LMFDB - Embedded newform 960.3.i.b.929.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(929,960)]

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, 1]))

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

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

chi := DirichletCharacter("960.929");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.i (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:	$64$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.17
Character	$\chi$	$=$	960.929
Dual form			960.3.i.b.929.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.31140 - 2.69819i) q^{3} +(-1.53895 - 4.75727i) q^{5} +12.2496i q^{7} +(-5.56048 + 7.07680i) q^{9} +O(q^{10})$	\(q+(-1.31140 - 2.69819i) q^{3} +(-1.53895 - 4.75727i) q^{5} +12.2496i q^{7} +(-5.56048 + 7.07680i) q^{9} -3.02475 q^{11} -13.7100 q^{13} +(-10.8178 + 10.3911i) q^{15} +25.8427 q^{17} -14.0811i q^{19} +(33.0516 - 16.0640i) q^{21} +17.8268 q^{23} +(-20.2632 + 14.6424i) q^{25} +(26.3866 + 5.72273i) q^{27} -27.5927 q^{29} +33.8371 q^{31} +(3.96664 + 8.16134i) q^{33} +(58.2744 - 18.8515i) q^{35} +14.9061 q^{37} +(17.9792 + 36.9921i) q^{39} -69.4754i q^{41} +65.9219 q^{43} +(42.2236 + 15.5618i) q^{45} +11.8232 q^{47} -101.052 q^{49} +(-33.8900 - 69.7285i) q^{51} +63.3827i q^{53} +(4.65495 + 14.3895i) q^{55} +(-37.9935 + 18.4659i) q^{57} +28.6025 q^{59} -2.88594i q^{61} +(-86.6876 - 68.1133i) q^{63} +(21.0990 + 65.2220i) q^{65} -69.0261 q^{67} +(-23.3780 - 48.1001i) q^{69} -69.9856i q^{71} -113.636i q^{73} +(66.0813 + 35.4720i) q^{75} -37.0518i q^{77} +109.941 q^{79} +(-19.1622 - 78.7008i) q^{81} -16.4401i q^{83} +(-39.7707 - 122.941i) q^{85} +(36.1849 + 74.4503i) q^{87} -66.4138i q^{89} -167.941i q^{91} +(-44.3739 - 91.2990i) q^{93} +(-66.9876 + 21.6702i) q^{95} -62.9956i q^{97} +(16.8190 - 21.4055i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q - 32 q^{9}+O(q^{10}) $ 64 * q - 32 * q^9	$ 64 q - 32 q^{9} - 32 q^{25} - 320 q^{49} + 448 q^{81}+O(q^{100}) $ 64 * q - 32 * q^9 - 32 * q^25 - 320 * q^49 + 448 * q^81

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
$3$	−1.31140	−	2.69819i	−0.437132	−	0.899397i
$3$	−1.31140	−	2.69819i	−0.437132	−	0.899397i
$4$		0			0
$5$	−1.53895	−	4.75727i	−0.307791	−	0.951454i
$5$	−1.53895	−	4.75727i	−0.307791	−	0.951454i
$6$		0			0
$7$			12.2496i			1.74994i	0.484181	+	0.874968i	$0.339118\pi$
$7$			12.2496i			1.74994i	−0.484181	+	0.874968i	$0.660882\pi$
$8$		0			0
$9$	−5.56048	+	7.07680i	−0.617831	+	0.786311i
$10$		0			0
$11$	−3.02475			−0.274977			−0.137488	−	0.990503i	$-0.543903\pi$
$11$	−3.02475			−0.274977			−0.137488	+	0.990503i	$0.543903\pi$
$12$		0			0
$13$	−13.7100			−1.05461			−0.527306	−	0.849675i	$-0.676798\pi$
$13$	−13.7100			−1.05461			−0.527306	+	0.849675i	$0.676798\pi$
$14$		0			0
$15$	−10.8178	+	10.3911i	−0.721190	+	0.692738i
$16$		0			0
$17$	25.8427			1.52016			0.760079	−	0.649830i	$-0.225160\pi$
$17$	25.8427			1.52016			0.760079	+	0.649830i	$0.225160\pi$
$18$		0			0
$19$		−	14.0811i		−	0.741111i	−0.928810	−	0.370556i	$-0.879167\pi$
$19$		−	14.0811i		−	0.741111i	0.928810	−	0.370556i	$-0.120833\pi$
$20$		0			0
$21$	33.0516	−	16.0640i	1.57389	−	0.764954i
$22$		0			0
$23$	17.8268			0.775078			0.387539	−	0.921853i	$-0.373325\pi$
$23$	17.8268			0.775078			0.387539	+	0.921853i	$0.373325\pi$
$24$		0			0
$25$	−20.2632	+	14.6424i	−0.810529	+	0.585698i
$26$		0			0
$27$	26.3866	+	5.72273i	0.977280	+	0.211953i
$28$		0			0
$29$	−27.5927			−0.951471			−0.475736	−	0.879588i	$-0.657818\pi$
$29$	−27.5927			−0.951471			−0.475736	+	0.879588i	$0.657818\pi$
$30$		0			0
$31$	33.8371			1.09152			0.545760	−	0.837942i	$-0.316241\pi$
$31$	33.8371			1.09152			0.545760	+	0.837942i	$0.316241\pi$
$32$		0			0
$33$	3.96664	+	8.16134i	0.120201	+	0.247313i
$34$		0			0
$35$	58.2744	−	18.8515i	1.66498	−	0.538615i
$36$		0			0
$37$	14.9061			0.402866			0.201433	−	0.979502i	$-0.435440\pi$
$37$	14.9061			0.402866			0.201433	+	0.979502i	$0.435440\pi$
$38$		0			0
$39$	17.9792	+	36.9921i	0.461005	+	0.948515i
$40$		0			0
$41$		−	69.4754i		−	1.69452i	−0.531178	−	0.847260i	$-0.678250\pi$
$41$		−	69.4754i		−	1.69452i	0.531178	−	0.847260i	$-0.321750\pi$
$42$		0			0
$43$	65.9219			1.53307			0.766534	−	0.642204i	$-0.221980\pi$
$43$	65.9219			1.53307			0.766534	+	0.642204i	$0.221980\pi$
$44$		0			0
$45$	42.2236	+	15.5618i	0.938302	+	0.345818i
$46$		0			0
$47$	11.8232			0.251557			0.125779	−	0.992058i	$-0.459857\pi$
$47$	11.8232			0.251557			0.125779	+	0.992058i	$0.459857\pi$
$48$		0			0
$49$	−101.052			−2.06228
$50$		0			0
$51$	−33.8900	−	69.7285i	−0.664510	−	1.36723i
$52$		0			0
$53$			63.3827i			1.19590i	0.801534	+	0.597950i	$0.204018\pi$
$53$			63.3827i			1.19590i	−0.801534	+	0.597950i	$0.795982\pi$
$54$		0			0
$55$	4.65495	+	14.3895i	0.0846354	+	0.261628i
$56$		0			0
$57$	−37.9935	+	18.4659i	−0.666553	+	0.323964i
$58$		0			0
$59$	28.6025			0.484788			0.242394	−	0.970178i	$-0.422067\pi$
$59$	28.6025			0.484788			0.242394	+	0.970178i	$0.422067\pi$
$60$		0			0
$61$		−	2.88594i		−	0.0473106i	−0.999720	−	0.0236553i	$-0.992470\pi$
$61$		−	2.88594i		−	0.0473106i	0.999720	−	0.0236553i	$-0.00753041\pi$
$62$		0			0
$63$	−86.6876	−	68.1133i	−1.37599	−	1.08116i
$64$		0			0
$65$	21.0990	+	65.2220i	0.324600	+	1.00342i
$66$		0			0
$67$	−69.0261			−1.03024			−0.515120	−	0.857118i	$-0.672253\pi$
$67$	−69.0261			−1.03024			−0.515120	+	0.857118i	$0.672253\pi$
$68$		0			0
$69$	−23.3780	−	48.1001i	−0.338811	−	0.697103i
$70$		0			0
$71$		−	69.9856i		−	0.985712i	−0.870111	−	0.492856i	$-0.835953\pi$
$71$		−	69.9856i		−	0.985712i	0.870111	−	0.492856i	$-0.164047\pi$
$72$		0			0
$73$		−	113.636i		−	1.55666i	−0.627857	−	0.778329i	$-0.716068\pi$
$73$		−	113.636i		−	1.55666i	0.627857	−	0.778329i	$-0.283932\pi$
$74$		0			0
$75$	66.0813	+	35.4720i	0.881084	+	0.472960i
$76$		0			0
$77$		−	37.0518i		−	0.481192i
$78$		0			0
$79$	109.941			1.39166			0.695829	−	0.718208i	$-0.255037\pi$
