LMFDB - Embedded newform 384.3.e.c.257.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(257,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.257");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	384.e (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:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 99x^{4} + 170x^{2} + 81 $ x^8 + 18x^6 + 99x^4 + 170*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.4
Root			$-2.98985i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.3.e.c.257.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.246559 + 2.98985i) q^{3} +6.63641i q^{5} -0.578158 q^{7} +(-8.87842 + 1.47435i) q^{9} +O(q^{10})$	\(q+(0.246559 + 2.98985i) q^{3} +6.63641i q^{5} -0.578158 q^{7} +(-8.87842 + 1.47435i) q^{9} +8.68786i q^{11} -17.9269 q^{13} +(-19.8419 + 1.63626i) q^{15} -19.0110i q^{17} +32.1769 q^{19} +(-0.142550 - 1.72861i) q^{21} +20.4836i q^{23} -19.0419 q^{25} +(-6.59713 - 26.1816i) q^{27} -22.0310i q^{29} -26.2344 q^{31} +(-25.9754 + 2.14207i) q^{33} -3.83689i q^{35} -53.3855 q^{37} +(-4.42003 - 53.5988i) q^{39} +35.6935i q^{41} +50.4895 q^{43} +(-9.78437 - 58.9208i) q^{45} -30.6265i q^{47} -48.6657 q^{49} +(56.8401 - 4.68732i) q^{51} +88.8962i q^{53} -57.6562 q^{55} +(7.93348 + 96.2040i) q^{57} +63.1939i q^{59} +33.5317 q^{61} +(5.13313 - 0.852405i) q^{63} -118.970i q^{65} -108.562 q^{67} +(-61.2430 + 5.05041i) q^{69} +59.3600i q^{71} -5.60477 q^{73} +(-4.69495 - 56.9325i) q^{75} -5.02296i q^{77} +78.9955 q^{79} +(76.6526 - 26.1797i) q^{81} +48.5283i q^{83} +126.165 q^{85} +(65.8694 - 5.43193i) q^{87} -58.7109i q^{89} +10.3646 q^{91} +(-6.46831 - 78.4369i) q^{93} +213.539i q^{95} +93.3544 q^{97} +(-12.8089 - 77.1345i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 8 q^{7}+O(q^{10}) $ 8 * q + 4 * q^3 - 8 * q^7	$ 8 q + 4 q^{3} - 8 q^{7} - 16 q^{15} - 24 q^{19} - 16 q^{21} - 40 q^{25} - 44 q^{27} + 56 q^{31} + 8 q^{33} - 32 q^{37} + 104 q^{39} + 136 q^{43} + 80 q^{45} + 72 q^{49} + 176 q^{51} - 192 q^{55} - 40 q^{57} + 160 q^{61} - 264 q^{63} - 280 q^{67} - 80 q^{69} - 80 q^{73} - 348 q^{75} + 408 q^{79} + 72 q^{81} - 192 q^{85} + 368 q^{87} + 336 q^{91} - 160 q^{93} + 96 q^{97} + 432 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 8 * q^7 - 16 * q^15 - 24 * q^19 - 16 * q^21 - 40 * q^25 - 44 * q^27 + 56 * q^31 + 8 * q^33 - 32 * q^37 + 104 * q^39 + 136 * q^43 + 80 * q^45 + 72 * q^49 + 176 * q^51 - 192 * q^55 - 40 * q^57 + 160 * q^61 - 264 * q^63 - 280 * q^67 - 80 * q^69 - 80 * q^73 - 348 * q^75 + 408 * q^79 + 72 * q^81 - 192 * q^85 + 368 * q^87 + 336 * q^91 - 160 * q^93 + 96 * q^97 + 432 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0.246559	+	2.98985i	0.0821862	+	0.996617i
