LMFDB - Embedded newform 960.3.n.a.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(161,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("960.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Character	$\chi$	$=$	960.161
Dual form			960.3.n.a.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.99223 + 0.215720i) q^{3} -2.23607 q^{5} -8.57552 q^{7} +(8.90693 - 1.29097i) q^{9} +O(q^{10})$	\(q+(-2.99223 + 0.215720i) q^{3} -2.23607 q^{5} -8.57552 q^{7} +(8.90693 - 1.29097i) q^{9} +15.7729 q^{11} +18.1183i q^{13} +(6.69084 - 0.482364i) q^{15} -5.58301i q^{17} -24.9510i q^{19} +(25.6600 - 1.84991i) q^{21} -40.7977i q^{23} +5.00000 q^{25} +(-26.3731 + 5.78428i) q^{27} +37.9035 q^{29} -30.6023 q^{31} +(-47.1963 + 3.40253i) q^{33} +19.1755 q^{35} +27.8697i q^{37} +(-3.90847 - 54.2141i) q^{39} -12.6340i q^{41} +77.0982i q^{43} +(-19.9165 + 2.88669i) q^{45} +61.8422i q^{47} +24.5396 q^{49} +(1.20437 + 16.7057i) q^{51} -19.2048 q^{53} -35.2693 q^{55} +(5.38242 + 74.6592i) q^{57} -11.4120 q^{59} +101.141i q^{61} +(-76.3816 + 11.0707i) q^{63} -40.5137i q^{65} -63.1963i q^{67} +(8.80088 + 122.076i) q^{69} +21.0444i q^{71} -61.5698 q^{73} +(-14.9612 + 1.07860i) q^{75} -135.261 q^{77} -114.761 q^{79} +(77.6668 - 22.9971i) q^{81} -100.898 q^{83} +12.4840i q^{85} +(-113.416 + 8.17654i) q^{87} +11.6579i q^{89} -155.374i q^{91} +(91.5692 - 6.60152i) q^{93} +55.7921i q^{95} +46.4906 q^{97} +(140.488 - 20.3623i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{9}+O(q^{10}) $ 24 * q - 8 * q^9	$ 24 q - 8 q^{9} + 120 q^{25} - 256 q^{33} + 104 q^{49} - 304 q^{57} + 400 q^{73} + 152 q^{81} + 208 q^{97}+O(q^{100}) $ 24 * q - 8 * q^9 + 120 * q^25 - 256 * q^33 + 104 * q^49 - 304 * q^57 + 400 * q^73 + 152 * q^81 + 208 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−2.99223	+	0.215720i	−0.997411	+	0.0719066i
$3$	−2.99223	+	0.215720i	−0.997411	+	0.0719066i
$4$		0			0
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$	−8.57552			−1.22507			−0.612537	−	0.790442i	$-0.709851\pi$
$7$	−8.57552			−1.22507			−0.612537	+	0.790442i	$0.709851\pi$
$8$		0			0
$9$	8.90693	−	1.29097i	0.989659	−	0.143441i
$10$		0			0
$11$	15.7729			1.43390			0.716951	−	0.697123i	$-0.245537\pi$
$11$	15.7729			1.43390			0.716951	+	0.697123i	$0.245537\pi$
$12$		0			0
$13$			18.1183i			1.39371i	0.717210	+	0.696857i	$0.245419\pi$
$13$			18.1183i			1.39371i	−0.717210	+	0.696857i	$0.754581\pi$
$14$		0			0
$15$	6.69084	−	0.482364i	0.446056	−	0.0321576i
$16$		0			0
$17$		−	5.58301i		−	0.328413i	−0.986426	−	0.164206i	$-0.947494\pi$
$17$		−	5.58301i		−	0.328413i	0.986426	−	0.164206i	$-0.0525063\pi$
$18$		0			0
$19$		−	24.9510i		−	1.31321i	−0.754235	−	0.656605i	$-0.771992\pi$
$19$		−	24.9510i		−	1.31321i	0.754235	−	0.656605i	$-0.228008\pi$
$20$		0			0
$21$	25.6600	−	1.84991i	1.22190	−	0.0880910i
$22$		0			0
$23$		−	40.7977i		−	1.77381i	−0.461948	−	0.886907i	$-0.652849\pi$
$23$		−	40.7977i		−	1.77381i	0.461948	−	0.886907i	$-0.347151\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$	−26.3731	+	5.78428i	−0.976783	+	0.214233i
$28$		0			0
$29$	37.9035			1.30702			0.653509	−	0.756918i	$-0.273296\pi$
$29$	37.9035			1.30702			0.653509	+	0.756918i	$0.273296\pi$
$30$		0			0
$31$	−30.6023			−0.987170			−0.493585	−	0.869697i	$-0.664314\pi$
$31$	−30.6023			−0.987170			−0.493585	+	0.869697i	$0.664314\pi$
$32$		0			0
$33$	−47.1963	+	3.40253i	−1.43019	+	0.103107i
$34$		0			0
$35$	19.1755			0.547870
$36$		0			0
$37$			27.8697i			0.753235i	0.926369	+	0.376617i	$0.122913\pi$
$37$			27.8697i			0.753235i	−0.926369	+	0.376617i	$0.877087\pi$
$38$		0			0
$39$	−3.90847	−	54.2141i	−0.100217	−	1.39011i
$40$		0			0
$41$		−	12.6340i		−	0.308147i	−0.988059	−	0.154073i	$-0.950761\pi$
$41$		−	12.6340i		−	0.308147i	0.988059	−	0.154073i	$-0.0492392\pi$
$42$		0			0
$43$			77.0982i			1.79298i	0.443062	+	0.896491i	$0.353892\pi$
$43$			77.0982i			1.79298i	−0.443062	+	0.896491i	$0.646108\pi$
$44$		0			0
$45$	−19.9165	+	2.88669i	−0.442589	+	0.0641487i
$46$		0			0
$47$			61.8422i			1.31579i	0.753109	+	0.657895i	$0.228553\pi$
$47$			61.8422i			1.31579i	−0.753109	+	0.657895i	$0.771447\pi$
$48$		0			0
$49$	24.5396			0.500809
$50$		0			0
$51$	1.20437	+	16.7057i	0.0236150	+	0.327562i
$52$		0			0
$53$	−19.2048			−0.362354			−0.181177	−	0.983450i	$-0.557991\pi$
$53$	−19.2048			−0.362354			−0.181177	+	0.983450i	$0.557991\pi$
$54$		0			0
$55$	−35.2693			−0.641261
$56$		0			0
$57$	5.38242	+	74.6592i	0.0944284	+	1.30981i
$58$		0			0
$59$	−11.4120			−0.193423			−0.0967115	−	0.995312i	$-0.530832\pi$
$59$	−11.4120			−0.193423			−0.0967115	+	0.995312i	$0.530832\pi$
$60$		0			0
$61$			101.141i			1.65805i	0.559213	+	0.829024i	$0.311103\pi$
$61$			101.141i			1.65805i	−0.559213	+	0.829024i	$0.688897\pi$
$62$		0			0
$63$	−76.3816	+	11.0707i	−1.21241	+	0.175726i
$64$		0			0
$65$		−	40.5137i		−	0.623288i
$66$		0			0
$67$		−	63.1963i		−	0.943228i	−0.881805	−	0.471614i	$-0.843672\pi$
$67$		−	63.1963i		−	0.943228i	0.881805	−	0.471614i	$-0.156328\pi$
$68$		0			0
$69$	8.80088	+	122.076i	0.127549	+	1.76922i
$70$		0			0
$71$			21.0444i			0.296401i	0.988957	+	0.148200i	$0.0473480\pi$
$71$			21.0444i			0.296401i	−0.988957	+	0.148200i	$0.952652\pi$
$72$		0			0
$73$	−61.5698			−0.843422			−0.421711	−	0.906730i	$-0.638570\pi$
$73$	−61.5698			−0.843422			−0.421711	+	0.906730i	$0.638570\pi$
$74$		0			0
$75$	−14.9612	+	1.07860i	−0.199482	+	0.0143813i
$76$		0			0
$77$	−135.261			−1.75664
$78$		0			0
$79$	−114.761			−1.45267			−0.726337	−	0.687339i	$-0.758779\pi$
$79$	−114.761			−1.45267			−0.726337	+	0.687339i	$0.758779\pi$
$80$		0			0
$81$	77.6668	−	22.9971i	0.958849	−	0.283915i