$79$	109.941			1.39166			0.695829	+	0.718208i	$0.255037\pi$
$80$		0			0
$81$	−19.1622	−	78.7008i	−0.236571	−	0.971614i
$82$		0			0
$83$		−	16.4401i		−	0.198073i	−0.995084	−	0.0990365i	$-0.968424\pi$
$83$		−	16.4401i		−	0.198073i	0.995084	−	0.0990365i	$-0.0315761\pi$
$84$		0			0
$85$	−39.7707	−	122.941i	−0.467891	−	1.44636i
$86$		0			0
$87$	36.1849	+	74.4503i	0.415919	+	0.855751i
$88$		0			0
$89$		−	66.4138i		−	0.746223i	−0.927787	−	0.373111i	$-0.878291\pi$
$89$		−	66.4138i		−	0.746223i	0.927787	−	0.373111i	$-0.121709\pi$
$90$		0			0
$91$		−	167.941i		−	1.84550i
$92$		0			0
$93$	−44.3739	−	91.2990i	−0.477138	−	0.981710i
$94$		0			0
$95$	−66.9876	+	21.6702i	−0.705133	+	0.228107i
$96$		0			0
$97$		−	62.9956i		−	0.649439i	−0.945810	−	0.324720i	$-0.894730\pi$
$97$		−	62.9956i		−	0.649439i	0.945810	−	0.324720i	$-0.105270\pi$
$98$		0			0
$99$	16.8190	−	21.4055i	0.169889	−	0.216217i
$100$		0			0
$101$	128.697			1.27422			0.637112	−	0.770772i	$-0.280129\pi$
$101$	128.697			1.27422			0.637112	+	0.770772i	$0.280129\pi$
$102$		0			0
$103$			84.1852i			0.817332i	0.912684	+	0.408666i	$0.134006\pi$
$103$			84.1852i			0.817332i	−0.912684	+	0.408666i	$0.865994\pi$
$104$		0			0
$105$	−127.286	−	132.514i	−1.21225	−	1.26204i
$106$		0			0
$107$			15.3572i			0.143525i	0.997422	+	0.0717626i	$0.0228624\pi$
$107$			15.3572i			0.143525i	−0.997422	+	0.0717626i	$0.977138\pi$
$108$		0			0
$109$			79.9342i			0.733341i	0.930351	+	0.366670i	$0.119502\pi$
$109$			79.9342i			0.733341i	−0.930351	+	0.366670i	$0.880498\pi$
$110$		0			0
$111$	−19.5478	−	40.2194i	−0.176106	−	0.362337i
$112$		0			0
$113$	16.9364			0.149879			0.0749396	−	0.997188i	$-0.476124\pi$
$113$	16.9364			0.149879			0.0749396	+	0.997188i	$0.476124\pi$
$114$		0			0
$115$	−27.4346	−	84.8068i	−0.238562	−	0.737451i
$116$		0			0
$117$	76.2339	−	97.0227i	0.651572	−	0.829253i
$118$		0			0
$119$			316.561i			2.66018i
$120$		0			0
$121$	−111.851			−0.924388
$122$		0			0
$123$	−187.458	+	91.1098i	−1.52405	+	0.740730i
$124$		0			0
$125$	100.842	+	73.8636i	0.806738	+	0.590909i
$126$		0			0
$127$		−	59.1817i		−	0.465997i	−0.972477	−	0.232999i	$-0.925146\pi$
$127$		−	59.1817i		−	0.465997i	0.972477	−	0.232999i	$-0.0748538\pi$
$128$		0			0
$129$	−86.4498	−	177.870i	−0.670154	−	1.37884i
$130$		0			0
$131$	62.0259			0.473480			0.236740	−	0.971573i	$-0.423921\pi$
$131$	62.0259			0.473480			0.236740	+	0.971573i	$0.423921\pi$
$132$		0			0
$133$	172.487			1.29690
$134$		0			0
$135$	−13.3831	−	134.335i	−0.0991343	−	0.995074i
$136$		0			0
$137$	212.368			1.55013			0.775067	−	0.631880i	$-0.217716\pi$
$137$	212.368			1.55013			0.775067	+	0.631880i	$0.217716\pi$
$138$		0			0
$139$		−	28.4083i		−	0.204377i	−0.994765	−	0.102188i	$-0.967416\pi$
$139$		−	28.4083i		−	0.204377i	0.994765	−	0.102188i	$-0.0325844\pi$
$140$		0			0
$141$	−15.5049	−	31.9012i	−0.109964	−	0.226250i
$142$		0			0
$143$	41.4691			0.289994
$144$		0			0
$145$	42.4639	+	131.266i	0.292854	+	0.905281i
$146$		0			0
$147$	132.519	+	272.656i	0.901488	+	1.85481i
$148$		0			0
$149$	−83.2918			−0.559006			−0.279503	−	0.960145i	$-0.590170\pi$
$149$	−83.2918			−0.559006			−0.279503	+	0.960145i	$0.590170\pi$
$150$		0			0
$151$	30.8984			0.204625			0.102313	−	0.994752i	$-0.467376\pi$
$151$	30.8984			0.204625			0.102313	+	0.994752i	$0.467376\pi$
$152$		0			0
$153$	−143.698	+	182.884i	−0.939200	+	1.19532i
$154$		0			0
$155$	−52.0738	−	160.972i	−0.335960	−	1.03853i
$156$		0			0
$157$	119.149			0.758912			0.379456	−	0.925210i	$-0.376111\pi$
$157$	119.149			0.758912			0.379456	+	0.925210i	$0.376111\pi$
$158$		0			0
$159$	171.019	−	83.1199i	1.07559	−	0.522766i
$160$		0			0
$161$			218.370i			1.35634i
$162$		0			0
$163$	217.566			1.33476			0.667380	−	0.744717i	$-0.267416\pi$
$163$	217.566			1.33476			0.667380	+	0.744717i	$0.267416\pi$
$164$		0			0
$165$	32.7212	−	31.4303i	0.198310	−	0.190487i
$166$		0			0
$167$	286.212			1.71384			0.856922	−	0.515446i	$-0.172374\pi$
$167$	286.212			1.71384			0.856922	+	0.515446i	$0.172374\pi$
$168$		0			0
$169$	18.9630			0.112207
$170$		0			0
$171$	99.6492	+	78.2977i	0.582744	+	0.457881i
$172$		0			0
$173$			180.392i			1.04273i	0.853333	+	0.521366i	$0.174577\pi$
$173$			180.392i			1.04273i	−0.853333	+	0.521366i	$0.825423\pi$
$174$		0			0
$175$	−179.363	−	248.216i	−1.02493	−	1.41837i
$176$		0			0
$177$	−37.5092	−	77.1750i	−0.211917	−	0.436017i
$178$		0			0
$179$	−67.6994			−0.378209			−0.189105	−	0.981957i	$-0.560559\pi$
$179$	−67.6994			−0.378209			−0.189105	+	0.981957i	$0.560559\pi$
$180$		0			0
$181$		−	222.756i		−	1.23070i	−0.788255	−	0.615349i	$-0.789015\pi$
$181$		−	222.756i		−	1.23070i	0.788255	−	0.615349i	$-0.210985\pi$
$182$		0			0
$183$	−7.78683	+	3.78462i	−0.0425510	+	0.0206810i
$184$		0			0
$185$	−22.9398	−	70.9121i	−0.123999	−	0.383309i
$186$		0			0
$187$	−78.1676			−0.418008
$188$		0			0
$189$	−70.1009	+	323.224i	−0.370904	+	1.71018i
$190$		0			0
$191$			218.719i			1.14513i	0.819860	+	0.572564i	$0.194051\pi$
$191$			218.719i			1.14513i	−0.819860	+	0.572564i	$0.805949\pi$
$192$		0			0
$193$		−	108.928i		−	0.564393i	−0.959357	−	0.282196i	$-0.908937\pi$
$193$		−	108.928i		−	0.564393i	0.959357	−	0.282196i	$-0.0910630\pi$
$194$		0			0
$195$	148.312	−	142.461i	0.760575	−	0.730570i
$196$		0			0
$197$		−	87.0371i		−	0.441813i	−0.975295	−	0.220906i	$-0.929098\pi$
$197$		−	87.0371i		−	0.441813i	0.975295	−	0.220906i	$-0.0709015\pi$
$198$		0			0
$199$	−76.1111			−0.382468			−0.191234	−	0.981544i	$-0.561249\pi$
$199$	−76.1111			−0.382468			−0.191234	+	0.981544i	$0.561249\pi$
$200$		0			0
$201$	90.5207	+	186.246i	0.450352	+	0.926596i
$202$		0			0
$203$		−	337.998i		−	1.66501i
$204$		0			0
$205$	−330.513	+	106.919i	−1.61226	+	0.521558i
$206$		0			0
$207$	−99.1254	+	126.157i	−0.478867	+	0.609452i
$208$		0			0
$209$			42.5918i			0.203788i
$210$		0			0
$211$			365.638i			1.73288i	0.499278	+	0.866442i	$0.333599\pi$
