LMFDB - Embedded newform 960.3.n.a.161.23

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			161.23
Character	$\chi$	$=$	960.161
Dual form			960.3.n.a.161.21

$q$-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	$-0.290149\pi$
$7$	8.57552			1.22507			0.612537	+	0.790442i	$0.290149\pi$
$8$		0			0
$9$	8.90693	+	1.29097i	0.989659	+	0.143441i
$10$		0			0
$11$	−15.7729			−1.43390			−0.716951	−	0.697123i	$-0.754463\pi$
$11$	−15.7729			−1.43390			−0.716951	+	0.697123i	$0.754463\pi$
$12$		0			0
$13$		−	18.1183i		−	1.39371i	−0.717210	−	0.696857i	$-0.754581\pi$
$13$		−	18.1183i		−	1.39371i	0.717210	−	0.696857i	$-0.245419\pi$
$14$		0			0
$15$	−6.69084	−	0.482364i	−0.446056	−	0.0321576i
$16$		0			0
$17$			5.58301i			0.328413i	0.986426	+	0.164206i	$0.0525063\pi$
$17$			5.58301i			0.328413i	−0.986426	+	0.164206i	$0.947494\pi$
$18$		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	$-0.335686\pi$
$31$	30.6023			0.987170			0.493585	+	0.869697i	$0.335686\pi$
$32$		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	$-0.877087\pi$
$37$		−	27.8697i		−	0.753235i	0.926369	−	0.376617i	$-0.122913\pi$
$38$		0			0
$39$	3.90847	−	54.2141i	0.100217	−	1.39011i
$40$		0			0
$41$			12.6340i			0.308147i	0.988059	+	0.154073i	$0.0492392\pi$
$41$			12.6340i			0.308147i	−0.988059	+	0.154073i	$0.950761\pi$
$42$		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	$-0.469168\pi$
$59$	11.4120			0.193423			0.0967115	+	0.995312i	$0.469168\pi$
$60$		0			0
$61$		−	101.141i		−	1.65805i	−0.559213	−	0.829024i	$-0.688897\pi$
$61$		−	101.141i		−	1.65805i	0.559213	−	0.829024i	$-0.311103\pi$
$62$		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	$-0.241221\pi$
$79$	114.761			1.45267			0.726337	+	0.687339i	$0.241221\pi$
$80$		0			0
$81$	77.6668	+	22.9971i	0.958849	+	0.283915i
$82$		0			0
$83$	100.898			1.21563			0.607817	−	0.794077i	$-0.292045\pi$
$83$	100.898			1.21563			0.607817	+	0.794077i	$0.292045\pi$
$84$		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	$-0.979138\pi$
$89$		−	11.6579i		−	0.130988i	0.997853	−	0.0654938i	$-0.0208623\pi$
$90$		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	$-0.486100\pi$
$103$	8.99283			0.0873090			0.0436545	+	0.999047i	$0.486100\pi$
$104$		0			0
$105$	−57.3775	−	4.13653i	−0.546452	−	0.0393955i
$106$		0			0
$107$	−81.0097			−0.757100			−0.378550	−	0.925581i	$-0.623577\pi$
$107$	−81.0097			−0.757100			−0.378550	+	0.925581i	$0.623577\pi$
$108$		0			0
$109$			41.8708i			0.384136i	0.981382	+	0.192068i	$0.0615194\pi$
$109$			41.8708i			0.384136i	−0.981382	+	0.192068i	$0.938481\pi$
$110$		0			0
$111$	6.01204	−	83.3926i	0.0541626	−	0.751285i
$112$		0			0
$113$		−	134.246i		−	1.18802i	−0.804459	−	0.594008i	$-0.797545\pi$
$113$		−	134.246i		−	1.18802i	0.804459	−	0.594008i	$-0.202455\pi$
$114$		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	$-0.515177\pi$
$127$	−12.1062			−0.0953247			−0.0476624	+	0.998864i	$0.515177\pi$
$128$		0			0
$129$	−16.6316	+	230.696i	−0.128927	+	1.78834i
$130$		0			0
$131$	97.3629			0.743229			0.371614	−	0.928387i	$-0.378804\pi$
$131$	97.3629			0.743229			0.371614	+	0.928387i	$0.378804\pi$
$132$		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	$0.160389\pi$
$137$			132.294i			0.965647i	−0.875718	+	0.482824i	$0.839611\pi$
$138$		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	$-0.253114\pi$
$151$	211.447			1.40031			0.700155	+	0.713991i	$0.253114\pi$
$152$		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	$0.165198\pi$
$157$			155.744i			0.992000i	−0.868323	+	0.496000i	$0.834802\pi$
