		0			0
$529$	−11.8736	−	11.8736i	−0.516244	−	0.516244i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.13953	+	7.57950i	−0.135988	+	0.328304i
$534$		0			0
$535$	−10.4743	+	10.4743i	−0.452845	+	0.452845i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.54094	+	2.29513i	−0.238665	+	0.0988584i
$540$		0			0
$541$	5.71438	+	13.7957i	0.245680	+	0.593125i	0.997828	−	0.0658697i	$-0.0209822\pi$
$541$	5.71438	+	13.7957i	0.245680	+	0.593125i	−0.752148	+	0.658994i	$0.770982\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.17724			−0.307439
$546$		0			0
$547$	8.00943	+	19.3365i	0.342458	+	0.826767i	0.997466	+	0.0711452i	$0.0226654\pi$
$547$	8.00943	+	19.3365i	0.342458	+	0.826767i	−0.655008	+	0.755622i	$0.727335\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.15136	−	2.96219i	−0.304658	−	0.126194i
$552$		0			0
$553$	2.06358	−	2.06358i	0.0877523	−	0.0877523i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	11.4512i		−	0.485203i	−0.970126	−	0.242602i	$-0.921999\pi$
$557$		−	11.4512i		−	0.485203i	0.970126	−	0.242602i	$-0.0780008\pi$
$558$		0			0
$559$	4.68023	+	4.68023i	0.197952	+	0.197952i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.1023	+	16.1023i	0.678629	+	0.678629i	0.959690	−	0.281061i	$-0.0906862\pi$
$563$	16.1023	+	16.1023i	0.678629	+	0.678629i	−0.281061	+	0.959690i	$0.590686\pi$
$564$		0			0
$565$			37.9831i			1.59796i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	22.4856	−	22.4856i	0.942644	−	0.942644i	−0.0557985	−	0.998442i	$-0.517770\pi$
$569$	22.4856	−	22.4856i	0.942644	−	0.942644i	0.998442	+	0.0557985i	$0.0177704\pi$
$570$		0			0
$571$	−36.9087	−	15.2881i	−1.54458	−	0.639787i	−0.562256	−	0.826963i	$-0.690067\pi$
$571$	−36.9087	−	15.2881i	−1.54458	−	0.639787i	−0.982326	+	0.187176i	$0.940067\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.72042	−	13.8103i	−0.238558	−	0.575930i
$576$		0			0
$577$	28.3973			1.18219			0.591097	−	0.806600i	$-0.298695\pi$
$577$	28.3973			1.18219			0.591097	+	0.806600i	$0.298695\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.80271	−	9.18055i	−0.157763	−	0.380873i
$582$		0			0
$583$	51.0181	−	21.1324i	2.11295	−	0.875214i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.76184	−	7.76184i	0.320365	−	0.320365i	−0.528542	−	0.848907i	$-0.677261\pi$
$587$	7.76184	−	7.76184i	0.320365	−	0.320365i	0.848907	+	0.528542i	$0.177261\pi$
$588$		0			0
$589$	−6.10453	+	14.7376i	−0.251533	+	0.607254i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.22864	+	2.22864i	0.0915194	+	0.0915194i	0.751384	−	0.659865i	$-0.229387\pi$
$593$	2.22864	+	2.22864i	0.0915194	+	0.0915194i	−0.659865	+	0.751384i	$0.729387\pi$
$594$		0			0
$595$	−21.2386	−	16.5039i	−0.870696	−	0.676594i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			15.4176i			0.629945i	0.949101	+	0.314972i	$0.101995\pi$
$599$			15.4176i			0.629945i	−0.949101	+	0.314972i	$0.898005\pi$
$600$		0			0
$601$	8.10132	−	19.5583i	0.330460	−	0.797801i	−0.668096	−	0.744075i	$-0.732890\pi$
$601$	8.10132	−	19.5583i	0.330460	−	0.797801i	0.998556	−	0.0537256i	$-0.0171096\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−32.4793	−	13.4534i	−1.32047	−	0.546957i
$606$		0			0
$607$	−14.0086	+	5.80253i	−0.568590	+	0.235518i	−0.648409	−	0.761292i	$-0.724565\pi$
$607$	−14.0086	+	5.80253i	−0.568590	+	0.235518i	0.0798199	+	0.996809i	$0.474565\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.77676			0.355070
$612$		0			0