$82$		0			0
$83$	−100.898			−1.21563			−0.607817	−	0.794077i	$-0.707955\pi$
$83$	−100.898			−1.21563			−0.607817	+	0.794077i	$0.707955\pi$
$84$		0			0
$85$			12.4840i			0.146871i
$86$		0			0
$87$	−113.416	+	8.17654i	−1.30364	+	0.0939833i
$88$		0			0
$89$			11.6579i			0.130988i	0.997853	+	0.0654938i	$0.0208623\pi$
$89$			11.6579i			0.130988i	−0.997853	+	0.0654938i	$0.979138\pi$
$90$		0			0
$91$		−	155.374i		−	1.70740i
$92$		0			0
$93$	91.5692	−	6.60152i	0.984615	−	0.0709841i
$94$		0			0
$95$			55.7921i			0.587285i
$96$		0			0
$97$	46.4906			0.479285			0.239642	−	0.970861i	$-0.422970\pi$
$97$	46.4906			0.479285			0.239642	+	0.970861i	$0.422970\pi$
$98$		0			0
$99$	140.488	−	20.3623i	1.41907	−	0.205680i
$100$		0			0
$101$	−103.146			−1.02125			−0.510623	−	0.859805i	$-0.670585\pi$
$101$	−103.146			−1.02125			−0.510623	+	0.859805i	$0.670585\pi$
$102$		0			0
$103$	−8.99283			−0.0873090			−0.0436545	−	0.999047i	$-0.513900\pi$
$103$	−8.99283			−0.0873090			−0.0436545	+	0.999047i	$0.513900\pi$
$104$		0			0
$105$	−57.3775	+	4.13653i	−0.546452	+	0.0393955i
$106$		0			0
$107$	81.0097			0.757100			0.378550	−	0.925581i	$-0.376423\pi$
$107$	81.0097			0.757100			0.378550	+	0.925581i	$0.376423\pi$
$108$		0			0
$109$		−	41.8708i		−	0.384136i	−0.981382	−	0.192068i	$-0.938481\pi$
$109$		−	41.8708i		−	0.384136i	0.981382	−	0.192068i	$-0.0615194\pi$
$110$		0			0
$111$	−6.01204	−	83.3926i	−0.0541626	−	0.751285i
$112$		0			0
$113$			134.246i			1.18802i	0.804459	+	0.594008i	$0.202455\pi$
$113$			134.246i			1.18802i	−0.804459	+	0.594008i	$0.797545\pi$
$114$		0			0
$115$			91.2265i			0.793274i
$116$		0			0
$117$	23.3901	+	161.378i	0.199916	+	1.37930i
$118$		0			0
$119$			47.8773i			0.402330i
$120$		0			0
$121$	127.785			1.05608
$122$		0			0
$123$	2.72541	+	37.8039i	0.0221578	+	0.307349i
$124$		0			0
$125$	−11.1803			−0.0894427
$126$		0			0
$127$	12.1062			0.0953247			0.0476624	−	0.998864i	$-0.484823\pi$
$127$	12.1062			0.0953247			0.0476624	+	0.998864i	$0.484823\pi$
$128$		0			0
$129$	−16.6316	−	230.696i	−0.128927	−	1.78834i
$130$		0			0
$131$	−97.3629			−0.743229			−0.371614	−	0.928387i	$-0.621196\pi$
$131$	−97.3629			−0.743229			−0.371614	+	0.928387i	$0.621196\pi$
$132$		0			0
$133$			213.968i			1.60878i
$134$		0			0
$135$	58.9721	−	12.9340i	0.436830	−	0.0958077i
$136$		0			0
$137$		−	132.294i		−	0.965647i	−0.875718	−	0.482824i	$-0.839611\pi$
$137$		−	132.294i		−	0.965647i	0.875718	−	0.482824i	$-0.160389\pi$
$138$		0			0
$139$			176.287i			1.26825i	0.773231	+	0.634125i	$0.218640\pi$
$139$			176.287i			1.26825i	−0.773231	+	0.634125i	$0.781360\pi$
$140$		0			0
$141$	−13.3406	−	185.046i	−0.0946141	−	1.31238i
$142$		0			0
$143$			285.778i			1.99845i
$144$		0			0
$145$	−84.7549			−0.584517
$146$		0			0
$147$	−73.4283	+	5.29368i	−0.499512	+	0.0360114i
$148$		0			0
$149$	−208.131			−1.39686			−0.698428	−	0.715680i	$-0.746117\pi$
$149$	−208.131			−1.39686			−0.698428	+	0.715680i	$0.746117\pi$
$150$		0			0
$151$	−211.447			−1.40031			−0.700155	−	0.713991i	$-0.746886\pi$
$151$	−211.447			−1.40031			−0.700155	+	0.713991i	$0.746886\pi$
$152$		0			0
$153$	−7.20749	−	49.7275i	−0.0471078	−	0.325016i
$154$		0			0
$155$	68.4288			0.441476
$156$		0			0
$157$		−	155.744i		−	0.992000i	−0.868323	−	0.496000i	$-0.834802\pi$
$157$		−	155.744i		−	0.992000i	0.868323	−	0.496000i	$-0.165198\pi$
$158$		0			0
$159$	57.4652	−	4.14285i	0.361416	−	0.0260557i
$160$		0			0
$161$			349.862i			2.17306i
$162$		0			0
$163$			99.2343i			0.608799i	0.952544	+	0.304400i	$0.0984558\pi$
$163$			99.2343i			0.608799i	−0.952544	+	0.304400i	$0.901544\pi$
$164$		0			0
$165$	105.534	−	7.60829i	0.639601	−	0.0461109i
$166$		0			0
$167$		−	36.1064i		−	0.216206i	−0.994140	−	0.108103i	$-0.965522\pi$
$167$		−	36.1064i		−	0.216206i	0.994140	−	0.108103i	$-0.0344776\pi$
$168$		0			0
$169$	−159.272			−0.942438
$170$		0			0
$171$	−32.2109	−	222.237i	−0.188368	−	1.29963i
$172$		0			0
$173$	−109.171			−0.631046			−0.315523	−	0.948918i	$-0.602180\pi$
$173$	−109.171			−0.631046			−0.315523	+	0.948918i	$0.602180\pi$
$174$		0			0
$175$	−42.8776			−0.245015
$176$		0			0
$177$	34.1473	−	2.46179i	0.192922	−	0.0139084i
$178$		0			0
$179$	197.188			1.10161			0.550806	−	0.834633i	$-0.314321\pi$
$179$	197.188			1.10161			0.550806	+	0.834633i	$0.314321\pi$
$180$		0			0
$181$			270.886i			1.49661i	0.663356	+	0.748304i	$0.269132\pi$
$181$			270.886i			1.49661i	−0.663356	+	0.748304i	$0.730868\pi$
$182$		0			0
$183$	−21.8181	−	302.637i	−0.119225	−	1.65376i
$184$		0			0
$185$		−	62.3185i		−	0.336857i
$186$		0			0
$187$		−	88.0604i		−	0.470911i
$188$		0			0
$189$	226.163	−	49.6032i	1.19663	−	0.262451i
$190$		0			0
$191$			64.1454i			0.335840i	0.985801	+	0.167920i	$0.0537050\pi$
$191$			64.1454i			0.335840i	−0.985801	+	0.167920i	$0.946295\pi$
$192$		0			0
$193$	−262.808			−1.36170			−0.680849	−	0.732424i	$-0.738389\pi$
$193$	−262.808			−1.36170			−0.680849	+	0.732424i	$0.738389\pi$
$194$		0			0
$195$	8.73961	+	121.226i	0.0448185	+	0.621674i
$196$		0			0
$197$	241.935			1.22809			0.614047	−	0.789269i	$-0.289540\pi$
$197$	241.935			1.22809			0.614047	+	0.789269i	$0.289540\pi$
$198$		0			0
$199$	105.427			0.529785			0.264892	−	0.964278i	$-0.414664\pi$
$199$	105.427			0.529785			0.264892	+	0.964278i	$0.414664\pi$
$200$		0			0
$201$	13.6327	+	189.098i	0.0678243	+	0.940786i
$202$		0			0
$203$	−325.043			−1.60120
$204$		0			0
$205$			28.2505i			0.137807i
$206$		0			0
$207$	−52.6686	−	363.383i	−0.254438	−	1.75547i
$208$		0			0
$209$		−	393.550i		−	1.88301i
$210$		0			0
$211$			2.10583i			0.00998025i	0.999988	+	0.00499012i	$0.00158841\pi$