$158$		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	$-0.685679\pi$
$179$	−197.188			−1.10161			−0.550806	+	0.834633i	$0.685679\pi$
$180$		0			0
$181$		−	270.886i		−	1.49661i	−0.663356	−	0.748304i	$-0.730868\pi$
$181$		−	270.886i		−	1.49661i	0.663356	−	0.748304i	$-0.269132\pi$
$182$		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	$-0.585336\pi$
$199$	−105.427			−0.529785			−0.264892	+	0.964278i	$0.585336\pi$
$200$		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	$-0.553409\pi$
$223$	−74.4836			−0.334007			−0.167004	+	0.985956i	$0.553409\pi$
$224$		0			0
$225$	44.5346	+	6.45484i	0.197932	+	0.0286882i
$226$		0			0
$227$	−88.4128			−0.389484			−0.194742	−	0.980855i	$-0.562387\pi$
$227$	−88.4128			−0.389484			−0.194742	+	0.980855i	$0.562387\pi$
$228$		0			0
$229$			77.6001i			0.338865i	0.985542	+	0.169433i	$0.0541935\pi$
$229$			77.6001i			0.338865i	−0.985542	+	0.169433i	$0.945807\pi$
$230$		0			0
$231$	−404.733	−	29.1785i	−1.75209	−	0.126314i
$232$		0			0
$233$		−	252.078i		−	1.08188i	−0.841061	−	0.540940i	$-0.818069\pi$
$233$		−	252.078i		−	1.08188i	0.841061	−	0.540940i	$-0.181931\pi$
$234$		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	$-0.344491\pi$
$251$	235.610			0.938686			0.469343	+	0.883016i	$0.344491\pi$
$252$		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	$-0.926145\pi$
$257$		−	118.192i		−	0.459890i	0.973204	−	0.229945i	$-0.0738546\pi$
$258$		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	$-0.635008\pi$
$271$	−223.054			−0.823076			−0.411538	+	0.911393i	$0.635008\pi$
$272$		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	$0.0411998\pi$
$277$			71.5059i			0.258144i	−0.991635	+	0.129072i	$0.958800\pi$
$278$		0			0
$279$	272.572	+	39.5066i	0.976962	+	0.141601i
$280$		0			0
$281$			388.811i			1.38367i	0.722056	+	0.691834i	$0.243197\pi$
$281$			388.811i			1.38367i	−0.722056	+	0.691834i	$0.756803\pi$
$282$		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	$-0.296763\pi$
$347$	413.612			1.19197			0.595983	+	0.802997i	$0.296763\pi$
$348$		0			0
$349$			209.754i			0.601016i	0.953779	+	0.300508i	$0.0971562\pi$
$349$			209.754i			0.601016i	−0.953779	+	0.300508i	$0.902844\pi$
$350$		0			0
$351$	104.801	−	477.836i	0.298579	−	1.36136i
$352$		0			0
$353$		−	201.928i		−	0.572033i	−0.958225	−	0.286016i	$-0.907669\pi$
$353$		−	201.928i		−	0.572033i	0.958225	−	0.286016i	$-0.0923312\pi$
$354$		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	$-0.409226\pi$
$367$	206.492			0.562649			0.281324	+	0.959613i	$0.409226\pi$
$368$		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	$-0.934888\pi$
$373$		−	151.537i		−	0.406265i	0.979151	−	0.203132i	$-0.0651122\pi$
$374$		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	$0.232528\pi$
$397$			529.795i			1.33450i	−0.744835	+	0.667249i	$0.767472\pi$
$398$		0			0
$399$	46.1571	−	640.242i	0.115682	−	1.60462i
$400$		0			0
$401$		−	507.162i		−	1.26474i	−0.774665	−	0.632372i	$-0.782081\pi$
$401$		−	507.162i		−	1.26474i	0.774665	−	0.632372i	$-0.217919\pi$
$402$		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	$-0.766040\pi$
$419$	−621.651			−1.48365			−0.741827	+	0.670592i	$0.766040\pi$
$420$		0			0
$421$		−	326.795i		−	0.776234i	−0.921610	−	0.388117i	$-0.873126\pi$
$421$		−	326.795i		−	0.776234i	0.921610	−	0.388117i	$-0.126874\pi$
$422$		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	$-0.576951\pi$
$439$	−210.195			−0.478804			−0.239402	+	0.970921i	$0.576951\pi$
$440$		0			0
$441$	218.573	+	31.6799i	0.495630	+	0.0718364i
$442$		0			0