$211$			365.638i			1.73288i	−0.499278	+	0.866442i	$0.666401\pi$
$212$		0			0
$213$	−188.834	+	91.7789i	−0.886547	+	0.430887i
$214$		0			0
$215$	−101.451	−	313.608i	−0.471864	−	1.45864i
$216$		0			0
$217$			414.489i			1.91009i
$218$		0			0
$219$	−306.612	+	149.022i	−1.40005	+	0.680466i
$220$		0			0
$221$	−354.302			−1.60318
$222$		0			0
$223$			165.412i			0.741757i	0.928681	+	0.370879i	$0.120943\pi$
$223$			165.412i			0.741757i	−0.928681	+	0.370879i	$0.879057\pi$
$224$		0			0
$225$	9.05155	−	224.818i	0.0402291	−	0.999190i
$226$		0			0
$227$		−	264.450i		−	1.16498i	−0.812839	−	0.582489i	$-0.802079\pi$
$227$		−	264.450i		−	1.16498i	0.812839	−	0.582489i	$-0.197921\pi$
$228$		0			0
$229$		−	147.492i		−	0.644068i	−0.946728	−	0.322034i	$-0.895634\pi$
$229$		−	147.492i		−	0.644068i	0.946728	−	0.322034i	$-0.104366\pi$
$230$		0			0
$231$	−99.9728	+	48.5896i	−0.432783	+	0.210345i
$232$		0			0
$233$	374.411			1.60691			0.803457	−	0.595363i	$-0.202992\pi$
$233$	374.411			1.60691			0.803457	+	0.595363i	$0.202992\pi$
$234$		0			0
$235$	−18.1954	−	56.2461i	−0.0774271	−	0.239345i
$236$		0			0
$237$	−144.176	−	296.642i	−0.608338	−	1.25165i
$238$		0			0
$239$		−	360.896i		−	1.51003i	−0.655710	−	0.755013i	$-0.727631\pi$
$239$		−	360.896i		−	1.51003i	0.655710	−	0.755013i	$-0.272369\pi$
$240$		0			0
$241$	102.174			0.423958			0.211979	−	0.977274i	$-0.432009\pi$
$241$	102.174			0.423958			0.211979	+	0.977274i	$0.432009\pi$
$242$		0			0
$243$	−187.220	+	154.911i	−0.770454	+	0.637495i
$244$		0			0
$245$	155.514	+	480.729i	0.634750	+	1.96216i
$246$		0			0
$247$			193.051i			0.781585i
$248$		0			0
$249$	−44.3584	+	21.5595i	−0.178146	+	0.0865841i
$250$		0			0
$251$	479.581			1.91068			0.955340	−	0.295510i	$-0.0954894\pi$
$251$	479.581			1.91068			0.955340	+	0.295510i	$0.0954894\pi$
$252$		0			0
$253$	−53.9215			−0.213128
$254$		0			0
$255$	−279.562	+	268.533i	−1.09632	+	1.05307i
$256$		0			0
$257$	295.439			1.14957			0.574785	−	0.818305i	$-0.305086\pi$
$257$	295.439			1.14957			0.574785	+	0.818305i	$0.305086\pi$
$258$		0			0
$259$			182.593i			0.704991i
$260$		0			0
$261$	153.428	−	195.268i	0.587848	−	0.748153i
$262$		0			0
$263$	−240.752			−0.915407			−0.457703	−	0.889105i	$-0.651328\pi$
$263$	−240.752			−0.915407			−0.457703	+	0.889105i	$0.651328\pi$
$264$		0			0
$265$	301.529	−	97.5431i	1.13784	−	0.368087i
$266$		0			0
$267$	−179.197	+	87.0949i	−0.671151	+	0.326198i
$268$		0			0
$269$	−309.312			−1.14986			−0.574930	−	0.818203i	$-0.694971\pi$
$269$	−309.312			−1.14986			−0.574930	+	0.818203i	$0.694971\pi$
$270$		0			0
$271$	−297.780			−1.09882			−0.549410	−	0.835553i	$-0.685148\pi$
$271$	−297.780			−1.09882			−0.549410	+	0.835553i	$0.685148\pi$
$272$		0			0
$273$	−453.137	+	220.237i	−1.65984	+	0.806730i
$274$		0			0
$275$	61.2911	−	44.2897i	0.222877	−	0.161053i
$276$		0			0
$277$	−67.6010			−0.244047			−0.122023	−	0.992527i	$-0.538938\pi$
$277$	−67.6010			−0.244047			−0.122023	+	0.992527i	$0.538938\pi$
$278$		0			0
$279$	−188.150	+	239.458i	−0.674374	+	0.858274i
$280$		0			0
$281$			296.540i			1.05530i	0.849461	+	0.527652i	$0.176927\pi$
$281$			296.540i			1.05530i	−0.849461	+	0.527652i	$0.823073\pi$
$282$		0			0
$283$	188.471			0.665976			0.332988	−	0.942931i	$-0.391943\pi$
$283$	188.471			0.665976			0.332988	+	0.942931i	$0.391943\pi$
$284$		0			0
$285$	146.318	+	152.327i	0.513396	+	0.534482i
$286$		0			0
$287$	851.042			2.96530
$288$		0			0
$289$	378.845			1.31088
$290$		0			0
$291$	−169.974	+	82.6122i	−0.584104	+	0.283891i
$292$		0			0
$293$		−	54.1870i		−	0.184938i	−0.995716	−	0.0924692i	$-0.970524\pi$
$293$		−	54.1870i		−	0.184938i	0.995716	−	0.0924692i	$-0.0294760\pi$
$294$		0			0
$295$	−44.0180	−	136.070i	−0.149213	−	0.461254i
$296$		0			0
$297$	−79.8126	−	17.3098i	−0.268729	−	0.0582822i
$298$		0			0
$299$	−244.404			−0.817406
$300$		0			0
$301$			807.514i			2.68277i
$302$		0			0
$303$	−168.772	−	347.248i	−0.557004	−	1.14603i
$304$		0			0
$305$	−13.7292	+	4.44134i	−0.0450138	+	0.0145618i
$306$		0			0
$307$	−285.099			−0.928662			−0.464331	−	0.885662i	$-0.653705\pi$
$307$	−285.099			−0.928662			−0.464331	+	0.885662i	$0.653705\pi$
$308$		0			0
$309$	227.148	−	110.400i	0.735106	−	0.357282i
$310$		0			0
$311$		−	276.078i		−	0.887709i	−0.896099	−	0.443855i	$-0.853611\pi$
$311$		−	276.078i		−	0.887709i	0.896099	−	0.443855i	$-0.146389\pi$
$312$		0			0
$313$		−	152.399i		−	0.486898i	−0.969914	−	0.243449i	$-0.921721\pi$
$313$		−	152.399i		−	0.486898i	0.969914	−	0.243449i	$-0.0782788\pi$
$314$		0			0
$315$	−190.625	+	517.220i	−0.605159	+	1.64197i
$316$		0			0
$317$		−	508.948i		−	1.60551i	−0.596307	−	0.802757i	$-0.703366\pi$
$317$		−	508.948i		−	1.60551i	0.596307	−	0.802757i	$-0.296634\pi$
$318$		0			0
$319$	83.4608			0.261633
$320$		0			0
$321$	41.4367	−	20.1394i	0.129086	−	0.0627395i
$322$		0			0
$323$		−	363.894i		−	1.12661i
$324$		0			0
$325$	277.808	−	200.747i	0.854794	−	0.617684i
$326$		0			0
$327$	215.678	−	104.825i	0.659565	−	0.320567i
$328$		0			0
$329$			144.829i			0.440209i
$330$		0			0
$331$			271.547i			0.820383i	0.911999	+	0.410191i	$0.134538\pi$
$331$			271.547i			0.820383i	−0.911999	+	0.410191i	$0.865462\pi$
$332$		0			0
$333$	−82.8848	+	105.487i	−0.248903	+	0.316778i
$334$		0			0
$335$	106.228	+	328.376i	0.317099	+	0.980227i
$336$		0			0
$337$			199.881i			0.593120i	0.955014	+	0.296560i	$0.0958395\pi$
$337$			199.881i			0.593120i	−0.955014	+	0.296560i	$0.904161\pi$
$338$		0			0
$339$	−22.2103	−	45.6975i	−0.0655171	−	0.134801i
$340$		0			0
$341$	−102.349			−0.300143
$342$		0			0
$343$		−	637.608i		−	1.85892i
$344$		0			0
$345$	−192.847	+	185.239i	−0.558978	+	0.536925i
$346$		0			0
$347$			609.297i			1.75590i	0.478753	+	0.877950i	$0.341089\pi$
$347$			609.297i			1.75590i	−0.478753	+	0.877950i	$0.658911\pi$
$348$		0			0
$349$			444.680i			1.27415i	0.770800	+	0.637077i	$0.219857\pi$
$349$			444.680i			1.27415i	−0.770800	+	0.637077i	$0.780143\pi$