$3$	0.246559	+	2.98985i	0.0821862	+	0.996617i
$4$		0			0
$5$			6.63641i			1.32728i	0.748051	+	0.663641i	$0.230990\pi$
$5$			6.63641i			1.32728i	−0.748051	+	0.663641i	$0.769010\pi$
$6$		0			0
$7$	−0.578158			−0.0825940			−0.0412970	−	0.999147i	$-0.513149\pi$
$7$	−0.578158			−0.0825940			−0.0412970	+	0.999147i	$0.513149\pi$
$8$		0			0
$9$	−8.87842	+	1.47435i	−0.986491	+	0.163816i
$10$		0			0
$11$			8.68786i			0.789806i	0.918723	+	0.394903i	$0.129222\pi$
$11$			8.68786i			0.789806i	−0.918723	+	0.394903i	$0.870778\pi$
$12$		0			0
$13$	−17.9269			−1.37899			−0.689497	−	0.724289i	$-0.742168\pi$
$13$	−17.9269			−1.37899			−0.689497	+	0.724289i	$0.742168\pi$
$14$		0			0
$15$	−19.8419	+	1.63626i	−1.32279	+	0.109084i
$16$		0			0
$17$		−	19.0110i		−	1.11829i	−0.829068	−	0.559147i	$-0.811129\pi$
$17$		−	19.0110i		−	1.11829i	0.829068	−	0.559147i	$-0.188871\pi$
$18$		0			0
$19$	32.1769			1.69352			0.846760	−	0.531976i	$-0.178550\pi$
$19$	32.1769			1.69352			0.846760	+	0.531976i	$0.178550\pi$
$20$		0			0
$21$	−0.142550	−	1.72861i	−0.00678808	−	0.0823146i
$22$		0			0
$23$			20.4836i			0.890592i	0.895383	+	0.445296i	$0.146902\pi$
$23$			20.4836i			0.890592i	−0.895383	+	0.445296i	$0.853098\pi$
$24$		0			0
$25$	−19.0419			−0.761677
$26$		0			0
$27$	−6.59713	−	26.1816i	−0.244338	−	0.969690i
$28$		0			0
$29$		−	22.0310i		−	0.759690i	−0.925050	−	0.379845i	$-0.875977\pi$
$29$		−	22.0310i		−	0.759690i	0.925050	−	0.379845i	$-0.124023\pi$
$30$		0			0
$31$	−26.2344			−0.846270			−0.423135	−	0.906067i	$-0.639070\pi$
$31$	−26.2344			−0.846270			−0.423135	+	0.906067i	$0.639070\pi$
$32$		0			0
$33$	−25.9754	+	2.14207i	−0.787134	+	0.0649111i
$34$		0			0
$35$		−	3.83689i		−	0.109625i
$36$		0			0
$37$	−53.3855			−1.44285			−0.721426	−	0.692492i	$-0.756513\pi$
$37$	−53.3855			−1.44285			−0.721426	+	0.692492i	$0.756513\pi$
$38$		0			0
$39$	−4.42003	−	53.5988i	−0.113334	−	1.37433i
$40$		0			0
$41$			35.6935i			0.870574i	0.900292	+	0.435287i	$0.143353\pi$
$41$			35.6935i			0.870574i	−0.900292	+	0.435287i	$0.856647\pi$
$42$		0			0
$43$	50.4895			1.17417			0.587087	−	0.809524i	$-0.300275\pi$
$43$	50.4895			1.17417			0.587087	+	0.809524i	$0.300275\pi$
$44$		0			0
$45$	−9.78437	−	58.9208i	−0.217430	−	1.30935i
$46$		0			0
$47$		−	30.6265i		−	0.651628i	−0.945434	−	0.325814i	$-0.894362\pi$
$47$		−	30.6265i		−	0.651628i	0.945434	−	0.325814i	$-0.105638\pi$
$48$		0			0
$49$	−48.6657			−0.993178
$50$		0			0
$51$	56.8401	−	4.68732i	1.11451	−	0.0919083i
$52$		0			0
$53$			88.8962i			1.67729i	0.544681	+	0.838644i	$0.316651\pi$
$53$			88.8962i			1.67729i	−0.544681	+	0.838644i	$0.683349\pi$
$54$		0			0
$55$	−57.6562			−1.04829
$56$		0			0