$211$			2.10583i			0.00998025i	−0.999988	+	0.00499012i	$0.998412\pi$
$212$		0			0
$213$	−4.53970	−	62.9699i	−0.0213132	−	0.295633i
$214$		0			0
$215$		−	172.397i		−	0.801846i
$216$		0			0
$217$	262.431			1.20936
$218$		0			0
$219$	184.231	−	13.2818i	0.841239	−	0.0606476i
$220$		0			0
$221$	101.155			0.457713
$222$		0			0
$223$	74.4836			0.334007			0.167004	−	0.985956i	$-0.446591\pi$
$223$	74.4836			0.334007			0.167004	+	0.985956i	$0.446591\pi$
$224$		0			0
$225$	44.5346	−	6.45484i	0.197932	−	0.0286882i
$226$		0			0
$227$	88.4128			0.389484			0.194742	−	0.980855i	$-0.437613\pi$
$227$	88.4128			0.389484			0.194742	+	0.980855i	$0.437613\pi$
$228$		0			0
$229$		−	77.6001i		−	0.338865i	−0.985542	−	0.169433i	$-0.945807\pi$
$229$		−	77.6001i		−	0.338865i	0.985542	−	0.169433i	$-0.0541935\pi$
$230$		0			0
$231$	404.733	−	29.1785i	1.75209	−	0.126314i
$232$		0			0
$233$			252.078i			1.08188i	0.841061	+	0.540940i	$0.181931\pi$
$233$			252.078i			1.08188i	−0.841061	+	0.540940i	$0.818069\pi$
$234$		0			0
$235$		−	138.283i		−	0.588440i
$236$		0			0
$237$	343.393	−	24.7563i	1.44891	−	0.104457i
$238$		0			0
$239$			192.521i			0.805527i	0.915304	+	0.402764i	$0.131950\pi$
$239$			192.521i			0.805527i	−0.915304	+	0.402764i	$0.868050\pi$
$240$		0			0
$241$	111.616			0.463135			0.231568	−	0.972819i	$-0.425615\pi$
$241$	111.616			0.463135			0.231568	+	0.972819i	$0.425615\pi$
$242$		0			0
$243$	−227.436	+	85.5671i	−0.935952	+	0.352128i
$244$		0			0
$245$	−54.8723			−0.223968
$246$		0			0
$247$	452.069			1.83024
$248$		0			0
$249$	301.909	−	21.7656i	1.21249	−	0.0874122i
$250$		0			0
$251$	−235.610			−0.938686			−0.469343	−	0.883016i	$-0.655509\pi$
$251$	−235.610			−0.938686			−0.469343	+	0.883016i	$0.655509\pi$
$252$		0			0
$253$		−	643.499i		−	2.54348i
$254$		0			0
$255$	−2.69305	−	37.3550i	−0.0105610	−	0.146490i
$256$		0			0
$257$			118.192i			0.459890i	0.973204	+	0.229945i	$0.0738546\pi$
$257$			118.192i			0.459890i	−0.973204	+	0.229945i	$0.926145\pi$
$258$		0			0
$259$		−	238.997i		−	0.922769i
$260$		0			0
$261$	337.604	−	48.9323i	1.29350	−	0.187480i
$262$		0			0
$263$		−	72.4069i		−	0.275311i	−0.990480	−	0.137656i	$-0.956043\pi$
$263$		−	72.4069i		−	0.275311i	0.990480	−	0.137656i	$-0.0439567\pi$
$264$		0			0
$265$	42.9432			0.162050
$266$		0			0
$267$	−2.51484	−	34.8832i	−0.00941887	−	0.130649i
$268$		0			0
$269$	−378.681			−1.40774			−0.703868	−	0.710331i	$-0.748545\pi$
$269$	−378.681			−1.40774			−0.703868	+	0.710331i	$0.748545\pi$
$270$		0			0
$271$	223.054			0.823076			0.411538	−	0.911393i	$-0.364992\pi$
$271$	223.054			0.823076			0.411538	+	0.911393i	$0.364992\pi$
$272$		0			0
$273$	33.5172	+	464.915i	0.122774	+	1.70298i
$274$		0			0
$275$	78.8646			0.286780
$276$		0			0
$277$		−	71.5059i		−	0.258144i	−0.991635	−	0.129072i	$-0.958800\pi$
$277$		−	71.5059i		−	0.258144i	0.991635	−	0.129072i	$-0.0411998\pi$
$278$		0			0
$279$	−272.572	+	39.5066i	−0.976962	+	0.141601i
$280$		0			0
$281$		−	388.811i		−	1.38367i	−0.722056	−	0.691834i	$-0.756803\pi$
$281$		−	388.811i		−	1.38367i	0.722056	−	0.691834i	$-0.243197\pi$
$282$		0			0
$283$			146.291i			0.516930i	0.966021	+	0.258465i	$0.0832166\pi$
$283$			146.291i			0.516930i	−0.966021	+	0.258465i	$0.916783\pi$
$284$		0			0
$285$	−12.0355	−	166.943i	−0.0422297	−	0.585765i
$286$		0			0
$287$			108.343i			0.377503i
$288$		0			0
$289$	257.830			0.892145
$290$		0			0
$291$	−139.111	+	10.0289i	−0.478044	+	0.0344637i
$292$		0			0
$293$	−228.079			−0.778426			−0.389213	−	0.921148i	$-0.627253\pi$
$293$	−228.079			−0.778426			−0.389213	+	0.921148i	$0.627253\pi$
$294$		0			0
$295$	25.5179			0.0865014
$296$		0			0
$297$	−415.981	+	91.2350i	−1.40061	+	0.307189i
$298$		0			0
$299$	739.185			2.47219
$300$		0			0
$301$		−	661.158i		−	2.19654i
$302$		0			0
$303$	308.637	−	22.2506i	1.01860	−	0.0734344i
$304$		0			0
$305$		−	226.158i		−	0.741502i
$306$		0			0
$307$			48.7665i			0.158849i	0.996841	+	0.0794243i	$0.0253082\pi$
$307$			48.7665i			0.158849i	−0.996841	+	0.0794243i	$0.974692\pi$
$308$		0			0
$309$	26.9086	−	1.93993i	0.0870830	−	0.00627809i
$310$		0			0
$311$			568.974i			1.82950i	0.404021	+	0.914750i	$0.367612\pi$
$311$			568.974i			1.82950i	−0.404021	+	0.914750i	$0.632388\pi$
$312$		0			0
$313$	−300.671			−0.960612			−0.480306	−	0.877101i	$-0.659474\pi$
$313$	−300.671			−0.960612			−0.480306	+	0.877101i	$0.659474\pi$
$314$		0			0
$315$	170.794	−	24.7549i	0.542205	−	0.0785870i
$316$		0			0
$317$	−243.420			−0.767886			−0.383943	−	0.923357i	$-0.625434\pi$
$317$	−243.420			−0.767886			−0.383943	+	0.923357i	$0.625434\pi$
$318$		0			0
$319$	597.850			1.87414
$320$		0			0
$321$	−242.400	+	17.4754i	−0.755140	+	0.0544405i
$322$		0			0
$323$	−139.302			−0.431274
$324$		0			0
$325$			90.5914i			0.278743i
$326$		0			0
$327$	9.03237	+	125.287i	0.0276219	+	0.383142i
$328$		0			0
$329$		−	530.329i		−	1.61194i
$330$		0			0
$331$			348.597i			1.05316i	0.850125	+	0.526581i	$0.176526\pi$
$331$			348.597i			1.05316i	−0.850125	+	0.526581i	$0.823474\pi$
$332$		0			0
$333$	35.9789	+	248.233i	0.108045	+	0.745446i
$334$		0			0
$335$			141.311i			0.421824i
$336$		0			0
$337$	−138.136			−0.409900			−0.204950	−	0.978772i	$-0.565703\pi$
$337$	−138.136			−0.409900			−0.204950	+	0.978772i	$0.565703\pi$
$338$		0			0
$339$	−28.9595	−	401.695i	−0.0854262	−	1.18494i
$340$		0			0
$341$	−482.687			−1.41551
$342$		0			0
$343$	209.761			0.611547
$344$		0			0
$345$	−19.6794	−	272.971i	−0.0570416	−	0.791220i
$346$		0			0
$347$	−413.612			−1.19197			−0.595983	−	0.802997i	$-0.703237\pi$
$347$	−413.612			−1.19197			−0.595983	+	0.802997i	$0.703237\pi$
$348$		0			0