$350$		0			0
$351$	−361.759	−	78.4584i	−1.03065	−	0.223528i
$352$		0			0
$353$	−221.451			−0.627341			−0.313671	−	0.949532i	$-0.601559\pi$
$353$	−221.451			−0.627341			−0.313671	+	0.949532i	$0.601559\pi$
$354$		0			0
$355$	−332.940	+	107.705i	−0.937860	+	0.303393i
$356$		0			0
$357$	854.143	−	415.138i	2.39256	−	1.16285i
$358$		0			0
$359$			11.5997i			0.0323111i	0.999869	+	0.0161556i	$0.00514270\pi$
$359$			11.5997i			0.0323111i	−0.999869	+	0.0161556i	$0.994857\pi$
$360$		0			0
$361$	162.722			0.450754
$362$		0			0
$363$	146.681	+	301.795i	0.404080	+	0.831392i
$364$		0			0
$365$	−540.597	+	174.881i	−1.48109	+	0.479125i
$366$		0			0
$367$			433.927i			1.18236i	0.806538	+	0.591182i	$0.201338\pi$
$367$			433.927i			1.18236i	−0.806538	+	0.591182i	$0.798662\pi$
$368$		0			0
$369$	491.663	+	386.316i	1.33242	+	1.04693i
$370$		0			0
$371$	−776.410			−2.09275
$372$		0			0
$373$	−210.700			−0.564879			−0.282440	−	0.959285i	$-0.591144\pi$
$373$	−210.700			−0.564879			−0.282440	+	0.959285i	$0.591144\pi$
$374$		0			0
$375$	67.0539	−	368.956i	0.178810	−	0.983884i
$376$		0			0
$377$	378.294			1.00343
$378$		0			0
$379$			290.509i			0.766514i	0.923642	+	0.383257i	$0.125198\pi$
$379$			290.509i			0.766514i	−0.923642	+	0.383257i	$0.874802\pi$
$380$		0			0
$381$	−159.683	+	77.6106i	−0.419117	+	0.203702i
$382$		0			0
$383$	28.4785			0.0743564			0.0371782	−	0.999309i	$-0.488163\pi$
$383$	28.4785			0.0743564			0.0371782	+	0.999309i	$0.488163\pi$
$384$		0			0
$385$	−176.265	+	57.0210i	−0.457832	+	0.148107i
$386$		0			0
$387$	−366.557	+	466.516i	−0.947176	+	1.20547i
$388$		0			0
$389$	463.706			1.19205			0.596024	−	0.802967i	$-0.296746\pi$
$389$	463.706			1.19205			0.596024	+	0.802967i	$0.296746\pi$
$390$		0			0
$391$	460.692			1.17824
$392$		0			0
$393$	−81.3405	−	167.358i	−0.206973	−	0.425847i
$394$		0			0
$395$	−169.194	−	523.019i	−0.428339	−	1.32410i
$396$		0			0
$397$	344.568			0.867930			0.433965	−	0.900930i	$-0.357114\pi$
$397$	344.568			0.867930			0.433965	+	0.900930i	$0.357114\pi$
$398$		0			0
$399$	−226.199	−	465.404i	−0.566916	−	1.16643i
$400$		0			0
$401$		−	300.230i		−	0.748704i	−0.927287	−	0.374352i	$-0.877865\pi$
$401$		−	300.230i		−	0.748704i	0.927287	−	0.374352i	$-0.122135\pi$
$402$		0			0
$403$	−463.905			−1.15113
$404$		0			0
$405$	−344.911	+	212.277i	−0.851632	+	0.524140i
$406$		0			0
$407$	−45.0870			−0.110779
$408$		0			0
$409$	113.232			0.276851			0.138425	−	0.990373i	$-0.455796\pi$
$409$	113.232			0.276851			0.138425	+	0.990373i	$0.455796\pi$
$410$		0			0
$411$	−278.499	−	573.010i	−0.677613	−	1.39419i
$412$		0			0
$413$			350.368i			0.848348i
$414$		0			0
$415$	−78.2098	+	25.3005i	−0.188457	+	0.0609651i
$416$		0			0
$417$	−76.6512	+	37.2546i	−0.183816	+	0.0893396i
$418$		0			0
$419$	608.930			1.45329			0.726647	−	0.687011i	$-0.241078\pi$
$419$	608.930			1.45329			0.726647	+	0.687011i	$0.241078\pi$
$420$		0			0
$421$			160.484i			0.381196i	0.981668	+	0.190598i	$0.0610428\pi$
$421$			160.484i			0.381196i	−0.981668	+	0.190598i	$0.938957\pi$
$422$		0			0
$423$	−65.7426	+	83.6704i	−0.155420	+	0.197802i
$424$		0			0
$425$	−523.657	+	378.400i	−1.23213	+	0.890354i
$426$		0			0
$427$	35.3515			0.0827905
$428$		0			0
$429$	−54.3825	−	111.892i	−0.126766	−	0.260820i
$430$		0			0
$431$			120.477i			0.279529i	0.990185	+	0.139764i	$0.0446345\pi$
$431$			120.477i			0.279529i	−0.990185	+	0.139764i	$0.955366\pi$
$432$		0			0
$433$		−	419.887i		−	0.969716i	−0.874593	−	0.484858i	$-0.838871\pi$
$433$		−	419.887i		−	0.969716i	0.874593	−	0.484858i	$-0.161129\pi$
$434$		0			0
$435$	298.493	−	286.717i	0.686191	−	0.659120i
$436$		0			0
$437$		−	251.021i		−	0.574419i
$438$		0			0
$439$	−655.375			−1.49288			−0.746440	−	0.665452i	$-0.768239\pi$
$439$	−655.375			−1.49288			−0.746440	+	0.665452i	$0.768239\pi$
$440$		0			0
$441$	561.895	−	715.122i	1.27414	−	1.62159i
$442$		0			0
$443$		−	207.128i		−	0.467558i	−0.972290	−	0.233779i	$-0.924891\pi$
$443$		−	207.128i		−	0.467558i	0.972290	−	0.233779i	$-0.0751092\pi$
$444$		0			0
$445$	−315.948	+	102.208i	−0.709997	+	0.229681i
$446$		0			0
$447$	109.229	+	224.737i	0.244359	+	0.502768i
$448$		0			0
$449$			167.158i			0.372289i	0.982522	+	0.186144i	$0.0595992\pi$
$449$			167.158i			0.372289i	−0.982522	+	0.186144i	$0.940401\pi$
$450$		0			0
$451$			210.145i			0.465954i
$452$		0			0
$453$	−40.5201	−	83.3698i	−0.0894483	−	0.184039i
$454$		0			0
$455$	−798.940	+	258.453i	−1.75591	+	0.568029i
$456$		0			0
$457$		−	212.777i		−	0.465595i	−0.972525	−	0.232797i	$-0.925212\pi$
$457$		−	212.777i		−	0.465595i	0.972525	−	0.232797i	$-0.0747879\pi$
$458$		0			0
$459$	681.900	+	147.891i	1.48562	+	0.322202i
$460$		0			0
$461$	657.831			1.42696			0.713482	−	0.700673i	$-0.247117\pi$
$461$	657.831			1.42696			0.713482	+	0.700673i	$0.247117\pi$
$462$		0			0
$463$			185.882i			0.401472i	0.979645	+	0.200736i	$0.0643334\pi$
$463$			185.882i			0.401472i	−0.979645	+	0.200736i	$0.935667\pi$
$464$		0			0
$465$	−366.045	+	351.604i	−0.787193	+	0.756137i
$466$		0			0
$467$		−	15.5127i		−	0.0332178i	−0.999862	−	0.0166089i	$-0.994713\pi$
$467$		−	15.5127i		−	0.0332178i	0.999862	−	0.0166089i	$-0.00528702\pi$
$468$		0			0
$469$		−	845.539i		−	1.80286i
$470$		0			0
$471$	−156.252	−	321.487i	−0.331745	−	0.682563i
$472$		0			0
$473$	−199.397			−0.421558
$474$		0			0
$475$	206.182	+	285.329i	0.434067	+	0.600692i
$476$		0			0
$477$	−448.547	−	352.438i	−0.940349	−	0.738863i
$478$		0			0
$479$			162.375i			0.338988i	0.985531	+	0.169494i	$0.0542133\pi$
$479$			162.375i			0.338988i	−0.985531	+	0.169494i	$0.945787\pi$
$480$		0			0
$481$	−204.361			−0.424868
$482$		0			0
$483$	589.204	−	286.370i	1.21988	−	0.592898i
$484$		0			0
$485$	−299.687	+	96.9474i	−0.617911	+	0.199892i
$486$		0			0
$487$			186.293i			0.382532i	0.981538	+	0.191266i	$0.0612593\pi$
$487$			186.293i			0.382532i	−0.981538	+	0.191266i	$0.938741\pi$
$488$		0			0