$57$	7.93348	+	96.2040i	0.139184	+	1.68779i
$58$		0			0
$59$			63.1939i			1.07108i	0.844509	+	0.535542i	$0.179893\pi$
$59$			63.1939i			1.07108i	−0.844509	+	0.535542i	$0.820107\pi$
$60$		0			0
$61$	33.5317			0.549700			0.274850	−	0.961487i	$-0.411372\pi$
$61$	33.5317			0.549700			0.274850	+	0.961487i	$0.411372\pi$
$62$		0			0
$63$	5.13313	−	0.852405i	0.0814782	−	0.0135302i
$64$		0			0
$65$		−	118.970i		−	1.83031i
$66$		0			0
$67$	−108.562			−1.62032			−0.810162	−	0.586206i	$-0.800621\pi$
$67$	−108.562			−1.62032			−0.810162	+	0.586206i	$0.800621\pi$
$68$		0			0
$69$	−61.2430	+	5.05041i	−0.887579	+	0.0731944i
$70$		0			0
$71$			59.3600i			0.836056i	0.908434	+	0.418028i	$0.137279\pi$
$71$			59.3600i			0.836056i	−0.908434	+	0.418028i	$0.862721\pi$
$72$		0			0
$73$	−5.60477			−0.0767777			−0.0383889	−	0.999263i	$-0.512223\pi$
$73$	−5.60477			−0.0767777			−0.0383889	+	0.999263i	$0.512223\pi$
$74$		0			0
$75$	−4.69495	−	56.9325i	−0.0625994	−	0.759101i
$76$		0			0
$77$		−	5.02296i		−	0.0652332i
$78$		0			0
$79$	78.9955			0.999943			0.499971	−	0.866042i	$-0.333344\pi$
$79$	78.9955			0.999943			0.499971	+	0.866042i	$0.333344\pi$
$80$		0			0
$81$	76.6526	−	26.1797i	0.946328	−	0.323207i
$82$		0			0
$83$			48.5283i			0.584679i	0.956315	+	0.292339i	$0.0944337\pi$
$83$			48.5283i			0.584679i	−0.956315	+	0.292339i	$0.905566\pi$
$84$		0			0
$85$	126.165			1.48429
$86$		0			0
$87$	65.8694	−	5.43193i	0.757120	−	0.0624360i
$88$		0			0
$89$		−	58.7109i		−	0.659672i	−0.944038	−	0.329836i	$-0.893006\pi$
$89$		−	58.7109i		−	0.659672i	0.944038	−	0.329836i	$-0.106994\pi$
$90$		0			0
$91$	10.3646			0.113897
$92$		0			0
$93$	−6.46831	−	78.4369i	−0.0695517	−	0.843407i
$94$		0			0
$95$			213.539i			2.24778i
$96$		0			0
$97$	93.3544			0.962416			0.481208	−	0.876606i	$-0.340198\pi$
$97$	93.3544			0.962416			0.481208	+	0.876606i	$0.340198\pi$
$98$		0			0
$99$	−12.8089	−	77.1345i	−0.129383	−	0.779136i
$100$		0			0
$101$			114.161i			1.13031i	0.824985	+	0.565155i	$0.191184\pi$
$101$			114.161i			1.13031i	−0.824985	+	0.565155i	$0.808816\pi$
$102$		0			0
$103$	−70.0355			−0.679956			−0.339978	−	0.940433i	$-0.610420\pi$
$103$	−70.0355			−0.679956			−0.339978	+	0.940433i	$0.610420\pi$
$104$		0			0
$105$	11.4717	−	0.946018i	0.109255	−	0.00900970i
$106$		0			0
$107$			50.9072i			0.475769i	0.971293	+	0.237884i	$0.0764539\pi$
$107$			50.9072i			0.475769i	−0.971293	+	0.237884i	$0.923546\pi$
$108$		0			0
$109$	−13.2178			−0.121264			−0.0606320	−	0.998160i	$-0.519312\pi$
$109$	−13.2178			−0.121264			−0.0606320	+	0.998160i	$0.519312\pi$
$110$		0			0
$111$	−13.1627	−	159.615i	−0.118582	−	1.43797i
$112$		0			0
$113$		−	62.9470i		−	0.557053i	−0.960429	−	0.278527i	$-0.910154\pi$