$349$		−	209.754i		−	0.601016i	−0.953779	−	0.300508i	$-0.902844\pi$
$349$		−	209.754i		−	0.601016i	0.953779	−	0.300508i	$-0.0971562\pi$
$350$		0			0
$351$	−104.801	−	477.836i	−0.298579	−	1.36136i
$352$		0			0
$353$			201.928i			0.572033i	0.958225	+	0.286016i	$0.0923312\pi$
$353$			201.928i			0.572033i	−0.958225	+	0.286016i	$0.907669\pi$
$354$		0			0
$355$		−	47.0568i		−	0.132554i
$356$		0			0
$357$	−10.3281	−	143.260i	−0.0289302	−	0.401288i
$358$		0			0
$359$			257.460i			0.717159i	0.933499	+	0.358580i	$0.116739\pi$
$359$			257.460i			0.717159i	−0.933499	+	0.358580i	$0.883261\pi$
$360$		0			0
$361$	−261.551			−0.724519
$362$		0			0
$363$	−382.363	+	27.5658i	−1.05334	+	0.0759388i
$364$		0			0
$365$	137.674			0.377190
$366$		0			0
$367$	−206.492			−0.562649			−0.281324	−	0.959613i	$-0.590774\pi$
$367$	−206.492			−0.562649			−0.281324	+	0.959613i	$0.590774\pi$
$368$		0			0
$369$	−16.3101	−	112.530i	−0.0442008	−	0.304960i
$370$		0			0
$371$	164.691			0.443911
$372$		0			0
$373$			151.537i			0.406265i	0.979151	+	0.203132i	$0.0651122\pi$
$373$			151.537i			0.406265i	−0.979151	+	0.203132i	$0.934888\pi$
$374$		0			0
$375$	33.4542	−	2.41182i	0.0892112	−	0.00643152i
$376$		0			0
$377$			686.747i			1.82161i
$378$		0			0
$379$			562.823i			1.48502i	0.669835	+	0.742510i	$0.266365\pi$
$379$			562.823i			1.48502i	−0.669835	+	0.742510i	$0.733635\pi$
$380$		0			0
$381$	−36.2247	+	2.61156i	−0.0950779	+	0.00685448i
$382$		0			0
$383$			270.631i			0.706609i	0.935508	+	0.353305i	$0.114942\pi$
$383$			270.631i			0.706609i	−0.935508	+	0.353305i	$0.885058\pi$
$384$		0			0
$385$	302.453			0.785592
$386$		0			0
$387$	99.5314	+	686.709i	0.257187	+	1.77444i
$388$		0			0
$389$	453.003			1.16453			0.582266	−	0.812998i	$-0.302166\pi$
$389$	453.003			1.16453			0.582266	+	0.812998i	$0.302166\pi$
$390$		0			0
$391$	−227.774			−0.582543
$392$		0			0
$393$	291.333	−	21.0031i	0.741305	−	0.0534430i
$394$		0			0
$395$	256.614			0.649656
$396$		0			0
$397$		−	529.795i		−	1.33450i	−0.744835	−	0.667249i	$-0.767472\pi$
$397$		−	529.795i		−	1.33450i	0.744835	−	0.667249i	$-0.232528\pi$
$398$		0			0
$399$	−46.1571	−	640.242i	−0.115682	−	1.60462i
$400$		0			0
$401$			507.162i			1.26474i	0.774665	+	0.632372i	$0.217919\pi$
$401$			507.162i			1.26474i	−0.774665	+	0.632372i	$0.782081\pi$
$402$		0			0
$403$		−	554.461i		−	1.37583i
$404$		0			0
$405$	−173.668	+	51.4231i	−0.428810	+	0.126971i
$406$		0			0
$407$			439.586i			1.08007i
$408$		0			0
$409$	−203.299			−0.497063			−0.248531	−	0.968624i	$-0.579948\pi$
$409$	−203.299			−0.497063			−0.248531	+	0.968624i	$0.579948\pi$
$410$		0			0
$411$	28.5384	+	395.854i	0.0694364	+	0.963147i
$412$		0			0
$413$	97.8636			0.236958
$414$		0			0
$415$	225.614			0.543648
$416$		0			0
$417$	−38.0285	−	527.491i	−0.0911955	−	1.26497i
$418$		0			0
$419$	621.651			1.48365			0.741827	−	0.670592i	$-0.233960\pi$
$419$	621.651			1.48365			0.741827	+	0.670592i	$0.233960\pi$
$420$		0			0
$421$			326.795i			0.776234i	0.921610	+	0.388117i	$0.126874\pi$
$421$			326.795i			0.776234i	−0.921610	+	0.388117i	$0.873126\pi$
$422$		0			0
$423$	79.8363	+	550.824i	0.188738	+	1.30218i
$424$		0			0
$425$		−	27.9151i		−	0.0656825i
$426$		0			0
$427$		−	867.337i		−	2.03123i
$428$		0			0
$429$	−61.6480	−	855.115i	−0.143702	−	1.99328i
$430$		0			0
$431$		−	403.286i		−	0.935698i	−0.883808	−	0.467849i	$-0.845029\pi$
$431$		−	403.286i		−	0.935698i	0.883808	−	0.467849i	$-0.154971\pi$
$432$		0			0
$433$	718.371			1.65906			0.829528	−	0.558465i	$-0.188609\pi$
$433$	718.371			1.65906			0.829528	+	0.558465i	$0.188609\pi$
$434$		0			0
$435$	253.607	−	18.2833i	0.583003	−	0.0420306i
$436$		0			0
$437$	−1017.94			−2.32939
$438$		0			0
$439$	210.195			0.478804			0.239402	−	0.970921i	$-0.423049\pi$
$439$	210.195			0.478804			0.239402	+	0.970921i	$0.423049\pi$
$440$		0			0
$441$	218.573	−	31.6799i	0.495630	−	0.0718364i
$442$		0			0
$443$	−581.438			−1.31250			−0.656250	−	0.754543i	$-0.727858\pi$
$443$	−581.438			−1.31250			−0.656250	+	0.754543i	$0.727858\pi$
$444$		0			0
$445$		−	26.0679i		−	0.0585794i
$446$		0			0
$447$	622.778	−	44.8981i	1.39324	−	0.100443i
$448$		0			0
$449$		−	849.129i		−	1.89116i	−0.325394	−	0.945578i	$-0.605497\pi$
$449$		−	849.129i		−	1.89116i	0.325394	−	0.945578i	$-0.394503\pi$
$450$		0			0
$451$		−	199.275i		−	0.441852i
$452$		0			0
$453$	632.699	−	45.6133i	1.39669	−	0.100692i
$454$		0			0
$455$			347.426i			0.763574i
$456$		0			0
$457$	395.661			0.865779			0.432890	−	0.901447i	$-0.357494\pi$
$457$	395.661			0.865779			0.432890	+	0.901447i	$0.357494\pi$
$458$		0			0
$459$	32.2937	+	147.242i	0.0703567	+	0.320788i
$460$		0			0
$461$	−216.351			−0.469307			−0.234654	−	0.972079i	$-0.575396\pi$
$461$	−216.351			−0.469307			−0.234654	+	0.972079i	$0.575396\pi$
$462$		0			0
$463$	−511.415			−1.10457			−0.552284	−	0.833656i	$-0.686244\pi$
$463$	−511.415			−1.10457			−0.552284	+	0.833656i	$0.686244\pi$
$464$		0			0
$465$	−204.755	+	14.7614i	−0.440333	+	0.0317450i
$466$		0			0
$467$	−865.595			−1.85352			−0.926761	−	0.375652i	$-0.877419\pi$
$467$	−865.595			−1.85352			−0.926761	+	0.375652i	$0.877419\pi$
$468$		0			0
$469$			541.941i			1.15553i
$470$		0			0
$471$	33.5971	+	466.022i	0.0713313	+	0.989432i
$472$		0			0
$473$			1216.06i			2.57096i
$474$		0			0
$475$		−	124.755i		−	0.262642i
$476$		0			0
$477$	−171.056	+	24.7928i	−0.358607	+	0.0519764i
$478$		0			0
$479$		−	59.5390i		−	0.124299i	−0.998067	−	0.0621493i	$-0.980205\pi$
$479$		−	59.5390i		−	0.124299i	0.998067	−	0.0621493i	$-0.0197955\pi$
$480$		0			0
$481$	−504.951			−1.04979
$482$		0			0
$483$	−75.4721	−	1046.87i	−0.156257	−	2.16743i
$484$		0			0
$485$	−103.956			−0.214343