$489$	−285.315	−	587.034i	−0.583467	−	1.20048i
$490$		0			0
$491$	−313.419			−0.638328			−0.319164	−	0.947699i	$-0.603402\pi$
$491$	−313.419			−0.638328			−0.319164	+	0.947699i	$0.603402\pi$
$492$		0			0
$493$	−713.069			−1.44639
$494$		0			0
$495$	−127.716	−	47.0705i	−0.258011	−	0.0950919i
$496$		0			0
$497$	857.292			1.72493
$498$		0			0
$499$		−	721.574i		−	1.44604i	−0.690827	−	0.723020i	$-0.742754\pi$
$499$		−	721.574i		−	1.44604i	0.690827	−	0.723020i	$-0.257246\pi$
$500$		0			0
$501$	−375.338	−	772.255i	−0.749177	−	1.54143i
$502$		0			0
$503$	−371.171			−0.737915			−0.368957	−	0.929446i	$-0.620285\pi$
$503$	−371.171			−0.737915			−0.368957	+	0.929446i	$0.620285\pi$
$504$		0			0
$505$	−198.058	−	612.244i	−0.392194	−	1.21236i
$506$		0			0
$507$	−24.8680	−	51.1658i	−0.0490493	−	0.100919i
$508$		0			0
$509$	289.124			0.568024			0.284012	−	0.958821i	$-0.408334\pi$
$509$	289.124			0.568024			0.284012	+	0.958821i	$0.408334\pi$
$510$		0			0
$511$	1391.99			2.72405
$512$		0			0
$513$	80.5824	−	371.552i	0.157081	−	0.724273i
$514$		0			0
$515$	400.492	−	129.557i	0.777654	−	0.251567i
$516$		0			0
$517$	−35.7622			−0.0691724
$518$		0			0
$519$	486.734	−	236.566i	0.937830	−	0.455812i
$520$		0			0
$521$		−	334.804i		−	0.642618i	−0.946974	−	0.321309i	$-0.895877\pi$
$521$		−	334.804i		−	0.642618i	0.946974	−	0.321309i	$-0.104123\pi$
$522$		0			0
$523$	−195.767			−0.374316			−0.187158	−	0.982330i	$-0.559928\pi$
$523$	−195.767			−0.374316			−0.187158	+	0.982330i	$0.559928\pi$
$524$		0			0
$525$	−434.517	+	809.466i	−0.827651	+	1.54184i
$526$		0			0
$527$	874.442			1.65928
$528$		0			0
$529$	−211.206			−0.399255
$530$		0			0
$531$	−159.044	+	202.414i	−0.299517	+	0.381194i
$532$		0			0
$533$			952.504i			1.78706i
$534$		0			0
$535$	73.0583	−	23.6340i	0.136558	−	0.0441758i
$536$		0			0
$537$	88.7808	+	182.666i	0.165327	+	0.340160i
$538$		0			0
$539$	305.655			0.567078
$540$		0			0
$541$			308.922i			0.571021i	0.958376	+	0.285510i	$0.0921631\pi$
$541$			308.922i			0.571021i	−0.958376	+	0.285510i	$0.907837\pi$
$542$		0			0
$543$	−601.039	+	292.122i	−1.10689	+	0.537978i
$544$		0			0
$545$	380.268	−	123.015i	0.697740	−	0.225716i
$546$		0			0
$547$	532.237			0.973011			0.486505	−	0.873678i	$-0.338271\pi$
$547$	532.237			0.973011			0.486505	+	0.873678i	$0.338271\pi$
$548$		0			0
$549$	20.4233	+	16.0472i	0.0372008	+	0.0292299i
$550$		0			0
$551$			388.535i			0.705146i
$552$		0			0
$553$			1346.73i			2.43531i
$554$		0			0
$555$	−161.251	+	154.890i	−0.290543	+	0.279081i
$556$		0			0
$557$		−	355.234i		−	0.637764i	−0.947794	−	0.318882i	$-0.896693\pi$
$557$		−	355.234i		−	0.637764i	0.947794	−	0.318882i	$-0.103307\pi$
$558$		0			0
$559$	−903.787			−1.61679
$560$		0			0
$561$	102.509	+	210.911i	0.182725	+	0.375956i
$562$		0			0
$563$			361.694i			0.642440i	0.947005	+	0.321220i	$0.104093\pi$
$563$			361.694i			0.642440i	−0.947005	+	0.321220i	$0.895907\pi$
$564$		0			0
$565$	−26.0643	−	80.5708i	−0.0461315	−	0.142603i
$566$		0			0
$567$	964.049	−	234.729i	1.70026	−	0.413984i
$568$		0			0
$569$			810.062i			1.42366i	0.702352	+	0.711829i	$0.252133\pi$
$569$			810.062i			1.42366i	−0.702352	+	0.711829i	$0.747867\pi$
$570$		0			0
$571$		−	781.618i		−	1.36886i	−0.729080	−	0.684429i	$-0.760052\pi$
$571$		−	781.618i		−	1.36886i	0.729080	−	0.684429i	$-0.239948\pi$
$572$		0			0
$573$	590.147	−	286.828i	1.02992	−	0.500572i
$574$		0			0
$575$	−361.228	+	261.028i	−0.628223	+	0.453961i
$576$		0			0
$577$			878.009i			1.52168i	0.648940	+	0.760839i	$0.275212\pi$
$577$			878.009i			1.52168i	−0.648940	+	0.760839i	$0.724788\pi$
$578$		0			0
$579$	−293.908	+	142.848i	−0.507613	+	0.246714i
$580$		0			0
$581$	201.383			0.346615
$582$		0			0
$583$		−	191.716i		−	0.328845i
$584$		0			0
$585$	−578.883	−	213.352i	−0.989544	−	0.364704i
$586$		0			0
$587$		−	138.105i		−	0.235272i	−0.993057	−	0.117636i	$-0.962468\pi$
$587$		−	138.105i		−	0.235272i	0.993057	−	0.117636i	$-0.0375316\pi$
$588$		0			0
$589$		−	476.464i		−	0.808937i
$590$		0			0
$591$	−234.843	+	114.140i	−0.397365	+	0.193131i
$592$		0			0
$593$	−273.578			−0.461345			−0.230673	−	0.973031i	$-0.574093\pi$
$593$	−273.578			−0.461345			−0.230673	+	0.973031i	$0.574093\pi$
$594$		0			0
$595$	1505.97	−	487.174i	2.53104	−	0.818779i
$596$		0			0
$597$	99.8119	+	205.362i	0.167189	+	0.343991i
$598$		0			0
$599$		−	821.780i		−	1.37192i	−0.727639	−	0.685960i	$-0.759382\pi$
$599$		−	821.780i		−	1.37192i	0.727639	−	0.685960i	$-0.240618\pi$
$600$		0			0
$601$	−933.331			−1.55296			−0.776482	−	0.630140i	$-0.782998\pi$
$601$	−933.331			−1.55296			−0.776482	+	0.630140i	$0.782998\pi$
$602$		0			0
$603$	383.818	−	488.484i	0.636514	−	0.810090i
$604$		0			0
$605$	172.134	+	532.105i	0.284518	+	0.879512i
$606$		0			0
$607$			268.120i			0.441714i	0.975306	+	0.220857i	$0.0708854\pi$
$607$			268.120i			0.441714i	−0.975306	+	0.220857i	$0.929115\pi$
$608$		0			0
$609$	−911.983	+	443.249i	−1.49751	+	0.727831i
$610$		0			0
$611$	−162.096			−0.265295
$612$		0			0
$613$	914.181			1.49132			0.745661	−	0.666325i	$-0.232134\pi$
$613$	914.181			1.49132			0.745661	+	0.666325i	$0.232134\pi$
$614$		0			0
$615$	721.923	+	751.574i	1.17386	+	1.22207i
$616$		0			0
$617$	−753.872			−1.22183			−0.610917	−	0.791695i	$-0.709199\pi$
$617$	−753.872			−1.22183			−0.610917	+	0.791695i	$0.709199\pi$
$618$		0			0
$619$		−	525.804i		−	0.849440i	−0.905325	−	0.424720i	$-0.860372\pi$
$619$		−	525.804i		−	0.849440i	0.905325	−	0.424720i	$-0.139628\pi$
$620$		0			0
$621$	470.387	+	102.018i	0.757468	+	0.164280i
$622$		0			0
$623$	813.540			1.30584
$624$		0			0
$625$	196.197	−	593.407i	0.313916	−	0.949451i
$626$		0			0
$627$	114.921	−	55.8547i	0.183287	−	0.0890825i
$628$		0			0
$629$	385.213			0.612421
$630$		0			0
$631$	171.651			0.272030			0.136015	−	0.990707i	$-0.456570\pi$
$631$	171.651			0.272030			0.136015	+	0.990707i	$0.456570\pi$
$632$		0			0
$633$	986.562	−	479.497i	1.55855	−	0.757499i