$113$		−	62.9470i		−	0.557053i	0.960429	−	0.278527i	$-0.0898460\pi$
$114$		0			0
$115$	−135.938			−1.18207
$116$		0			0
$117$	159.163	−	26.4305i	1.36036	−	0.225902i
$118$		0			0
$119$			10.9914i			0.0923644i
$120$		0			0
$121$	45.5210			0.376207
$122$		0			0
$123$	−106.718	+	8.80054i	−0.867629	+	0.0715491i
$124$		0			0
$125$			39.5402i			0.316321i
$126$		0			0
$127$	236.659			1.86345			0.931727	−	0.363160i	$-0.118302\pi$
$127$	236.659			1.86345			0.931727	+	0.363160i	$0.118302\pi$
$128$		0			0
$129$	12.4486	+	150.956i	0.0965009	+	1.17020i
$130$		0			0
$131$			121.212i			0.925279i	0.886546	+	0.462640i	$0.153098\pi$
$131$			121.212i			0.925279i	−0.886546	+	0.462640i	$0.846902\pi$
$132$		0			0
$133$	−18.6033			−0.139875
$134$		0			0
$135$	173.752	−	43.7812i	1.28705	−	0.324305i
$136$		0			0
$137$			136.677i			0.997645i	0.866704	+	0.498822i	$0.166234\pi$
$137$			136.677i			0.997645i	−0.866704	+	0.498822i	$0.833766\pi$
$138$		0			0
$139$	−29.5616			−0.212673			−0.106337	−	0.994330i	$-0.533912\pi$
$139$	−29.5616			−0.212673			−0.106337	+	0.994330i	$0.533912\pi$
$140$		0			0
$141$	91.5687	−	7.55123i	0.649424	−	0.0535548i
$142$		0			0
$143$		−	155.747i		−	1.08914i
$144$		0			0
$145$	146.207			1.00832
$146$		0			0
$147$	−11.9990	−	145.503i	−0.0816255	−	0.989818i
$148$		0			0
$149$			17.5092i			0.117511i	0.998272	+	0.0587556i	$0.0187133\pi$
$149$			17.5092i			0.117511i	−0.998272	+	0.0587556i	$0.981287\pi$
$150$		0			0
$151$	119.607			0.792099			0.396050	−	0.918229i	$-0.370381\pi$
$151$	119.607			0.792099			0.396050	+	0.918229i	$0.370381\pi$
$152$		0			0
$153$	28.0288	+	168.788i	0.183195	+	1.10319i
$154$		0			0
$155$		−	174.102i		−	1.12324i
$156$		0			0
$157$	189.488			1.20693			0.603466	−	0.797388i	$-0.293786\pi$
$157$	189.488			1.20693			0.603466	+	0.797388i	$0.293786\pi$
$158$		0			0
$159$	−265.786	+	21.9181i	−1.67161	+	0.137850i
$160$		0			0
$161$		−	11.8428i		−	0.0735576i
$162$		0			0
$163$	104.654			0.642051			0.321026	−	0.947071i	$-0.395972\pi$
$163$	104.654			0.642051			0.321026	+	0.947071i	$0.395972\pi$
$164$		0			0
$165$	−14.2156	−	172.384i	−0.0861553	−	1.04475i
$166$		0			0
$167$		−	142.664i		−	0.854278i	−0.904186	−	0.427139i	$-0.859522\pi$
$167$		−	142.664i		−	0.854278i	0.904186	−	0.427139i	$-0.140478\pi$
$168$		0			0
$169$	152.374			0.901623
$170$		0			0
$171$	−285.680	+	47.4399i	−1.67064	+	0.277426i
$172$		0			0
$173$			11.6594i			0.0673952i	0.999432	+	0.0336976i	$0.0107283\pi$
$173$			11.6594i			0.0673952i	−0.999432	+	0.0336976i	$0.989272\pi$
$174$		0			0
$175$	11.0092			0.0629100
$176$		0			0
$177$	−188.940	+	15.5810i	−1.06746	+	0.0880283i
$178$		0			0
$179$			40.3979i			0.225687i	0.993613	+	0.112843i	$0.0359958\pi$