$486$		0			0
$487$	−326.211			−0.669838			−0.334919	−	0.942247i	$-0.608709\pi$
$487$	−326.211			−0.669838			−0.334919	+	0.942247i	$0.608709\pi$
$488$		0			0
$489$	−21.4068	−	296.932i	−0.0437767	−	0.607223i
$490$		0			0
$491$	−307.774			−0.626831			−0.313416	−	0.949616i	$-0.601473\pi$
$491$	−307.774			−0.626831			−0.313416	+	0.949616i	$0.601473\pi$
$492$		0			0
$493$		−	211.616i		−	0.429241i
$494$		0			0
$495$	−314.141	+	45.5316i	−0.634629	+	0.0919830i
$496$		0			0
$497$		−	180.467i		−	0.363113i
$498$		0			0
$499$			423.290i			0.848276i	0.905597	+	0.424138i	$0.139423\pi$
$499$			423.290i			0.848276i	−0.905597	+	0.424138i	$0.860577\pi$
$500$		0			0
$501$	7.78886	+	108.039i	0.0155466	+	0.215646i
$502$		0			0
$503$		−	460.655i		−	0.915815i	−0.889000	−	0.457908i	$-0.848599\pi$
$503$		−	460.655i		−	0.915815i	0.889000	−	0.457908i	$-0.151401\pi$
$504$		0			0
$505$	230.641			0.456715
$506$		0			0
$507$	476.579	−	34.3581i	0.939999	−	0.0677675i
$508$		0			0
$509$	314.299			0.617483			0.308742	−	0.951146i	$-0.400092\pi$
$509$	314.299			0.617483			0.308742	+	0.951146i	$0.400092\pi$
$510$		0			0
$511$	527.994			1.03326
$512$		0			0
$513$	144.323	+	658.036i	0.281332	+	1.28272i
$514$		0			0
$515$	20.1086			0.0390458
$516$		0			0
$517$			975.432i			1.88672i
$518$		0			0
$519$	326.665	−	23.5503i	0.629412	−	0.0453764i
$520$		0			0
$521$			155.654i			0.298760i	0.988780	+	0.149380i	$0.0477278\pi$
$521$			155.654i			0.298760i	−0.988780	+	0.149380i	$0.952272\pi$
$522$		0			0
$523$		−	333.460i		−	0.637591i	−0.947824	−	0.318795i	$-0.896722\pi$
$523$		−	333.460i		−	0.637591i	0.947824	−	0.318795i	$-0.103278\pi$
$524$		0			0
$525$	128.300	−	9.24955i	0.244381	−	0.0176182i
$526$		0			0
$527$			170.853i			0.324199i
$528$		0			0
$529$	−1135.45			−2.14642
$530$		0			0
$531$	−101.646	+	14.7325i	−0.191423	+	0.0277448i
$532$		0			0
$533$	228.907			0.429468
$534$		0			0
$535$	−181.143			−0.338586
$536$		0			0
$537$	−590.034	+	42.5375i	−1.09876	+	0.0792131i
$538$		0			0
$539$	387.062			0.718110
$540$		0			0
$541$		−	163.096i		−	0.301472i	−0.988574	−	0.150736i	$-0.951836\pi$
$541$		−	163.096i		−	0.301472i	0.988574	−	0.150736i	$-0.0481643\pi$
$542$		0			0
$543$	−58.4355	−	810.555i	−0.107616	−	1.49273i
$544$		0			0
$545$			93.6261i			0.171791i
$546$		0			0
$547$			912.005i			1.66729i	0.552304	+	0.833643i	$0.313749\pi$
$547$			912.005i			1.66729i	−0.552304	+	0.833643i	$0.686251\pi$
$548$		0			0
$549$	130.570	+	900.855i	0.237832	+	1.64090i
$550$		0			0
$551$		−	945.731i		−	1.71639i
$552$		0			0
$553$	984.138			1.77963
$554$		0			0
$555$	13.4433	+	186.472i	0.0242222	+	0.335985i
$556$		0			0
$557$	933.496			1.67594			0.837968	−	0.545720i	$-0.183744\pi$
$557$	933.496			1.67594			0.837968	+	0.545720i	$0.183744\pi$
$558$		0			0
$559$	−1396.89			−2.49890
$560$		0			0
$561$	18.9964	+	263.497i	0.0338616	+	0.469692i
$562$		0			0
$563$	−462.673			−0.821799			−0.410899	−	0.911681i	$-0.634785\pi$
$563$	−462.673			−0.821799			−0.410899	+	0.911681i	$0.634785\pi$
$564$		0			0
$565$		−	300.183i		−	0.531297i
$566$		0			0
$567$	−666.034	+	197.212i	−1.17466	+	0.347817i
$568$		0			0
$569$		−	637.202i		−	1.11986i	−0.828539	−	0.559932i	$-0.810827\pi$
$569$		−	637.202i		−	1.11986i	0.828539	−	0.559932i	$-0.189173\pi$
$570$		0			0
$571$			551.797i			0.966369i	0.875518	+	0.483185i	$0.160520\pi$
$571$			551.797i			0.966369i	−0.875518	+	0.483185i	$0.839480\pi$
$572$		0			0
$573$	−13.8374	−	191.938i	−0.0241491	−	0.334970i
$574$		0			0
$575$		−	203.989i		−	0.354763i
$576$		0			0
$577$	166.512			0.288583			0.144291	−	0.989535i	$-0.453910\pi$
$577$	166.512			0.288583			0.144291	+	0.989535i	$0.453910\pi$
$578$		0			0
$579$	786.382	−	56.6928i	1.35817	−	0.0979150i
$580$		0			0
$581$	865.251			1.48924
$582$		0			0
$583$	−302.915			−0.519581
$584$		0			0
$585$	−52.3019	−	360.853i	−0.0894050	−	0.616842i
$586$		0			0
$587$	559.999			0.954001			0.477001	−	0.878903i	$-0.341724\pi$
$587$	559.999			0.954001			0.477001	+	0.878903i	$0.341724\pi$
$588$		0			0
$589$			763.557i			1.29636i
$590$		0			0
$591$	−723.925	+	52.1901i	−1.22492	+	0.0883081i
$592$		0			0
$593$		−	777.469i		−	1.31108i	−0.755161	−	0.655539i	$-0.772441\pi$
$593$		−	777.469i		−	1.31108i	0.755161	−	0.655539i	$-0.227559\pi$
$594$		0			0
$595$		−	107.057i		−	0.179927i
$596$		0			0
$597$	−315.463	+	22.7427i	−0.528413	+	0.0380950i
$598$		0			0
$599$		−	966.387i		−	1.61333i	−0.591007	−	0.806667i	$-0.701269\pi$
$599$		−	966.387i		−	1.61333i	0.591007	−	0.806667i	$-0.298731\pi$
$600$		0			0
$601$	−676.869			−1.12624			−0.563119	−	0.826376i	$-0.690399\pi$
$601$	−676.869			−1.12624			−0.563119	+	0.826376i	$0.690399\pi$
$602$		0			0
$603$	−81.5844	−	562.885i	−0.135297	−	0.933474i
$604$		0			0
$605$	−285.736			−0.472291
$606$		0			0
$607$	−193.447			−0.318694			−0.159347	−	0.987223i	$-0.550939\pi$
$607$	−193.447			−0.318694			−0.159347	+	0.987223i	$0.550939\pi$
$608$		0			0
$609$	972.604	−	70.1182i	1.59705	−	0.115137i
$610$		0			0
$611$	−1120.47			−1.83384
$612$		0			0
$613$			777.986i			1.26915i	0.772863	+	0.634573i	$0.218824\pi$
$613$			777.986i			1.26915i	−0.772863	+	0.634573i	$0.781176\pi$
$614$		0			0
$615$	−6.09419	−	84.5321i	−0.00990926	−	0.137451i
$616$		0			0
$617$			96.3008i			0.156079i	0.996950	+	0.0780396i	$0.0248661\pi$
$617$			96.3008i			0.156079i	−0.996950	+	0.0780396i	$0.975134\pi$
$618$		0			0
$619$			575.607i			0.929898i	0.885337	+	0.464949i	$0.153927\pi$
$619$			575.607i			0.929898i	−0.885337	+	0.464949i	$0.846073\pi$
$620$		0			0
$621$	235.986	+	1075.96i	0.380009	+	1.73263i
$622$		0			0
$623$		−	99.9726i		−	0.160470i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	84.8965	+	1177.59i	0.135401	+	1.87814i