$634$		0			0
$635$	−281.543	+	91.0779i	−0.443375	+	0.143430i
$636$		0			0
$637$	1385.41			2.17490
$638$		0			0
$639$	495.274	+	389.153i	0.775077	+	0.609003i
$640$		0			0
$641$		−	626.169i		−	0.976862i	−0.872602	−	0.488431i	$-0.837569\pi$
$641$		−	626.169i		−	0.976862i	0.872602	−	0.488431i	$-0.162431\pi$
$642$		0			0
$643$	265.568			0.413013			0.206507	−	0.978445i	$-0.433790\pi$
$643$	265.568			0.413013			0.206507	+	0.978445i	$0.433790\pi$
$644$		0			0
$645$	−713.133	+	684.999i	−1.10563	+	1.06201i
$646$		0			0
$647$	25.6635			0.0396654			0.0198327	−	0.999803i	$-0.493687\pi$
$647$	25.6635			0.0396654			0.0198327	+	0.999803i	$0.493687\pi$
$648$		0			0
$649$	−86.5153			−0.133306
$650$		0			0
$651$	1118.37	−	543.560i	1.71793	−	0.834962i
$652$		0			0
$653$		−	621.080i		−	0.951119i	−0.879684	−	0.475559i	$-0.842246\pi$
$653$		−	621.080i		−	0.951119i	0.879684	−	0.475559i	$-0.157754\pi$
$654$		0			0
$655$	−95.4550	−	295.074i	−0.145733	−	0.450494i
$656$		0			0
$657$	804.180	+	631.870i	1.22402	+	0.961751i
$658$		0			0
$659$	460.999			0.699543			0.349771	−	0.936835i	$-0.386259\pi$
$659$	460.999			0.699543			0.349771	+	0.936835i	$0.386259\pi$
$660$		0			0
$661$			9.42461i			0.0142581i	0.999975	+	0.00712905i	$0.00226927\pi$
$661$			9.42461i			0.0142581i	−0.999975	+	0.00712905i	$0.997731\pi$
$662$		0			0
$663$	464.631	+	955.975i	0.700801	+	1.44189i
$664$		0			0
$665$	−265.450	−	820.569i	−0.399173	−	1.23394i
$666$		0			0
$667$	−491.888			−0.737464
$668$		0			0
$669$	446.313	−	216.921i	0.667135	−	0.324246i
$670$		0			0
$671$			8.72925i			0.0130093i
$672$		0			0
$673$			254.588i			0.378288i	0.981949	+	0.189144i	$0.0605712\pi$
$673$			254.588i			0.378288i	−0.981949	+	0.189144i	$0.939429\pi$
$674$		0			0
$675$	−618.472	+	270.403i	−0.916255	+	0.400597i
$676$		0			0
$677$		−	1280.94i		−	1.89209i	−0.324040	−	0.946043i	$-0.605041\pi$
$677$		−	1280.94i		−	1.89209i	0.324040	−	0.946043i	$-0.394959\pi$
$678$		0			0
$679$	771.668			1.13648
$680$		0			0
$681$	−713.537	+	346.799i	−1.04778	+	0.509250i
$682$		0			0
$683$			484.089i			0.708768i	0.935100	+	0.354384i	$0.115309\pi$
$683$			484.089i			0.708768i	−0.935100	+	0.354384i	$0.884691\pi$
$684$		0			0
$685$	−326.825	−	1010.29i	−0.477117	−	1.47488i
$686$		0			0
$687$	−397.960	+	193.420i	−0.579273	+	0.281543i
$688$		0			0
$689$		−	868.974i		−	1.26121i
$690$		0			0
$691$			542.386i			0.784929i	0.919767	+	0.392464i	$0.128377\pi$
$691$			542.386i			0.784929i	−0.919767	+	0.392464i	$0.871623\pi$
$692$		0			0
$693$	262.208	+	206.026i	0.378367	+	0.297295i
$694$		0			0
$695$	−135.146	+	43.7192i	−0.194455	+	0.0629053i
$696$		0			0
$697$		−	1795.43i		−	2.57594i
$698$		0			0
$699$	−491.001	−	1010.23i	−0.702434	−	1.44525i
$700$		0			0
$701$	−1233.95			−1.76028			−0.880138	−	0.474718i	$-0.842550\pi$
$701$	−1233.95			−1.76028			−0.880138	+	0.474718i	$0.842550\pi$
$702$		0			0
$703$		−	209.894i		−	0.298569i
$704$		0			0
$705$	−127.901	+	122.856i	−0.181421	+	0.174263i
$706$		0			0
$707$			1576.48i			2.22981i
$708$		0			0
$709$		−	708.293i		−	0.999002i	−0.866313	−	0.499501i	$-0.833517\pi$
$709$		−	708.293i		−	0.999002i	0.866313	−	0.499501i	$-0.166483\pi$
$710$		0			0
$711$	−611.324	+	778.030i	−0.859808	+	1.09428i
$712$		0			0
$713$	603.207			0.846012
$714$		0			0
$715$	−63.8191	−	197.280i	−0.0892575	−	0.275916i
$716$		0			0
$717$	−973.767	+	473.278i	−1.35811	+	0.660081i
$718$		0			0
$719$			399.429i			0.555535i	0.960648	+	0.277767i	$0.0895944\pi$
$719$			399.429i			0.555535i	−0.960648	+	0.277767i	$0.910406\pi$
$720$		0			0
$721$	−1031.23			−1.43028
$722$		0			0
$723$	−133.991	−	275.685i	−0.185326	−	0.381307i
$724$		0			0
$725$	559.117	−	404.024i	0.771195	−	0.557275i
$726$		0			0
$727$			509.611i			0.700978i	0.936567	+	0.350489i	$0.113985\pi$
$727$			509.611i			0.700978i	−0.936567	+	0.350489i	$0.886015\pi$
$728$		0			0
$729$	663.501	+	302.006i	0.910152	+	0.414275i
$730$		0			0
$731$	1703.60			2.33051
$732$		0			0
$733$	−717.150			−0.978376			−0.489188	−	0.872178i	$-0.662707\pi$
$733$	−717.150			−0.978376			−0.489188	+	0.872178i	$0.662707\pi$
$734$		0			0
$735$	1093.16	−	1050.03i	1.48729	−	1.42862i
$736$		0			0
$737$	208.786			0.283292
$738$		0			0
$739$		−	1361.02i		−	1.84171i	−0.389905	−	0.920855i	$-0.627492\pi$
$739$		−	1361.02i		−	1.84171i	0.389905	−	0.920855i	$-0.372508\pi$
$740$		0			0
$741$	520.890	−	253.167i	0.702955	−	0.341656i
$742$		0			0
$743$	−1230.60			−1.65625			−0.828127	−	0.560540i	$-0.810594\pi$
$743$	−1230.60			−1.65625			−0.828127	+	0.560540i	$0.810594\pi$
$744$		0			0
$745$	128.182	+	396.242i	0.172057	+	0.531868i
$746$		0			0
$747$	116.343	+	91.4146i	0.155747	+	0.122376i
$748$		0			0
$749$	−188.119			−0.251160
$750$		0			0
$751$	−477.996			−0.636479			−0.318239	−	0.948010i	$-0.603092\pi$
$751$	−477.996			−0.636479			−0.318239	+	0.948010i	$0.603092\pi$
$752$		0			0
$753$	−628.921	−	1294.00i	−0.835220	−	1.71846i
$754$		0			0
$755$	−47.5513	−	146.992i	−0.0629818	−	0.194692i
$756$		0			0
$757$	−710.649			−0.938770			−0.469385	−	0.882994i	$-0.655524\pi$
$757$	−710.649			−0.938770			−0.469385	+	0.882994i	$0.655524\pi$
$758$		0			0
$759$	70.7125	+	145.490i	0.0931653	+	0.191687i
$760$		0			0
$761$			134.002i			0.176087i	0.996117	+	0.0880433i	$0.0280614\pi$
$761$			134.002i			0.176087i	−0.996117	+	0.0880433i	$0.971939\pi$
$762$		0			0
$763$	−979.158			−1.28330
$764$		0			0
$765$	1091.17	+	402.159i	1.42637	+	0.525698i
$766$		0			0
$767$	−392.139			−0.511264
$768$		0			0
$769$	−301.919			−0.392612			−0.196306	−	0.980543i	$-0.562895\pi$
$769$	−301.919			−0.392612			−0.196306	+	0.980543i	$0.562895\pi$
$770$		0			0
$771$	−387.438	−	797.152i	−0.502514	−	1.03392i
$772$		0			0
$773$		−	151.914i		−	0.196525i	−0.995161	−	0.0982625i	$-0.968672\pi$
$773$		−	151.914i		−	0.196525i	0.995161	−	0.0982625i	$-0.0313285\pi$
$774$		0			0
$775$	−685.649	+	495.458i	−0.884709	+	0.639301i
$776$		0			0