$179$			40.3979i			0.225687i	−0.993613	+	0.112843i	$0.964004\pi$
$180$		0			0
$181$	−121.175			−0.669472			−0.334736	−	0.942312i	$-0.608647\pi$
$181$	−121.175			−0.669472			−0.334736	+	0.942312i	$0.608647\pi$
$182$		0			0
$183$	8.26752	+	100.255i	0.0451777	+	0.547840i
$184$		0			0
$185$		−	354.288i		−	1.91507i
$186$		0			0
$187$	165.165			0.883235
$188$		0			0
$189$	3.81418	+	15.1371i	0.0201808	+	0.0800906i
$190$		0			0
$191$			337.287i			1.76590i	0.469465	+	0.882951i	$0.344447\pi$
$191$			337.287i			1.76590i	−0.469465	+	0.882951i	$0.655553\pi$
$192$		0			0
$193$	−227.937			−1.18102			−0.590511	−	0.807029i	$-0.701074\pi$
$193$	−227.937			−1.18102			−0.590511	+	0.807029i	$0.701074\pi$
$194$		0			0
$195$	355.704	−	29.3332i	1.82412	−	0.150426i
$196$		0			0
$197$		−	172.276i		−	0.874495i	−0.899341	−	0.437248i	$-0.855953\pi$
$197$		−	172.276i		−	0.874495i	0.899341	−	0.437248i	$-0.144047\pi$
$198$		0			0
$199$	−0.598837			−0.00300923			−0.00150461	−	0.999999i	$-0.500479\pi$
$199$	−0.598837			−0.00300923			−0.00150461	+	0.999999i	$0.500479\pi$
$200$		0			0
$201$	−26.7668	−	324.583i	−0.133168	−	1.61484i
$202$		0			0
$203$			12.7374i			0.0627458i
$204$		0			0
$205$	−236.877			−1.15550
$206$		0			0
$207$	−30.2000	−	181.862i	−0.145894	−	0.878561i
$208$		0			0
$209$			279.548i			1.33755i
$210$		0			0
$211$	−243.040			−1.15185			−0.575923	−	0.817504i	$-0.695357\pi$
$211$	−243.040			−1.15185			−0.575923	+	0.817504i	$0.695357\pi$
$212$		0			0
$213$	−177.478	+	14.6357i	−0.833228	+	0.0687123i
$214$		0			0
$215$			335.069i			1.55846i
$216$		0			0
$217$	15.1676			0.0698968
$218$		0			0
$219$	−1.38190	−	16.7574i	−0.00631007	−	0.0765180i
$220$		0			0
$221$			340.809i			1.54212i
$222$		0			0
$223$	−191.174			−0.857284			−0.428642	−	0.903474i	$-0.641008\pi$
$223$	−191.174			−0.857284			−0.428642	+	0.903474i	$0.641008\pi$
$224$		0			0
$225$	169.062	−	28.0744i	0.751388	−	0.124775i
$226$		0			0
$227$			8.76834i			0.0386271i	0.999813	+	0.0193135i	$0.00614807\pi$
$227$			8.76834i			0.0386271i	−0.999813	+	0.0193135i	$0.993852\pi$
$228$		0			0
$229$	131.740			0.575283			0.287641	−	0.957738i	$-0.407129\pi$
$229$	131.740			0.575283			0.287641	+	0.957738i	$0.407129\pi$
$230$		0			0
$231$	15.0179	−	1.23845i	0.0650125	−	0.00536127i
$232$		0			0
$233$		−	96.0149i		−	0.412081i	−0.978543	−	0.206040i	$-0.933942\pi$
$233$		−	96.0149i		−	0.412081i	0.978543	−	0.206040i	$-0.0660579\pi$
$234$		0			0
$235$	203.250			0.864894
$236$		0			0
$237$	19.4770	+	236.185i	0.0821815	+	0.996560i
$238$		0			0
$239$		−	317.978i		−	1.33045i	−0.746643	−	0.665225i	$-0.768336\pi$
$239$		−	317.978i		−	1.33045i	0.746643	−	0.665225i	$-0.231664\pi$
$240$		0			0