$443$	581.438			1.31250			0.656250	−	0.754543i	$-0.272142\pi$
$443$	581.438			1.31250			0.656250	+	0.754543i	$0.272142\pi$
$444$		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	$0.394503\pi$
$449$			849.129i			1.89116i	−0.325394	+	0.945578i	$0.605497\pi$
$450$		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	$-0.313756\pi$
$463$	511.415			1.10457			0.552284	+	0.833656i	$0.313756\pi$
$464$		0			0
$465$	−204.755	−	14.7614i	−0.440333	−	0.0317450i
$466$		0			0
$467$	865.595			1.85352			0.926761	−	0.375652i	$-0.122581\pi$
$467$	865.595			1.85352			0.926761	+	0.375652i	$0.122581\pi$
$468$		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	$-0.391291\pi$
$487$	326.211			0.669838			0.334919	+	0.942247i	$0.391291\pi$
$488$		0			0
$489$	−21.4068	+	296.932i	−0.0437767	+	0.607223i
$490$		0			0
$491$	307.774			0.626831			0.313416	−	0.949616i	$-0.398527\pi$
$491$	307.774			0.626831			0.313416	+	0.949616i	$0.398527\pi$
$492$		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	$-0.952272\pi$
$521$		−	155.654i		−	0.298760i	0.988780	−	0.149380i	$-0.0477278\pi$
$522$		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	$0.0481643\pi$
$541$			163.096i			0.301472i	−0.988574	+	0.150736i	$0.951836\pi$
$542$		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	$-0.365215\pi$
$563$	462.673			0.821799			0.410899	+	0.911681i	$0.365215\pi$
$564$		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	$0.189173\pi$
$569$			637.202i			1.11986i	−0.828539	+	0.559932i	$0.810827\pi$
$570$		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	$-0.658276\pi$
$587$	−559.999			−0.954001			−0.477001	+	0.878903i	$0.658276\pi$
$588$		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	$0.227559\pi$
$593$			777.469i			1.31108i	−0.755161	+	0.655539i	$0.772441\pi$
$594$		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	$-0.449061\pi$
$607$	193.447			0.318694			0.159347	+	0.987223i	$0.449061\pi$
$608$		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	$-0.781176\pi$
$613$		−	777.986i		−	1.26915i	0.772863	−	0.634573i	$-0.218824\pi$
$614$		0			0
$615$	6.09419	−	84.5321i	0.00990926	−	0.137451i
$616$		0			0
$617$		−	96.3008i		−	0.156079i	−0.996950	−	0.0780396i	$-0.975134\pi$
$617$		−	96.3008i		−	0.156079i	0.996950	−	0.0780396i	$-0.0248661\pi$
$618$		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	$-0.535876\pi$
$631$	−141.935			−0.224937			−0.112468	+	0.993655i	$0.535876\pi$
$632$		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	$-0.822084\pi$
$641$		−	679.828i		−	1.06057i	0.847818	−	0.530287i	$-0.177916\pi$
$642$		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	$-0.744389\pi$
$659$	−915.395			−1.38907			−0.694534	+	0.719460i	$0.744389\pi$
$660$		0			0
$661$			279.058i			0.422176i	0.977467	+	0.211088i	$0.0677007\pi$
$661$			279.058i			0.422176i	−0.977467	+	0.211088i	$0.932299\pi$
$662$		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	$-0.630429\pi$
$683$	−544.192			−0.796767			−0.398384	+	0.917219i	$0.630429\pi$
$684$		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	$0.311944\pi$
$709$			1177.65i			1.66100i	−0.557021	+	0.830499i	$0.688056\pi$
$710$		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	$-0.350611\pi$
$727$	657.613			0.904557			0.452279	+	0.891877i	$0.350611\pi$
$728$		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	$-0.765611\pi$
$733$		−	984.554i		−	1.34318i	0.740921	−	0.671592i	$-0.234389\pi$
$734$		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	$-0.304094\pi$
$751$	867.150			1.15466			0.577331	+	0.816511i	$0.304094\pi$