$777$	492.670	−	239.451i	0.634067	−	0.308174i
$778$		0			0
$779$	−978.290			−1.25583
$780$		0			0
$781$			211.689i			0.271048i
$782$		0			0
$783$	−728.075	−	157.905i	−0.929854	−	0.201667i
$784$		0			0
$785$	−183.365	−	566.825i	−0.233586	−	0.722070i
$786$		0			0
$787$	99.6831			0.126662			0.0633311	−	0.997993i	$-0.479828\pi$
$787$	99.6831			0.126662			0.0633311	+	0.997993i	$0.479828\pi$
$788$		0			0
$789$	315.721	+	649.595i	0.400154	+	0.823314i
$790$		0			0
$791$			207.463i			0.262279i
$792$		0			0
$793$			39.5662i			0.0498943i
$794$		0			0
$795$	−658.614	−	685.664i	−0.828445	−	0.862471i
$796$		0			0
$797$		−	72.9028i		−	0.0914715i	−0.998954	−	0.0457358i	$-0.985437\pi$
$797$		−	72.9028i		−	0.0914715i	0.998954	−	0.0457358i	$-0.0145632\pi$
$798$		0			0
$799$	305.543			0.382407
$800$		0			0
$801$	469.997	+	369.292i	0.586763	+	0.461039i
$802$		0			0
$803$			343.720i			0.428045i
$804$		0			0
$805$	1038.85	−	336.062i	1.29049	−	0.417468i
$806$		0			0
$807$	405.631	+	834.584i	0.502641	+	1.03418i
$808$		0			0
$809$		−	973.309i		−	1.20310i	−0.798835	−	0.601551i	$-0.794550\pi$
$809$		−	973.309i		−	1.20310i	0.798835	−	0.601551i	$-0.205450\pi$
$810$		0			0
$811$			215.941i			0.266266i	0.991098	+	0.133133i	$0.0425037\pi$
$811$			215.941i			0.266266i	−0.991098	+	0.133133i	$0.957496\pi$
$812$		0			0
$813$	390.508	+	803.468i	0.480330	+	0.988276i
$814$		0			0
$815$	−334.824	−	1035.02i	−0.410827	−	1.26996i
$816$		0			0
$817$		−	928.254i		−	1.13617i
$818$		0			0
$819$	1188.48	+	933.831i	1.45114	+	1.14021i
$820$		0			0
$821$	−626.689			−0.763324			−0.381662	−	0.924302i	$-0.624648\pi$
$821$	−626.689			−0.763324			−0.381662	+	0.924302i	$0.624648\pi$
$822$		0			0
$823$		−	153.375i		−	0.186361i	−0.995649	−	0.0931804i	$-0.970297\pi$
$823$		−	153.375i		−	0.186361i	0.995649	−	0.0931804i	$-0.0297033\pi$
$824$		0			0
$825$	−199.879	−	107.294i	−0.242278	−	0.130053i
$826$		0			0
$827$			44.0953i			0.0533196i	0.999645	+	0.0266598i	$0.00848709\pi$
$827$			44.0953i			0.0533196i	−0.999645	+	0.0266598i	$0.991513\pi$
$828$		0			0
$829$		−	333.433i		−	0.402211i	−0.979570	−	0.201105i	$-0.935547\pi$
$829$		−	333.433i		−	0.402211i	0.979570	−	0.201105i	$-0.0644533\pi$
$830$		0			0
$831$	88.6517	+	182.400i	0.106681	+	0.219495i
$832$		0			0
$833$	−2611.44			−3.13499
$834$		0			0
$835$	−440.467	−	1361.59i	−0.527506	−	1.63064i
$836$		0			0
$837$	892.845	+	193.641i	1.06672	+	0.231351i
$838$		0			0
$839$		−	1404.88i		−	1.67447i	−0.546840	−	0.837237i	$-0.684169\pi$
$839$		−	1404.88i		−	1.67447i	0.546840	−	0.837237i	$-0.315831\pi$
$840$		0			0
$841$	−79.6447			−0.0947024
$842$		0			0
$843$	800.122	−	388.882i	0.949137	−	0.461307i
$844$		0			0
$845$	−29.1832	−	90.2120i	−0.0345363	−	0.106760i
$846$		0			0
$847$		−	1370.12i		−	1.61762i
$848$		0			0
$849$	−247.161	−	508.532i	−0.291120	−	0.598977i
$850$		0			0
$851$	265.727			0.312253
$852$		0			0
$853$	−640.386			−0.750746			−0.375373	−	0.926874i	$-0.622485\pi$
$853$	−640.386			−0.750746			−0.375373	+	0.926874i	$0.622485\pi$
$854$		0			0
$855$	219.128	−	594.555i	0.256290	−	0.695386i
$856$		0			0
$857$	−529.587			−0.617955			−0.308978	−	0.951069i	$-0.599987\pi$
$857$	−529.587			−0.617955			−0.308978	+	0.951069i	$0.599987\pi$
$858$		0			0
$859$			1275.46i			1.48482i	0.669945	+	0.742410i	$0.266318\pi$
$859$			1275.46i			1.48482i	−0.669945	+	0.742410i	$0.733682\pi$
$860$		0			0
$861$	−1116.05	−	2296.27i	−1.29623	−	2.66699i
$862$		0			0
$863$	−180.285			−0.208905			−0.104453	−	0.994530i	$-0.533309\pi$
$863$	−180.285			−0.208905			−0.104453	+	0.994530i	$0.533309\pi$
$864$		0			0
$865$	858.176	−	277.616i	0.992111	−	0.320943i
$866$		0			0
$867$	−496.816	−	1022.20i	−0.573029	−	1.17900i
$868$		0			0
$869$	−332.543			−0.382674
$870$		0			0
$871$	946.346			1.08650
$872$		0			0
$873$	445.807	+	350.285i	0.510661	+	0.401243i
$874$		0			0
$875$	−904.796	+	1235.27i	−1.03405	+	1.41174i
$876$		0			0
$877$	1172.33			1.33675			0.668375	−	0.743825i	$-0.266990\pi$
$877$	1172.33			1.33675			0.668375	+	0.743825i	$0.266990\pi$
$878$		0			0
$879$	−146.207	+	71.0606i	−0.166333	+	0.0808426i
$880$		0			0
$881$			1361.82i			1.54576i	0.634551	+	0.772881i	$0.281185\pi$
$881$			1361.82i			1.54576i	−0.634551	+	0.772881i	$0.718815\pi$
$882$		0			0
$883$	−242.461			−0.274588			−0.137294	−	0.990530i	$-0.543841\pi$
$883$	−242.461			−0.274588			−0.137294	+	0.990530i	$0.543841\pi$
$884$		0			0
$885$	−309.417	+	297.210i	−0.349624	+	0.335831i
$886$		0			0
$887$	−768.568			−0.866481			−0.433240	−	0.901278i	$-0.642630\pi$
$887$	−768.568			−0.866481			−0.433240	+	0.901278i	$0.642630\pi$
$888$		0			0
$889$	724.949			0.815465
$890$		0			0
$891$	57.9608	+	238.050i	0.0650514	+	0.267171i
$892$		0			0
$893$		−	166.484i		−	0.186432i
$894$		0			0
$895$	104.186	+	322.065i	0.116409	+	0.359849i
$896$		0			0
$897$	320.511	+	659.450i	0.357315	+	0.735173i
$898$		0			0
$899$	−933.656			−1.03855
$900$		0			0
$901$			1637.98i			1.81796i
$902$		0			0
$903$	2178.83	−	1058.97i	2.41288	−	1.17273i
$904$		0			0
$905$	−1059.71	+	342.812i	−1.17095	+	0.378798i
$906$		0			0
$907$	390.266			0.430282			0.215141	−	0.976583i	$-0.430979\pi$
$907$	390.266			0.430282			0.215141	+	0.976583i	$0.430979\pi$
$908$		0			0
$909$	−715.614	+	910.760i	−0.787254	+	1.00194i
$910$		0			0
$911$		−	1007.79i		−	1.10624i	−0.833101	−	0.553121i	$-0.813437\pi$
$911$		−	1007.79i		−	1.10624i	0.833101	−	0.553121i	$-0.186563\pi$
$912$		0			0
$913$			49.7270i			0.0544655i
$914$		0			0
$915$	29.9880	+	31.2197i	0.0327738	+	0.0341199i
$916$		0			0
$917$			759.789i			0.828560i
$918$		0			0
$919$	86.2167			0.0938158			0.0469079	−	0.998899i	$-0.485063\pi$
$919$	86.2167			0.0938158			0.0469079	+	0.998899i	$0.485063\pi$
$920$		0			0
$921$	373.878	+	769.252i	0.405948	+	0.835236i
$922$		0			0
$923$			959.499i			1.03954i
$924$		0			0
$925$	−302.045	+	218.261i	−0.326535	+	0.235958i
$926$		0			0
$927$	−595.762	−	468.110i	−0.642677	−	0.504973i
$928$		0			0