$613$	43.4543			1.75510			0.877551	−	0.479484i	$-0.159176\pi$
$613$	43.4543			1.75510			0.877551	+	0.479484i	$0.159176\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	36.3420	−	15.0534i	1.46307	−	0.606025i	0.497807	−	0.867288i	$-0.334139\pi$
$617$	36.3420	−	15.0534i	1.46307	−	0.606025i	0.965268	+	0.261263i	$0.0841388\pi$
$618$		0			0
$619$	9.69556	+	4.01603i	0.389697	+	0.161418i	0.568924	−	0.822390i	$-0.307360\pi$
$619$	9.69556	+	4.01603i	0.389697	+	0.161418i	−0.179227	+	0.983808i	$0.557360\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.72783	−	6.58557i	0.109288	−	0.263845i
$624$		0			0
$625$			31.2330i			1.24932i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.65017	+	16.9403i	−0.185414	+	0.675455i
$630$		0			0
$631$	−32.2236	−	32.2236i	−1.28280	−	1.28280i	−0.939067	−	0.343734i	$-0.888308\pi$
$631$	−32.2236	−	32.2236i	−1.28280	−	1.28280i	−0.343734	−	0.939067i	$-0.611692\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.1074	−	36.4725i	0.599519	−	1.44737i
$636$		0			0
$637$	−0.927666	+	0.927666i	−0.0367555	+	0.0367555i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13.8538	+	5.73842i	−0.547191	+	0.226654i	−0.639114	−	0.769112i	$-0.720699\pi$
$641$	−13.8538	+	5.73842i	−0.547191	+	0.226654i	0.0919230	+	0.995766i	$0.470699\pi$
$642$		0			0
$643$	7.89107	+	19.0507i	0.311193	+	0.751287i	0.999661	+	0.0260206i	$0.00828356\pi$
$643$	7.89107	+	19.0507i	0.311193	+	0.751287i	−0.688468	+	0.725267i	$0.741716\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−26.8160			−1.05424			−0.527122	−	0.849790i	$-0.676729\pi$
$647$	−26.8160			−1.05424			−0.527122	+	0.849790i	$0.676729\pi$
$648$		0			0
$649$	−23.5690	−	56.9007i	−0.925166	−	2.23355i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.6601	+	5.24399i	0.495428	+	0.205213i	0.616385	−	0.787445i	$-0.288596\pi$
$653$	12.6601	+	5.24399i	0.495428	+	0.205213i	−0.120957	+	0.992658i	$0.538596\pi$
$654$		0			0
$655$	4.63504	−	4.63504i	0.181106	−	0.181106i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	20.2130i		−	0.787388i	−0.919241	−	0.393694i	$-0.871197\pi$
$659$		−	20.2130i		−	0.787388i	0.919241	−	0.393694i	$-0.128803\pi$
$660$		0			0
$661$	5.04254	+	5.04254i	0.196132	+	0.196132i	0.798340	−	0.602208i	$-0.205712\pi$
$661$	5.04254	+	5.04254i	0.196132	+	0.196132i	−0.602208	+	0.798340i	$0.705712\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.5206	+	10.5206i	0.407972	+	0.407972i
$666$		0			0
$667$			21.4089i			0.828957i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.45443	+	9.45443i	−0.364984	+	0.364984i
$672$		0			0
$673$	−34.6300	−	14.3442i	−1.33489	−	0.552929i	−0.402844	−	0.915269i	$-0.631978\pi$
$673$	−34.6300	−	14.3442i	−1.33489	−	0.552929i	−0.932046	+	0.362339i	$0.881978\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.0912	−	34.0193i	−0.541571	−	1.30747i	−0.923614	−	0.383323i	$-0.874780\pi$
$677$	−14.0912	−	34.0193i	−0.541571	−	1.30747i	0.382044	−	0.924144i	$-0.375220\pi$
$678$		0			0
$679$	43.8681			1.68350
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.42199	−	15.5041i	−0.245731	−	0.593246i	0.752102	−	0.659047i	$-0.229040\pi$
$683$	−6.42199	−	15.5041i	−0.245731	−	0.593246i	−0.997833	+	0.0658003i	$0.979040\pi$
$684$		0			0
$685$	3.56609	−	1.47712i	0.136253	−	0.0564379i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.54146	−	8.54146i	0.325404	−	0.325404i
$690$		0			0
$691$	5.24785	−	12.6694i	0.199638	−	0.481968i	−0.792078	−	0.610420i	$-0.791001\pi$