$628$		0			0
$629$	155.597			0.247372
$630$		0			0
$631$	141.935			0.224937			0.112468	−	0.993655i	$-0.464124\pi$
$631$	141.935			0.224937			0.112468	+	0.993655i	$0.464124\pi$
$632$		0			0
$633$	−0.454270	−	6.30114i	−0.000717646	−	0.00995441i
$634$		0			0
$635$	−27.0704			−0.0426305
$636$		0			0
$637$			444.616i			0.697984i
$638$		0			0
$639$	27.1677	+	187.441i	0.0425160	+	0.293336i
$640$		0			0
$641$			679.828i			1.06057i	0.847818	+	0.530287i	$0.177916\pi$
$641$			679.828i			1.06057i	−0.847818	+	0.530287i	$0.822084\pi$
$642$		0			0
$643$			475.778i			0.739934i	0.929045	+	0.369967i	$0.120631\pi$
$643$			475.778i			0.739934i	−0.929045	+	0.369967i	$0.879369\pi$
$644$		0			0
$645$	37.1894	+	515.852i	0.0576580	+	0.799770i
$646$		0			0
$647$		−	818.949i		−	1.26576i	−0.774248	−	0.632882i	$-0.781872\pi$
$647$		−	818.949i		−	1.26576i	0.774248	−	0.632882i	$-0.218128\pi$
$648$		0			0
$649$	−180.000			−0.277350
$650$		0			0
$651$	−785.254	+	56.6115i	−1.20623	+	0.0869608i
$652$		0			0
$653$	−511.275			−0.782963			−0.391482	−	0.920186i	$-0.628037\pi$
$653$	−511.275			−0.782963			−0.391482	+	0.920186i	$0.628037\pi$
$654$		0			0
$655$	217.710			0.332382
$656$		0			0
$657$	−548.398	+	79.4847i	−0.834700	+	0.120981i
$658$		0			0
$659$	915.395			1.38907			0.694534	−	0.719460i	$-0.255611\pi$
$659$	915.395			1.38907			0.694534	+	0.719460i	$0.255611\pi$
$660$		0			0
$661$		−	279.058i		−	0.422176i	−0.977467	−	0.211088i	$-0.932299\pi$
$661$		−	279.058i		−	0.422176i	0.977467	−	0.211088i	$-0.0677007\pi$
$662$		0			0
$663$	−302.678	+	21.8210i	−0.456528	+	0.0329126i
$664$		0			0
$665$		−	478.446i		−	0.719468i
$666$		0			0
$667$		−	1546.38i		−	2.31841i
$668$		0			0
$669$	−222.872	+	16.0676i	−0.333143	+	0.0240173i
$670$		0			0
$671$			1595.29i			2.37748i
$672$		0			0
$673$	−440.649			−0.654753			−0.327377	−	0.944894i	$-0.606165\pi$
$673$	−440.649			−0.654753			−0.327377	+	0.944894i	$0.606165\pi$
$674$		0			0
$675$	−131.866	+	28.9214i	−0.195357	+	0.0428465i
$676$		0			0
$677$	629.644			0.930050			0.465025	−	0.885297i	$-0.346045\pi$
$677$	629.644			0.930050			0.465025	+	0.885297i	$0.346045\pi$
$678$		0			0
$679$	−398.681			−0.587159
$680$		0			0
$681$	−264.552	+	19.0724i	−0.388475	+	0.0280064i
$682$		0			0
$683$	544.192			0.796767			0.398384	−	0.917219i	$-0.369571\pi$
$683$	544.192			0.796767			0.398384	+	0.917219i	$0.369571\pi$
$684$		0			0
$685$			295.818i			0.431850i
$686$		0			0
$687$	16.7399	+	232.198i	0.0243666	+	0.337988i
$688$		0			0
$689$		−	347.958i		−	0.505018i
$690$		0			0
$691$		−	44.5813i		−	0.0645170i	−0.999480	−	0.0322585i	$-0.989730\pi$
$691$		−	44.5813i		−	0.0645170i	0.999480	−	0.0322585i	$-0.0102700\pi$
$692$		0			0
$693$	−1204.76	+	174.618i	−1.73847	+	0.251974i
$694$		0			0
$695$		−	394.189i		−	0.567179i
$696$		0			0
$697$	−70.5359			−0.101199
$698$		0			0
$699$	−54.3782	−	754.276i	−0.0777942	−	1.07908i
$700$		0			0
$701$	269.459			0.384392			0.192196	−	0.981357i	$-0.438439\pi$
$701$	269.459			0.384392			0.192196	+	0.981357i	$0.438439\pi$
$702$		0			0
$703$	695.376			0.989155
$704$		0			0
$705$	29.8304	+	413.776i	0.0423127	+	0.586916i
$706$		0			0
$707$	884.530			1.25110
$708$		0			0
$709$		−	1177.65i		−	1.66100i	−0.557021	−	0.830499i	$-0.688056\pi$
$709$		−	1177.65i		−	1.66100i	0.557021	−	0.830499i	$-0.311944\pi$
$710$		0			0
$711$	−1022.17	+	148.153i	−1.43765	+	0.208373i
$712$		0			0
$713$			1248.50i			1.75106i
$714$		0			0
$715$		−	639.020i		−	0.893734i
$716$		0			0
$717$	−41.5306	−	576.068i	−0.0579227	−	0.803442i
$718$		0			0
$719$			597.547i			0.831081i	0.909575	+	0.415541i	$0.136408\pi$
$719$			597.547i			0.831081i	−0.909575	+	0.415541i	$0.863592\pi$
$720$		0			0
$721$	77.1182			0.106960
$722$		0			0
$723$	−333.980	+	24.0777i	−0.461937	+	0.0333025i
$724$		0			0
$725$	189.518			0.261404
$726$		0			0
$727$	−657.613			−0.904557			−0.452279	−	0.891877i	$-0.649389\pi$
$727$	−657.613			−0.904557			−0.452279	+	0.891877i	$0.649389\pi$
$728$		0			0
$729$	662.084	−	305.099i	0.908209	−	0.418517i
$730$		0			0
$731$	430.440			0.588838
$732$		0			0
$733$			984.554i			1.34318i	0.740921	+	0.671592i	$0.234389\pi$
$733$			984.554i			1.34318i	−0.740921	+	0.671592i	$0.765611\pi$
$734$		0			0
$735$	164.191	−	11.8370i	0.223389	−	0.0161048i
$736$		0			0
$737$		−	996.790i		−	1.35250i
$738$		0			0
$739$		−	158.250i		−	0.214141i	−0.994251	−	0.107070i	$-0.965853\pi$
$739$		−	158.250i		−	0.214141i	0.994251	−	0.107070i	$-0.0341470\pi$
$740$		0			0
$741$	−1352.70	+	97.5202i	−1.82550	+	0.131606i
$742$		0			0
$743$			382.521i			0.514833i	0.966301	+	0.257416i	$0.0828712\pi$
$743$			382.521i			0.514833i	−0.966301	+	0.257416i	$0.917129\pi$
$744$		0			0
$745$	465.396			0.624693
$746$		0			0
$747$	−898.689	+	130.256i	−1.20306	+	0.174372i
$748$		0			0
$749$	−694.701			−0.927505
$750$		0			0
$751$	−867.150			−1.15466			−0.577331	−	0.816511i	$-0.695906\pi$
$751$	−867.150			−1.15466			−0.577331	+	0.816511i	$0.695906\pi$
$752$		0			0
$753$	705.001	−	50.8258i	0.936256	−	0.0674977i
$754$		0			0
$755$	472.810			0.626238
$756$		0			0
$757$			271.673i			0.358881i	0.983769	+	0.179441i	$0.0574288\pi$
$757$			271.673i			0.358881i	−0.983769	+	0.179441i	$0.942571\pi$
$758$		0			0
$759$	138.816	+	1925.50i	0.182893	+	2.53689i
$760$		0			0
$761$		−	955.082i		−	1.25504i	−0.778602	−	0.627518i	$-0.784071\pi$
$761$		−	955.082i		−	1.25504i	0.778602	−	0.627518i	$-0.215929\pi$
$762$		0			0
$763$			359.064i			0.470596i
$764$		0			0
$765$	16.1164	+	111.194i	0.0210672	+	0.145352i
$766$		0			0
$767$		−	206.765i		−	0.269576i
$768$		0			0
$769$	476.597			0.619762			0.309881	−	0.950775i	$-0.399711\pi$