$752$		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	$-0.942571\pi$
$757$		−	271.673i		−	0.358881i	0.983769	−	0.179441i	$-0.0574288\pi$
$758$		0			0
$759$	−138.816	+	1925.50i	−0.182893	+	2.53689i
$760$		0			0
$761$			955.082i			1.25504i	0.778602	+	0.627518i	$0.215929\pi$
$761$			955.082i			1.25504i	−0.778602	+	0.627518i	$0.784071\pi$
$762$		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	$-0.918249\pi$
$809$		−	410.994i		−	0.508027i	0.967201	−	0.254013i	$-0.0817508\pi$
$810$		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	$-0.441547\pi$
$823$	300.567			0.365209			0.182604	+	0.983186i	$0.441547\pi$
$824$		0			0
$825$	−235.981	−	17.0127i	−0.286038	−	0.0206214i
$826$		0			0
$827$	179.865			0.217491			0.108746	−	0.994070i	$-0.465317\pi$
$827$	179.865			0.217491			0.108746	+	0.994070i	$0.465317\pi$
$828$		0			0
$829$		−	811.633i		−	0.979050i	−0.871989	−	0.489525i	$-0.837170\pi$
$829$		−	811.633i		−	0.979050i	0.871989	−	0.489525i	$-0.162830\pi$
$830$		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	$-0.986985\pi$
$853$		−	69.7357i		−	0.0817535i	0.999164	−	0.0408767i	$-0.0130151\pi$
$854$		0			0
$855$	−72.0258	+	496.936i	−0.0842407	+	0.581212i
$856$		0			0
$857$		−	246.998i		−	0.288212i	−0.989562	−	0.144106i	$-0.953969\pi$
$857$		−	246.998i		−	0.288212i	0.989562	−	0.144106i	$-0.0460306\pi$
$858$		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	$-0.706308\pi$
$877$		−	1398.31i		−	1.59442i	0.603702	−	0.797210i	$-0.293692\pi$
$878$		0			0
$879$	−682.466	−	49.2011i	−0.776411	−	0.0559740i
$880$		0			0
$881$		−	1494.26i		−	1.69609i	−0.529924	−	0.848045i	$-0.677779\pi$
$881$		−	1494.26i		−	1.69609i	0.529924	−	0.848045i	$-0.322221\pi$
$882$		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	$-0.261280\pi$
$919$	1252.80			1.36322			0.681610	+	0.731716i	$0.261280\pi$
$920$		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	$-0.736832\pi$
$929$		−	1367.02i		−	1.47149i	0.677258	−	0.735745i	$-0.263168\pi$
$930$		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	$-0.407738\pi$
$947$	541.322			0.571618			0.285809	+	0.958287i	$0.407738\pi$
$948$		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	$0.219455\pi$
$953$			1212.42i			1.27221i	−0.771603	+	0.636105i	$0.780545\pi$
$954$		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	$-0.720614\pi$
$967$	−1235.65			−1.27782			−0.638909	+	0.769282i	$0.720614\pi$
$968$		0			0
$969$	416.823	+	30.0501i	0.430158	+	0.0310115i
$970$		0			0
$971$	612.350			0.630638			0.315319	−	0.948986i	$-0.397888\pi$
$971$	612.350			0.630638			0.315319	+	0.948986i	$0.397888\pi$
$972$		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	$0.131219\pi$
$977$			782.891i			0.801322i	−0.916226	+	0.400661i	$0.868781\pi$
$978$		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	$-0.847629\pi$
$991$	−1759.22			−1.77520			−0.887600	+	0.460614i	$0.847629\pi$
$992$		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	$0.368283\pi$
$997$			1825.71i			1.83120i	−0.402092	+	0.915599i	$0.631717\pi$
$998$		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.161.23	yes	24
3.2	odd	2	inner	960.3.n.a.161.4	yes	24
4.3	odd	2	inner	960.3.n.a.161.1	✓	24
8.3	odd	2	inner	960.3.n.a.161.24	yes	24
8.5	even	2	inner	960.3.n.a.161.2	yes	24
12.11	even	2	inner	960.3.n.a.161.22	yes	24
24.5	odd	2	inner	960.3.n.a.161.21	yes	24
24.11	even	2	inner	960.3.n.a.161.3	yes	24

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
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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