$691$	5.24785	−	12.6694i	0.199638	−	0.481968i	0.991716	+	0.128452i	$0.0410007\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	29.8172	+	29.8172i	1.13103	+	1.13103i
$696$		0			0
$697$	15.6316	+	27.4606i	0.592091	+	1.04014i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	11.5629i		−	0.436724i	−0.975868	−	0.218362i	$-0.929929\pi$
$701$		−	11.5629i		−	0.436724i	0.975868	−	0.218362i	$-0.0700713\pi$
$702$		0			0
$703$	3.71867	−	8.97766i	0.140252	−	0.338599i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	20.0809	+	8.31776i	0.755218	+	0.312822i
$708$		0			0
$709$	−13.9142	+	5.76346i	−0.522560	+	0.216451i	−0.628341	−	0.777938i	$-0.716266\pi$
$709$	−13.9142	+	5.76346i	−0.522560	+	0.216451i	0.105781	+	0.994389i	$0.466266\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	44.1199			1.65230
$714$		0			0
$715$	−14.2222			−0.531880
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.76251	+	0.730055i	−0.0657305	+	0.0272264i	−0.415306	−	0.909682i	$-0.636326\pi$
$719$	−1.76251	+	0.730055i	−0.0657305	+	0.0272264i	0.349576	+	0.936908i	$0.386326\pi$
$720$		0			0
$721$	−4.96342	−	2.05592i	−0.184847	−	0.0765663i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.07772	+	7.43027i	−0.114304	+	0.275953i
$726$		0			0
$727$		−	4.61649i		−	0.171216i	−0.996329	−	0.0856082i	$-0.972717\pi$
$727$		−	4.61649i		−	0.171216i	0.996329	−	0.0856082i	$-0.0272833\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	25.2945	−	3.17311i	0.935550	−	0.117362i
$732$		0			0
$733$	4.76113	+	4.76113i	0.175856	+	0.175856i	0.789547	−	0.613691i	$-0.210316\pi$
$733$	4.76113	+	4.76113i	0.175856	+	0.175856i	−0.613691	+	0.789547i	$0.710316\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	23.9741	−	57.8786i	0.883097	−	2.13199i
$738$		0			0
$739$	6.49918	−	6.49918i	0.239076	−	0.239076i	−0.577391	−	0.816468i	$-0.695929\pi$
$739$	6.49918	−	6.49918i	0.239076	−	0.239076i	0.816468	+	0.577391i	$0.195929\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.90655	+	3.27500i	−0.290063	+	0.120148i	−0.522970	−	0.852351i	$-0.675176\pi$
$743$	−7.90655	+	3.27500i	−0.290063	+	0.120148i	0.232907	+	0.972499i	$0.425176\pi$
$744$		0			0
$745$	16.8703	+	40.7286i	0.618081	+	1.49218i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.1122			−0.479108
$750$		0			0
$751$	3.59617	+	8.68192i	0.131226	+	0.316808i	0.975812	−	0.218613i	$-0.0701532\pi$
$751$	3.59617	+	8.68192i	0.131226	+	0.316808i	−0.844586	+	0.535420i	$0.820153\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.81554	+	3.65151i	0.320830	+	0.132892i
$756$		0			0
$757$	−30.9701	+	30.9701i	−1.12563	+	1.12563i	−0.134750	+	0.990880i	$0.543023\pi$
$757$	−30.9701	+	30.9701i	−1.12563	+	1.12563i	−0.990880	+	0.134750i	$0.956977\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			7.85599i			0.284779i	0.989811	+	0.142390i	$0.0454786\pi$
$761$			7.85599i			0.284779i	−0.989811	+	0.142390i	$0.954521\pi$
$762$		0			0
$763$	−4.49236	−	4.49236i	−0.162634	−	0.162634i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.52632	−	9.52632i	−0.343976	−	0.343976i
$768$		0			0
$769$		−	33.7233i		−	1.21609i	−0.793901	−	0.608047i	$-0.791953\pi$
$769$		−	33.7233i		−	1.21609i	0.793901	−	0.608047i	$-0.208047\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.9064	+	19.9064i	−0.715983	+	0.715983i	−0.967780	−	0.251797i	$-0.918978\pi$
$773$	−19.9064	+	19.9064i	−0.715983	+	0.715983i	0.251797	+	0.967780i	$0.418978\pi$