$241$	−114.189			−0.473812			−0.236906	−	0.971533i	$-0.576133\pi$
$241$	−114.189			−0.473812			−0.236906	+	0.971533i	$0.576133\pi$
$242$		0			0
$243$	97.1728	+	222.725i	0.399888	+	0.916564i
$244$		0			0
$245$		−	322.966i		−	1.31823i
$246$		0			0
$247$	−576.832			−2.33535
$248$		0			0
$249$	−145.093	+	11.9651i	−0.582701	+	0.0480525i
$250$		0			0
$251$			376.853i			1.50140i	0.660641	+	0.750702i	$0.270285\pi$
$251$			376.853i			1.50140i	−0.660641	+	0.750702i	$0.729715\pi$
$252$		0			0
$253$	−177.959			−0.703395
$254$		0			0
$255$	31.1070	+	377.214i	0.121988	+	1.47927i
$256$		0			0
$257$			152.041i			0.591599i	0.955250	+	0.295800i	$0.0955861\pi$
$257$			152.041i			0.591599i	−0.955250	+	0.295800i	$0.904414\pi$
$258$		0			0
$259$	30.8653			0.119171
$260$		0			0
$261$	32.4813	+	195.600i	0.124450	+	0.749427i
$262$		0			0
$263$		−	495.013i		−	1.88218i	−0.338158	−	0.941089i	$-0.609804\pi$
$263$		−	495.013i		−	1.88218i	0.338158	−	0.941089i	$-0.390196\pi$
$264$		0			0
$265$	−589.952			−2.22623
$266$		0			0
$267$	175.537	−	14.4757i	0.657441	−	0.0542160i
$268$		0			0
$269$			173.556i			0.645190i	0.946537	+	0.322595i	$0.104555\pi$
$269$			173.556i			0.645190i	−0.946537	+	0.322595i	$0.895445\pi$
$270$		0			0
$271$	125.415			0.462784			0.231392	−	0.972861i	$-0.425672\pi$
$271$	125.415			0.462784			0.231392	+	0.972861i	$0.425672\pi$
$272$		0			0
$273$	2.55548	+	30.9886i	0.00936072	+	0.113511i
$274$		0			0
$275$		−	165.434i		−	0.601577i
$276$		0			0
$277$	−226.300			−0.816967			−0.408483	−	0.912766i	$-0.633942\pi$
$277$	−226.300			−0.816967			−0.408483	+	0.912766i	$0.633942\pi$
$278$		0			0
$279$	232.920	−	38.6786i	0.834838	−	0.138633i
$280$		0			0
$281$		−	13.8834i		−	0.0494073i	−0.999695	−	0.0247036i	$-0.992136\pi$
$281$		−	13.8834i		−	0.0494073i	0.999695	−	0.0247036i	$-0.00786421\pi$
$282$		0			0
$283$	306.461			1.08290			0.541450	−	0.840733i	$-0.317875\pi$
$283$	306.461			1.08290			0.541450	+	0.840733i	$0.317875\pi$
$284$		0			0
$285$	−638.449	+	52.6498i	−2.24017	+	0.184736i
$286$		0			0
$287$		−	20.6365i		−	0.0719042i
$288$		0			0
$289$	−72.4183			−0.250582
$290$		0			0
$291$	23.0173	+	279.116i	0.0790973	+	0.959160i
$292$		0			0
$293$		−	121.869i		−	0.415935i	−0.978136	−	0.207968i	$-0.933315\pi$
$293$		−	121.869i		−	0.415935i	0.978136	−	0.207968i	$-0.0666848\pi$
$294$		0			0
$295$	−419.381			−1.42163
$296$		0			0
$297$	227.462	−	57.3149i	0.765867	−	0.192980i
$298$		0			0
$299$		−	367.208i		−	1.22812i
$300$		0			0
$301$	−29.1909			−0.0969797
$302$		0			0
$303$	−341.325	+	28.1474i	−1.12649	+	0.0928959i
$304$		0			0
$305$			222.530i			0.729607i
$306$		0			0
$307$	−19.0431			−0.0620298			−0.0310149	−	0.999519i	$-0.509874\pi$
$307$	−19.0431			−0.0620298			−0.0310149	+	0.999519i	$0.509874\pi$