$769$	476.597			0.619762			0.309881	+	0.950775i	$0.399711\pi$
$770$		0			0
$771$	−25.4963	−	353.657i	−0.0330691	−	0.458699i
$772$		0			0
$773$	323.748			0.418821			0.209410	−	0.977828i	$-0.432846\pi$
$773$	323.748			0.418821			0.209410	+	0.977828i	$0.432846\pi$
$774$		0			0
$775$	−153.011			−0.197434
$776$		0			0
$777$	51.5564	+	715.136i	0.0663532	+	0.920380i
$778$		0			0
$779$	−315.231			−0.404661
$780$		0			0
$781$			331.932i			0.425010i
$782$		0			0
$783$	−999.635	+	219.245i	−1.27667	+	0.280006i
$784$		0			0
$785$			348.254i			0.443636i
$786$		0			0
$787$		−	143.837i		−	0.182767i	−0.995816	−	0.0913833i	$-0.970871\pi$
$787$		−	143.837i		−	0.182767i	0.995816	−	0.0913833i	$-0.0291288\pi$
$788$		0			0
$789$	15.6196	+	216.658i	0.0197967	+	0.274599i
$790$		0			0
$791$		−	1151.23i		−	1.45541i
$792$		0			0
$793$	−1832.50			−2.31084
$794$		0			0
$795$	−128.496	+	9.26370i	−0.161630	+	0.0116524i
$796$		0			0
$797$	−140.307			−0.176044			−0.0880218	−	0.996119i	$-0.528055\pi$
$797$	−140.307			−0.176044			−0.0880218	+	0.996119i	$0.528055\pi$
$798$		0			0
$799$	345.266			0.432122
$800$		0			0
$801$	15.0500	+	103.836i	0.0187890	+	0.129633i
$802$		0			0
$803$	−971.136			−1.20939
$804$		0			0
$805$		−	782.315i		−	0.971820i
$806$		0			0
$807$	1133.10	−	81.6890i	1.40409	−	0.101226i
$808$		0			0
$809$			410.994i			0.508027i	0.967201	+	0.254013i	$0.0817508\pi$
$809$			410.994i			0.508027i	−0.967201	+	0.254013i	$0.918249\pi$
$810$		0			0
$811$			796.990i			0.982725i	0.870955	+	0.491362i	$0.163501\pi$
$811$			796.990i			0.982725i	−0.870955	+	0.491362i	$0.836499\pi$
$812$		0			0
$813$	−667.429	+	48.1171i	−0.820946	+	0.0591846i
$814$		0			0
$815$		−	221.895i		−	0.272263i
$816$		0			0
$817$	1923.68			2.35456
$818$		0			0
$819$	−200.583	−	1383.90i	−0.244912	−	1.68975i
$820$		0			0
$821$	372.638			0.453883			0.226941	−	0.973908i	$-0.427127\pi$
$821$	372.638			0.453883			0.226941	+	0.973908i	$0.427127\pi$
$822$		0			0
$823$	−300.567			−0.365209			−0.182604	−	0.983186i	$-0.558453\pi$
$823$	−300.567			−0.365209			−0.182604	+	0.983186i	$0.558453\pi$
$824$		0			0
$825$	−235.981	+	17.0127i	−0.286038	+	0.0206214i
$826$		0			0
$827$	−179.865			−0.217491			−0.108746	−	0.994070i	$-0.534683\pi$
$827$	−179.865			−0.217491			−0.108746	+	0.994070i	$0.534683\pi$
$828$		0			0
$829$			811.633i			0.979050i	0.871989	+	0.489525i	$0.162830\pi$
$829$			811.633i			0.979050i	−0.871989	+	0.489525i	$0.837170\pi$
$830$		0			0
$831$	15.4252	+	213.962i	0.0185623	+	0.257476i
$832$		0			0
$833$		−	137.005i		−	0.164472i
$834$		0			0
$835$			80.7363i			0.0966902i
$836$		0			0
$837$	807.078	−	177.012i	0.964251	−	0.211484i
$838$		0			0
$839$		−	699.139i		−	0.833300i	−0.909067	−	0.416650i	$-0.863204\pi$
$839$		−	699.139i		−	0.833300i	0.909067	−	0.416650i	$-0.136796\pi$
$840$		0			0
$841$	595.679			0.708298
$842$		0			0
$843$	83.8742	+	1163.41i	0.0994949	+	1.38009i
$844$		0			0
$845$	356.143			0.421471
$846$		0			0
$847$	−1095.82			−1.29377
$848$		0			0
$849$	−31.5579	−	437.737i	−0.0371707	−	0.515592i
$850$		0			0
$851$	1137.02			1.33610
$852$		0			0
$853$			69.7357i			0.0817535i	0.999164	+	0.0408767i	$0.0130151\pi$
$853$			69.7357i			0.0817535i	−0.999164	+	0.0408767i	$0.986985\pi$
$854$		0			0
$855$	72.0258	+	496.936i	0.0842407	+	0.581212i
$856$		0			0
$857$			246.998i			0.288212i	0.989562	+	0.144106i	$0.0460306\pi$
$857$			246.998i			0.288212i	−0.989562	+	0.144106i	$0.953969\pi$
$858$		0			0
$859$			872.243i			1.01542i	0.861529	+	0.507708i	$0.169507\pi$
$859$			872.243i			1.01542i	−0.861529	+	0.507708i	$0.830493\pi$
$860$		0			0
$861$	−23.3718	−	324.188i	−0.0271449	−	0.376526i
$862$		0			0
$863$		−	204.850i		−	0.237370i	−0.992932	−	0.118685i	$-0.962132\pi$
$863$		−	204.850i		−	0.237370i	0.992932	−	0.118685i	$-0.0378679\pi$
$864$		0			0
$865$	244.114			0.282212
$866$		0			0
$867$	−771.488	+	55.6190i	−0.889836	+	0.0641511i
$868$		0			0
$869$	−1810.12			−2.08299
$870$		0			0
$871$	1145.01			1.31459
$872$		0			0
$873$	414.089	−	60.0179i	0.474328	−	0.0687490i
$874$		0			0
$875$	95.8773			0.109574
$876$		0			0
$877$			1398.31i			1.59442i	0.603702	+	0.797210i	$0.293692\pi$
$877$			1398.31i			1.59442i	−0.603702	+	0.797210i	$0.706308\pi$
$878$		0			0
$879$	682.466	−	49.2011i	0.776411	−	0.0559740i
$880$		0			0
$881$			1494.26i			1.69609i	0.529924	+	0.848045i	$0.322221\pi$
$881$			1494.26i			1.69609i	−0.529924	+	0.848045i	$0.677779\pi$
$882$		0			0
$883$			456.849i			0.517382i	0.965960	+	0.258691i	$0.0832912\pi$
$883$			456.849i			0.517382i	−0.965960	+	0.258691i	$0.916709\pi$
$884$		0			0
$885$	−76.3556	+	5.50472i	−0.0862775	+	0.00622002i
$886$		0			0
$887$		−	1289.76i		−	1.45407i	−0.686602	−	0.727033i	$-0.740899\pi$
$887$		−	1289.76i		−	1.45407i	0.686602	−	0.727033i	$-0.259101\pi$
$888$		0			0
$889$	−103.817			−0.116780
$890$		0			0
$891$	1225.03	−	362.732i	1.37490	−	0.407107i
$892$		0			0
$893$	1543.02			1.72791
$894$		0			0
$895$	−440.927			−0.492656
$896$		0			0
$897$	−2211.81	+	159.457i	−2.46579	+	0.177767i
$898$		0			0
$899$	−1159.93			−1.29025
$900$		0			0
$901$			107.221i			0.119002i
$902$		0			0
$903$	142.625	+	1978.34i	0.157946	+	2.19085i
$904$		0			0
$905$		−	605.720i		−	0.669303i
$906$		0			0
$907$		−	367.633i		−	0.405329i	−0.979248	−	0.202664i	$-0.935040\pi$
$907$		−	367.633i		−	0.405329i	0.979248	−	0.202664i	$-0.0649601\pi$
$908$		0			0
$909$	−918.713	+	133.158i	−1.01069	+	0.146489i
$910$		0			0
$911$			333.826i			0.366439i	0.983072	+	0.183219i	$0.0586518\pi$
$911$			333.826i			0.366439i	−0.983072	+	0.183219i	$0.941348\pi$
$912$		0			0
$913$	−1591.45			−1.74310
$914$		0			0