$774$		0			0
$775$	15.3124	+	6.34261i	0.550038	+	0.227833i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.68883	−	16.1483i	−0.239652	−	0.578571i
$780$		0			0
$781$	20.7971			0.744180
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.2220	+	53.6488i	0.793139	+	1.91481i
$786$		0			0
$787$	31.9518	−	13.2349i	1.13896	−	0.471772i	0.268141	−	0.963380i	$-0.413591\pi$
$787$	31.9518	−	13.2349i	1.13896	−	0.471772i	0.870817	+	0.491608i	$0.163591\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−23.7743	+	23.7743i	−0.845317	+	0.845317i
$792$		0			0
$793$	−1.11925	+	2.70211i	−0.0397458	+	0.0959549i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−31.8112	−	31.8112i	−1.12681	−	1.12681i	−0.990692	−	0.136119i	$-0.956537\pi$
$797$	−31.8112	−	31.8112i	−1.12681	−	1.12681i	−0.136119	−	0.990692i	$-0.543463\pi$
$798$		0			0
$799$	20.7419	−	26.6924i	0.733797	−	0.944310i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			50.1964i			1.77139i
$804$		0			0
$805$	15.7477	−	38.0184i	0.555035	−	1.33997i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.7452	−	11.9066i	−1.01063	−	0.418615i	−0.184943	−	0.982749i	$-0.559210\pi$
$809$	−28.7452	−	11.9066i	−1.01063	−	0.418615i	−0.825683	+	0.564134i	$0.809210\pi$
$810$		0			0
$811$	44.6958	−	18.5136i	1.56948	−	0.650101i	0.582779	−	0.812631i	$-0.301965\pi$
$811$	44.6958	−	18.5136i	1.56948	−	0.650101i	0.986704	+	0.162529i	$0.0519653\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	53.9495			1.88977
$816$		0			0
$817$	−14.1016			−0.493351
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.2625	−	8.80722i	0.742067	−	0.307374i	0.0205669	−	0.999788i	$-0.493453\pi$
$821$	21.2625	−	8.80722i	0.742067	−	0.307374i	0.721500	+	0.692414i	$0.243453\pi$
$822$		0			0
$823$	10.0413	+	4.15925i	0.350018	+	0.144982i	0.550764	−	0.834661i	$-0.314336\pi$
$823$	10.0413	+	4.15925i	0.350018	+	0.144982i	−0.200746	+	0.979643i	$0.564336\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.2934	+	29.6790i	−0.427485	+	1.03204i	0.552598	+	0.833448i	$0.313637\pi$
$827$	−12.2934	+	29.6790i	−0.427485	+	1.03204i	−0.980082	+	0.198591i	$0.936363\pi$
$828$		0			0
$829$			21.0657i			0.731642i	0.930685	+	0.365821i	$0.119212\pi$
$829$			21.0657i			0.731642i	−0.930685	+	0.365821i	$0.880788\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.628942	+	5.01361i	0.0217915	+	0.173711i
$834$		0			0
$835$	−15.2374	−	15.2374i	−0.527313	−	0.527313i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.3781	−	39.5402i	0.565435	−	1.36508i	−0.339932	−	0.940450i	$-0.610404\pi$
$839$	16.3781	−	39.5402i	0.565435	−	1.36508i	0.905367	−	0.424631i	$-0.139596\pi$
$840$		0			0
$841$	−12.3613	+	12.3613i	−0.426251	+	0.426251i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	29.7307	−	12.3149i	1.02277	−	0.423644i
$846$		0			0
$847$	−11.9087	−	28.7500i	−0.409186	−	0.987863i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−26.8763			−0.921307
$852$		0			0
$853$	5.75449	+	13.8926i	0.197030	+	0.475673i	0.991256	−	0.131949i	$-0.0421236\pi$
$853$	5.75449	+	13.8926i	0.197030	+	0.475673i	−0.794226	+	0.607622i	$0.792124\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.2462	−	4.65833i	−0.384163	−	0.159126i	0.182240	−	0.983254i	$-0.441665\pi$
$857$	−11.2462	−	4.65833i	−0.384163	−	0.159126i	−0.566403	+	0.824129i	$0.691665\pi$
$858$		0			0
$859$	9.12775	−	9.12775i	0.311435	−	0.311435i	−0.534030	−	0.845465i	$-0.679323\pi$