$308$		0			0
$309$	−17.2678	−	209.396i	−0.0558830	−	0.677656i
$310$		0			0
$311$			223.159i			0.717552i	0.933424	+	0.358776i	$0.116806\pi$
$311$			223.159i			0.717552i	−0.933424	+	0.358776i	$0.883194\pi$
$312$		0			0
$313$	−194.708			−0.622070			−0.311035	−	0.950399i	$-0.600676\pi$
$313$	−194.708			−0.622070			−0.311035	+	0.950399i	$0.600676\pi$
$314$		0			0
$315$	5.65691	+	34.0655i	0.0179584	+	0.108145i
$316$		0			0
$317$		−	532.192i		−	1.67884i	−0.543483	−	0.839420i	$-0.682895\pi$
$317$		−	532.192i		−	1.67884i	0.543483	−	0.839420i	$-0.317105\pi$
$318$		0			0
$319$	191.402			0.600007
$320$		0			0
$321$	−152.205	+	12.5516i	−0.474159	+	0.0391016i
$322$		0			0
$323$		−	611.715i		−	1.89385i
$324$		0			0
$325$	341.363			1.05035
$326$		0			0
$327$	−3.25896	−	39.5192i	−0.00996623	−	0.120854i
$328$		0			0
$329$			17.7070i			0.0538206i
$330$		0			0
$331$	91.7273			0.277122			0.138561	−	0.990354i	$-0.455752\pi$
$331$	91.7273			0.277122			0.138561	+	0.990354i	$0.455752\pi$
$332$		0			0
$333$	473.979	−	78.7088i	1.42336	−	0.236363i
$334$		0			0
$335$		−	720.460i		−	2.15063i
$336$		0			0
$337$	82.1658			0.243815			0.121908	−	0.992541i	$-0.461099\pi$
$337$	82.1658			0.243815			0.121908	+	0.992541i	$0.461099\pi$
$338$		0			0
$339$	188.202	−	15.5201i	0.555169	−	0.0457821i
$340$		0			0
$341$		−	227.921i		−	0.668389i
$342$		0			0
$343$	56.4662			0.164625
$344$		0			0
$345$	−33.5166	−	406.433i	−0.0971496	−	1.17807i
$346$		0			0
$347$		−	323.122i		−	0.931189i	−0.884998	−	0.465594i	$-0.845841\pi$
$347$		−	323.122i		−	0.931189i	0.884998	−	0.465594i	$-0.154159\pi$
$348$		0			0
$349$	255.324			0.731586			0.365793	−	0.930696i	$-0.380798\pi$
$349$	255.324			0.731586			0.365793	+	0.930696i	$0.380798\pi$
$350$		0			0
$351$	118.266	+	469.356i	0.336941	+	1.33720i
$352$		0			0
$353$			344.332i			0.975444i	0.872999	+	0.487722i	$0.162172\pi$
$353$			344.332i			0.975444i	−0.872999	+	0.487722i	$0.837828\pi$
$354$		0			0
$355$	−393.937			−1.10968
$356$		0			0
$357$	−32.8625	+	2.71001i	−0.0920519	+	0.00759107i
$358$		0			0
$359$			119.962i			0.334157i	0.985944	+	0.167079i	$0.0534334\pi$
$359$			119.962i			0.334157i	−0.985944	+	0.167079i	$0.946567\pi$
$360$		0			0
$361$	674.351			1.86801
$362$		0			0
$363$	11.2236	+	136.101i	0.0309190	+	0.374934i
$364$		0			0
$365$		−	37.1956i		−	0.101906i
$366$		0			0
$367$	398.726			1.08645			0.543223	−	0.839589i	$-0.317204\pi$
$367$	398.726			1.08645			0.543223	+	0.839589i	$0.317204\pi$
$368$		0			0
$369$	−52.6246	−	316.902i	−0.142614	−	0.858813i
$370$		0			0
$371$		−	51.3960i		−	0.138534i
$372$		0			0
$373$	−606.592			−1.62625			−0.813126	−	0.582088i	$-0.802236\pi$
$373$	−606.592			−1.62625			−0.813126	+	0.582088i	$0.802236\pi$