$929$		−	950.507i		−	1.02315i	−0.859238	−	0.511575i	$-0.829062\pi$
$929$		−	950.507i		−	1.02315i	0.859238	−	0.511575i	$-0.170938\pi$
$930$		0			0
$931$			1422.92i			1.52838i
$932$		0			0
$933$	−744.910	+	362.047i	−0.798403	+	0.388046i
$934$		0			0
$935$	120.296	+	371.864i	0.128659	+	0.397716i
$936$		0			0
$937$			1522.58i			1.62495i	0.582993	+	0.812477i	$0.301882\pi$
$937$			1522.58i			1.62495i	−0.582993	+	0.812477i	$0.698118\pi$
$938$		0			0
$939$	−411.202	+	199.856i	−0.437915	+	0.212839i
$940$		0			0
$941$	1788.60			1.90075			0.950374	−	0.311111i	$-0.100701\pi$
$941$	1788.60			1.90075			0.950374	+	0.311111i	$0.100701\pi$
$942$		0			0
$943$		−	1238.52i		−	1.31339i
$944$		0			0
$945$	1645.54	−	163.937i	1.74132	−	0.173479i
$946$		0			0
$947$		−	195.826i		−	0.206786i	−0.994641	−	0.103393i	$-0.967030\pi$
$947$		−	195.826i		−	0.206786i	0.994641	−	0.103393i	$-0.0329699\pi$
$948$		0			0
$949$			1557.95i			1.64167i
$950$		0			0
$951$	−1373.24	+	667.432i	−1.44399	+	0.701822i
$952$		0			0
$953$	1532.78			1.60837			0.804186	−	0.594378i	$-0.202602\pi$
$953$	1532.78			1.60837			0.804186	+	0.594378i	$0.202602\pi$
$954$		0			0
$955$	1040.51	−	336.599i	1.08954	−	0.352460i
$956$		0			0
$957$	−109.450	−	225.193i	−0.114368	−	0.235312i
$958$		0			0
$959$			2601.42i			2.71263i
$960$		0			0
$961$	183.950			0.191415
$962$		0			0
$963$	−108.680	−	85.3933i	−0.112855	−	0.0886743i
$964$		0			0
$965$	−518.199	+	167.635i	−0.536994	+	0.173715i
$966$		0			0
$967$			232.608i			0.240546i	0.992741	+	0.120273i	$0.0383770\pi$
$967$			232.608i			0.240546i	−0.992741	+	0.120273i	$0.961623\pi$
$968$		0			0
$969$	−981.855	+	477.209i	−1.01327	+	0.492476i
$970$		0			0
$971$	−500.031			−0.514965			−0.257482	−	0.966283i	$-0.582893\pi$
$971$	−500.031			−0.514965			−0.257482	+	0.966283i	$0.582893\pi$
$972$		0			0
$973$	347.990			0.357646
$974$		0			0
$975$	−905.972	−	486.320i	−0.929202	−	0.498790i
$976$		0			0
$977$	−1764.27			−1.80580			−0.902900	−	0.429850i	$-0.858566\pi$
$977$	−1764.27			−1.80580			−0.902900	+	0.429850i	$0.858566\pi$
$978$		0			0
$979$			200.885i			0.205194i
$980$		0			0
$981$	−565.678	−	444.472i	−0.576634	−	0.453080i
$982$		0			0
$983$	−601.403			−0.611804			−0.305902	−	0.952063i	$-0.598958\pi$
$983$	−601.403			−0.611804			−0.305902	+	0.952063i	$0.598958\pi$
$984$		0			0
$985$	−414.059	+	133.946i	−0.420365	+	0.135986i
$986$		0			0
$987$	390.776	−	189.928i	0.395923	−	0.192430i
$988$		0			0
$989$	1175.18			1.18825
$990$		0			0
$991$	1824.90			1.84148			0.920738	−	0.390181i	$-0.127588\pi$
$991$	1824.90			1.84148			0.920738	+	0.390181i	$0.127588\pi$
$992$		0			0
$993$	732.685	−	356.106i	0.737850	−	0.358616i
$994$		0			0
$995$	117.132	+	362.081i	0.117720	+	0.363901i
$996$		0			0
$997$	−1345.70			−1.34975			−0.674877	−	0.737930i	$-0.735803\pi$
$997$	−1345.70			−1.34975			−0.674877	+	0.737930i	$0.735803\pi$
$998$		0			0
$999$	393.320	+	85.3034i	0.393713	+	0.0853888i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.i.b.929.17	✓	64
3.2	odd	2	inner	960.3.i.b.929.24	yes	64
4.3	odd	2	inner	960.3.i.b.929.45	yes	64
5.4	even	2	inner	960.3.i.b.929.46	yes	64
8.3	odd	2	inner	960.3.i.b.929.20	yes	64
8.5	even	2	inner	960.3.i.b.929.48	yes	64
12.11	even	2	inner	960.3.i.b.929.44	yes	64
15.14	odd	2	inner	960.3.i.b.929.43	yes	64
20.19	odd	2	inner	960.3.i.b.929.18	yes	64
24.5	odd	2	inner	960.3.i.b.929.41	yes	64
24.11	even	2	inner	960.3.i.b.929.21	yes	64
40.19	odd	2	inner	960.3.i.b.929.47	yes	64
40.29	even	2	inner	960.3.i.b.929.19	yes	64
60.59	even	2	inner	960.3.i.b.929.23	yes	64
120.29	odd	2	inner	960.3.i.b.929.22	yes	64
120.59	even	2	inner	960.3.i.b.929.42	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.3.i.b.929.17	✓	64	1.1	even	1	trivial
960.3.i.b.929.18	yes	64	20.19	odd	2	inner
960.3.i.b.929.19	yes	64	40.29	even	2	inner
960.3.i.b.929.20	yes	64	8.3	odd	2	inner
960.3.i.b.929.21	yes	64	24.11	even	2	inner
960.3.i.b.929.22	yes	64	120.29	odd	2	inner
960.3.i.b.929.23	yes	64	60.59	even	2	inner
960.3.i.b.929.24	yes	64	3.2	odd	2	inner
960.3.i.b.929.41	yes	64	24.5	odd	2	inner
960.3.i.b.929.42	yes	64	120.59	even	2	inner
960.3.i.b.929.43	yes	64	15.14	odd	2	inner
960.3.i.b.929.44	yes	64	12.11	even	2	inner
960.3.i.b.929.45	yes	64	4.3	odd	2	inner
960.3.i.b.929.46	yes	64	5.4	even	2	inner
960.3.i.b.929.47	yes	64	40.19	odd	2	inner
960.3.i.b.929.48	yes	64	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.31140 - 2.69819i) q^{3} +(-1.53895 - 4.75727i) q^{5} +12.2496i q^{7} +(-5.56048 + 7.07680i) q^{9} +O(q^{10})\)	\(q+(-1.31140 - 2.69819i) q^{3} +(-1.53895 - 4.75727i) q^{5} +12.2496i q^{7} +(-5.56048 + 7.07680i) q^{9} -3.02475 q^{11} -13.7100 q^{13} +(-10.8178 + 10.3911i) q^{15} +25.8427 q^{17} -14.0811i q^{19} +(33.0516 - 16.0640i) q^{21} +17.8268 q^{23} +(-20.2632 + 14.6424i) q^{25} +(26.3866 + 5.72273i) q^{27} -27.5927 q^{29} +33.8371 q^{31} +(3.96664 + 8.16134i) q^{33} +(58.2744 - 18.8515i) q^{35} +14.9061 q^{37} +(17.9792 + 36.9921i) q^{39} -69.4754i q^{41} +65.9219 q^{43} +(42.2236 + 15.5618i) q^{45} +11.8232 q^{47} -101.052 q^{49} +(-33.8900 - 69.7285i) q^{51} +63.3827i q^{53} +(4.65495 + 14.3895i) q^{55} +(-37.9935 + 18.4659i) q^{57} +28.6025 q^{59} -2.88594i q^{61} +(-86.6876 - 68.1133i) q^{63} +(21.0990 + 65.2220i) q^{65} -69.0261 q^{67} +(-23.3780 - 48.1001i) q^{69} -69.9856i q^{71} -113.636i q^{73} +(66.0813 + 35.4720i) q^{75} -37.0518i q^{77} +109.941 q^{79} +(-19.1622 - 78.7008i) q^{81} -16.4401i q^{83} +(-39.7707 - 122.941i) q^{85} +(36.1849 + 74.4503i) q^{87} -66.4138i q^{89} -167.941i q^{91} +(-44.3739 - 91.2990i) q^{93} +(-66.9876 + 21.6702i) q^{95} -62.9956i q^{97} +(16.8190 - 21.4055i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{9}+O(q^{10}) \) 64 * q - 32 * q^9	\( 64 q - 32 q^{9} - 32 q^{25} - 320 q^{49} + 448 q^{81}+O(q^{100}) \) 64 * q - 32 * q^9 - 32 * q^25 - 320 * q^49 + 448 * q^81

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