$859$	9.12775	−	9.12775i	0.311435	−	0.311435i	0.845465	+	0.534030i	$0.179323\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			21.4811i			0.731226i	0.930767	+	0.365613i	$0.119141\pi$
$863$			21.4811i			0.731226i	−0.930767	+	0.365613i	$0.880859\pi$
$864$		0			0
$865$	−46.0488	−	46.0488i	−1.56571	−	1.56571i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.20258	−	4.20258i	−0.142563	−	0.142563i
$870$		0			0
$871$		−	13.7038i		−	0.464335i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.1332	+	12.1332i	−0.410175	+	0.410175i
$876$		0			0
$877$	−47.3960	−	19.6321i	−1.60045	−	0.662928i	−0.608971	−	0.793192i	$-0.708417\pi$
$877$	−47.3960	−	19.6321i	−1.60045	−	0.662928i	−0.991479	+	0.130264i	$0.958417\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.5491	+	37.5388i	0.523861	+	1.26471i	0.935487	+	0.353361i	$0.114961\pi$
$881$	15.5491	+	37.5388i	0.523861	+	1.26471i	−0.411626	+	0.911353i	$0.635039\pi$
$882$		0			0
$883$	3.59541			0.120995			0.0604975	−	0.998168i	$-0.480731\pi$
$883$	3.59541			0.120995			0.0604975	+	0.998168i	$0.480731\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	15.3637	+	37.0913i	0.515864	+	1.24540i	0.940424	+	0.340005i	$0.110429\pi$
$887$	15.3637	+	37.0913i	0.515864	+	1.24540i	−0.424560	+	0.905400i	$0.639571\pi$
$888$		0			0
$889$	32.2848	−	13.3728i	1.08280	−	0.448509i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.2222	+	13.2222i	−0.442465	+	0.442465i
$894$		0			0
$895$	−25.0979	+	60.5917i	−0.838930	+	2.02536i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−16.7850	−	16.7850i	−0.559810	−	0.559810i
$900$		0			0
$901$	−5.79096	−	46.1627i	−0.192925	−	1.53790i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	33.0423i		−	1.09836i
$906$		0			0
$907$	16.9424	−	40.9026i	0.562563	−	1.35815i	−0.345147	−	0.938549i	$-0.612171\pi$
$907$	16.9424	−	40.9026i	0.562563	−	1.35815i	0.907710	−	0.419599i	$-0.137829\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.08964	−	3.35084i	−0.268022	−	0.111018i	0.244625	−	0.969618i	$-0.421335\pi$
$911$	−8.08964	−	3.35084i	−0.268022	−	0.111018i	−0.512647	+	0.858599i	$0.671335\pi$
$912$		0			0
$913$	−18.6966	+	7.74440i	−0.618768	+	0.256302i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.80230			0.191609
$918$		0			0
$919$	−32.6562			−1.07723			−0.538615	−	0.842552i	$-0.681052\pi$
$919$	−32.6562			−1.07723			−0.538615	+	0.842552i	$0.681052\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.20298	−	1.74093i	0.138343	−	0.0573034i
$924$		0			0
$925$	−9.32780	−	3.86370i	−0.306696	−	0.127038i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.8006	−	45.3888i	0.616829	−	1.48916i	−0.238537	−	0.971133i	$-0.576668\pi$
$929$	18.8006	−	45.3888i	0.616829	−	1.48916i	0.855366	−	0.518024i	$-0.173332\pi$
$930$		0			0
$931$		−	2.79506i		−	0.0916045i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−33.6110	+	43.2534i	−1.09920	+	1.41454i
$936$		0			0
$937$	28.6009	+	28.6009i	0.934352	+	0.934352i	0.997974	−	0.0636224i	$-0.0202653\pi$
$937$	28.6009	+	28.6009i	0.934352	+	0.934352i	−0.0636224	+	0.997974i	$0.520265\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.10983	+	2.67938i	−0.0361796	+	0.0873452i	−0.940937	−	0.338583i	$-0.890052\pi$
$941$	−1.10983	+	2.67938i	−0.0361796	+	0.0873452i	0.904757	+	0.425928i	$0.140052\pi$
$942$		0			0
$943$	−34.1835	+	34.1835i	−1.11317	+	1.11317i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.7742	−	17.3034i	1.35748	−	0.562286i	0.419114	−	0.907934i	$-0.362341\pi$