$915$	48.7868	+	676.718i	0.0533189	+	0.739582i
$916$		0			0
$917$	834.938			0.910511
$918$		0			0
$919$	−1252.80			−1.36322			−0.681610	−	0.731716i	$-0.738720\pi$
$919$	−1252.80			−1.36322			−0.681610	+	0.731716i	$0.738720\pi$
$920$		0			0
$921$	−10.5199	−	145.921i	−0.0114223	−	0.158437i
$922$		0			0
$923$	−381.289			−0.413098
$924$		0			0
$925$			139.348i			0.150647i
$926$		0			0
$927$	−80.0985	+	11.6095i	−0.0864061	+	0.0125237i
$928$		0			0
$929$			1367.02i			1.47149i	0.677258	+	0.735745i	$0.263168\pi$
$929$			1367.02i			1.47149i	−0.677258	+	0.735745i	$0.736832\pi$
$930$		0			0
$931$		−	612.288i		−	0.657667i
$932$		0			0
$933$	−122.739	−	1702.50i	−0.131553	−	1.82476i
$934$		0			0
$935$			196.909i			0.210598i
$936$		0			0
$937$	145.207			0.154970			0.0774852	−	0.996994i	$-0.475311\pi$
$937$	145.207			0.154970			0.0774852	+	0.996994i	$0.475311\pi$
$938$		0			0
$939$	899.680	−	64.8608i	0.958125	−	0.0690743i
$940$		0			0
$941$	−1463.12			−1.55485			−0.777426	−	0.628975i	$-0.783475\pi$
$941$	−1463.12			−1.55485			−0.777426	+	0.628975i	$0.783475\pi$
$942$		0			0
$943$	−515.439			−0.546595
$944$		0			0
$945$	−505.717	+	110.916i	−0.535150	+	0.117372i
$946$		0			0
$947$	−541.322			−0.571618			−0.285809	−	0.958287i	$-0.592262\pi$
$947$	−541.322			−0.571618			−0.285809	+	0.958287i	$0.592262\pi$
$948$		0			0
$949$		−	1115.54i		−	1.17549i
$950$		0			0
$951$	728.370	−	52.5105i	0.765899	−	0.0552161i
$952$		0			0
$953$		−	1212.42i		−	1.27221i	−0.771603	−	0.636105i	$-0.780545\pi$
$953$		−	1212.42i		−	1.27221i	0.771603	−	0.636105i	$-0.219455\pi$
$954$		0			0
$955$		−	143.433i		−	0.150192i
$956$		0			0
$957$	−1788.91	+	128.968i	−1.86929	+	0.134763i
$958$		0			0
$959$			1134.49i			1.18299i
$960$		0			0
$961$	−24.5007			−0.0254950
$962$		0			0
$963$	721.548	−	104.581i	0.749271	−	0.108599i
$964$		0			0
$965$	587.656			0.608969
$966$		0			0
$967$	1235.65			1.27782			0.638909	−	0.769282i	$-0.279386\pi$
$967$	1235.65			1.27782			0.638909	+	0.769282i	$0.279386\pi$
$968$		0			0
$969$	416.823	−	30.0501i	0.430158	−	0.0310115i
$970$		0			0
$971$	−612.350			−0.630638			−0.315319	−	0.948986i	$-0.602112\pi$
$971$	−612.350			−0.630638			−0.315319	+	0.948986i	$0.602112\pi$
$972$		0			0
$973$		−	1511.75i		−	1.55370i
$974$		0			0
$975$	−19.5424	−	271.071i	−0.0200434	−	0.278021i
$976$		0			0
$977$		−	782.891i		−	0.801322i	−0.916226	−	0.400661i	$-0.868781\pi$
$977$		−	782.891i		−	0.801322i	0.916226	−	0.400661i	$-0.131219\pi$
$978$		0			0
$979$			183.879i			0.187823i
$980$		0			0
$981$	−54.0539	−	372.941i	−0.0551008	−	0.380164i
$982$		0			0
$983$		−	389.928i		−	0.396671i	−0.980134	−	0.198336i	$-0.936446\pi$
$983$		−	389.928i		−	0.396671i	0.980134	−	0.198336i	$-0.0635536\pi$
$984$		0			0
$985$	−540.982			−0.549221
$986$		0			0
$987$	114.402	+	1586.87i	0.115909	+	1.60777i
$988$		0			0
$989$	3145.43			3.18042
$990$		0			0
$991$	1759.22			1.77520			0.887600	−	0.460614i	$-0.152371\pi$
$991$	1759.22			1.77520			0.887600	+	0.460614i	$0.152371\pi$
$992$		0			0
$993$	−75.1992	−	1043.08i	−0.0757293	−	1.05044i
$994$		0			0
$995$	−235.742			−0.236927
$996$		0			0
$997$		−	1825.71i		−	1.83120i	−0.402092	−	0.915599i	$-0.631717\pi$
$997$		−	1825.71i		−	1.83120i	0.402092	−	0.915599i	$-0.368283\pi$
$998$		0			0
$999$	−161.206	−	735.011i	−0.161367	−	0.735747i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.n.a.161.3	yes	24
3.2	odd	2	inner	960.3.n.a.161.24	yes	24
4.3	odd	2	inner	960.3.n.a.161.21	yes	24
8.3	odd	2	inner	960.3.n.a.161.4	yes	24
8.5	even	2	inner	960.3.n.a.161.22	yes	24
12.11	even	2	inner	960.3.n.a.161.2	yes	24
24.5	odd	2	inner	960.3.n.a.161.1	✓	24
24.11	even	2	inner	960.3.n.a.161.23	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.3.n.a.161.1	✓	24	24.5	odd	2	inner
960.3.n.a.161.2	yes	24	12.11	even	2	inner
960.3.n.a.161.3	yes	24	1.1	even	1	trivial
960.3.n.a.161.4	yes	24	8.3	odd	2	inner
960.3.n.a.161.21	yes	24	4.3	odd	2	inner
960.3.n.a.161.22	yes	24	8.5	even	2	inner
960.3.n.a.161.23	yes	24	24.11	even	2	inner
960.3.n.a.161.24	yes	24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.99223 + 0.215720i) q^{3} -2.23607 q^{5} -8.57552 q^{7} +(8.90693 - 1.29097i) q^{9} +O(q^{10})\)	\(q+(-2.99223 + 0.215720i) q^{3} -2.23607 q^{5} -8.57552 q^{7} +(8.90693 - 1.29097i) q^{9} +15.7729 q^{11} +18.1183i q^{13} +(6.69084 - 0.482364i) q^{15} -5.58301i q^{17} -24.9510i q^{19} +(25.6600 - 1.84991i) q^{21} -40.7977i q^{23} +5.00000 q^{25} +(-26.3731 + 5.78428i) q^{27} +37.9035 q^{29} -30.6023 q^{31} +(-47.1963 + 3.40253i) q^{33} +19.1755 q^{35} +27.8697i q^{37} +(-3.90847 - 54.2141i) q^{39} -12.6340i q^{41} +77.0982i q^{43} +(-19.9165 + 2.88669i) q^{45} +61.8422i q^{47} +24.5396 q^{49} +(1.20437 + 16.7057i) q^{51} -19.2048 q^{53} -35.2693 q^{55} +(5.38242 + 74.6592i) q^{57} -11.4120 q^{59} +101.141i q^{61} +(-76.3816 + 11.0707i) q^{63} -40.5137i q^{65} -63.1963i q^{67} +(8.80088 + 122.076i) q^{69} +21.0444i q^{71} -61.5698 q^{73} +(-14.9612 + 1.07860i) q^{75} -135.261 q^{77} -114.761 q^{79} +(77.6668 - 22.9971i) q^{81} -100.898 q^{83} +12.4840i q^{85} +(-113.416 + 8.17654i) q^{87} +11.6579i q^{89} -155.374i q^{91} +(91.5692 - 6.60152i) q^{93} +55.7921i q^{95} +46.4906 q^{97} +(140.488 - 20.3623i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{9}+O(q^{10}) \) 24 * q - 8 * q^9	\( 24 q - 8 q^{9} + 120 q^{25} - 256 q^{33} + 104 q^{49} - 304 q^{57} + 400 q^{73} + 152 q^{81} + 208 q^{97}+O(q^{100}) \) 24 * q - 8 * q^9 + 120 * q^25 - 256 * q^33 + 104 * q^49 - 304 * q^57 + 400 * q^73 + 152 * q^81 + 208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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