$947$	41.7742	−	17.3034i	1.35748	−	0.562286i	0.938364	+	0.345648i	$0.112341\pi$
$948$		0			0
$949$	4.20194	+	10.1444i	0.136401	+	0.329301i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−17.8145			−0.577069			−0.288534	−	0.957470i	$-0.593168\pi$
$953$	−17.8145			−0.577069			−0.288534	+	0.957470i	$0.593168\pi$
$954$		0			0
$955$	−3.40549	−	8.22157i	−0.110199	−	0.266044i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.15663	+	1.30752i	0.101933	+	0.0422221i
$960$		0			0
$961$	−12.6704	+	12.6704i	−0.408722	+	0.408722i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			56.3320i			1.81339i
$966$		0			0
$967$	19.1664	+	19.1664i	0.616349	+	0.616349i	0.944593	−	0.328244i	$-0.106457\pi$
$967$	19.1664	+	19.1664i	0.616349	+	0.616349i	−0.328244	+	0.944593i	$0.606457\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.02700	−	1.02700i	−0.0329582	−	0.0329582i	0.690436	−	0.723394i	$-0.257419\pi$
$971$	−1.02700	−	1.02700i	−0.0329582	−	0.0329582i	−0.723394	+	0.690436i	$0.757419\pi$
$972$		0			0
$973$			37.3262i			1.19662i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−34.2951	+	34.2951i	−1.09720	+	1.09720i	−0.102462	+	0.994737i	$0.532672\pi$
$977$	−34.2951	+	34.2951i	−1.09720	+	1.09720i	−0.994737	+	0.102462i	$0.967328\pi$
$978$		0			0
$979$	−13.4118	−	5.55536i	−0.428644	−	0.177550i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.76821	−	16.3399i	−0.215872	−	0.521162i	0.778433	−	0.627727i	$-0.216015\pi$
$983$	−6.76821	−	16.3399i	−0.215872	−	0.521162i	−0.994306	+	0.106565i	$0.966015\pi$
$984$		0			0
$985$	−53.5184			−1.70524
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.9255	+	36.0333i	0.474603	+	1.14579i
$990$		0			0
$991$	−26.3051	+	10.8959i	−0.835610	+	0.346121i	−0.759121	−	0.650950i	$-0.774371\pi$
$991$	−26.3051	+	10.8959i	−0.835610	+	0.346121i	−0.0764888	+	0.997070i	$0.524371\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.2826	+	29.2826i	−0.928322	+	0.928322i
$996$		0			0
$997$	−4.92753	+	11.8961i	−0.156056	+	0.376753i	−0.982499	−	0.186267i	$-0.940361\pi$
$997$	−4.92753	+	11.8961i	−0.156056	+	0.376753i	0.826443	+	0.563021i	$0.190361\pi$
$998$		0			0
$999$		0			0

Display

a_p

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up to: 50 250 1000 (See

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a_n

instead) (See

a_n

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a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.2.bq.e.865.4		16
3.2	odd	2		408.2.ba.a.49.1	yes	16
12.11	even	2		816.2.bq.f.49.3		16
17.8	even	8	inner	1224.2.bq.e.433.4		16
51.5	even	16		6936.2.a.bo.1.7		8
51.8	odd	8		408.2.ba.a.25.1	✓	16
51.29	even	16		6936.2.a.bl.1.2		8
204.59	even	8		816.2.bq.f.433.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.ba.a.25.1	✓	16	51.8	odd	8
408.2.ba.a.49.1	yes	16	3.2	odd	2
816.2.bq.f.49.3		16	12.11	even	2
816.2.bq.f.433.3		16	204.59	even	8
1224.2.bq.e.433.4		16	17.8	even	8	inner
1224.2.bq.e.865.4		16	1.1	even	1	trivial
6936.2.a.bl.1.2		8	51.29	even	16
6936.2.a.bo.1.7		8	51.5	even	16

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

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Newspace parameters

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1224.2.bq.e.865.4

Level	$1224$
Weight	$2$
Character	1224.865
Analytic conductor	$9.774$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$