$374$		0			0
$375$	−118.219	+	9.74896i	−0.315251	+	0.0259972i
$376$		0			0
$377$			394.948i			1.04761i
$378$		0			0
$379$	−266.492			−0.703146			−0.351573	−	0.936161i	$-0.614353\pi$
$379$	−266.492			−0.703146			−0.351573	+	0.936161i	$0.614353\pi$
$380$		0			0
$381$	58.3502	+	707.574i	0.153150	+	1.85715i
$382$		0			0
$383$			406.790i			1.06212i	0.847336	+	0.531058i	$0.178205\pi$
$383$			406.790i			1.06212i	−0.847336	+	0.531058i	$0.821795\pi$
$384$		0			0
$385$	33.3344			0.0865828
$386$		0			0
$387$	−448.267	+	74.4390i	−1.15831	+	0.192349i
$388$		0			0
$389$		−	217.748i		−	0.559763i	−0.960035	−	0.279881i	$-0.909705\pi$
$389$		−	217.748i		−	0.559763i	0.960035	−	0.279881i	$-0.0902951\pi$
$390$		0			0
$391$	389.414			0.995944
$392$		0			0
$393$	−362.405	+	29.8858i	−0.922149	+	0.0760452i
$394$		0			0
$395$			524.246i			1.32721i
$396$		0			0
$397$	−53.9651			−0.135932			−0.0679661	−	0.997688i	$-0.521651\pi$
$397$	−53.9651			−0.135932			−0.0679661	+	0.997688i	$0.521651\pi$
$398$		0			0
$399$	−4.58680	−	55.6211i	−0.0114958	−	0.139401i
$400$		0			0
$401$		−	294.291i		−	0.733893i	−0.930242	−	0.366946i	$-0.880403\pi$
$401$		−	294.291i		−	0.733893i	0.930242	−	0.366946i	$-0.119597\pi$
$402$		0			0
$403$	470.302			1.16700
$404$		0			0
$405$	173.739	+	508.698i	0.428986	+	1.25604i
$406$		0			0
$407$		−	463.806i		−	1.13957i
$408$		0			0
$409$	569.204			1.39170			0.695849	−	0.718188i	$-0.255028\pi$
$409$	569.204			1.39170			0.695849	+	0.718188i	$0.255028\pi$
$410$		0			0
$411$	−408.645	+	33.6990i	−0.994270	+	0.0819926i
$412$		0			0
$413$		−	36.5361i		−	0.0884651i
$414$		0			0
$415$	−322.054			−0.776034
$416$		0			0
$417$	−7.28866	−	88.3847i	−0.0174788	−	0.211954i
$418$		0			0
$419$			590.728i			1.40985i	0.709281	+	0.704926i	$0.249020\pi$
$419$			590.728i			1.40985i	−0.709281	+	0.704926i	$0.750980\pi$
$420$		0			0
$421$	475.989			1.13061			0.565307	−	0.824880i	$-0.308758\pi$
$421$	475.989			1.13061			0.565307	+	0.824880i	$0.308758\pi$
$422$		0			0
$423$	45.1541	+	271.915i	0.106747	+	0.642825i
$424$		0			0
$425$			362.006i			0.851780i
$426$		0			0
$427$	−19.3866			−0.0454019
$428$		0			0
$429$	465.659	−	38.4007i	1.08545	−	0.0895120i
$430$		0			0
$431$		−	341.539i		−	0.792435i	−0.918157	−	0.396217i	$-0.870323\pi$
$431$		−	341.539i		−	0.792435i	0.918157	−	0.396217i	$-0.129677\pi$
$432$		0			0
$433$	−88.2258			−0.203755			−0.101877	−	0.994797i	$-0.532485\pi$
$433$	−88.2258			−0.203755			−0.101877	+	0.994797i	$0.532485\pi$
$434$		0			0
$435$	36.0485	+	437.136i	0.0828702	+	1.00491i
$436$		0			0
$437$			659.099i			1.50824i
$438$		0			0
$439$	735.647			1.67573			0.837867	−	0.545875i	$-0.183803\pi$
$439$	735.647			1.67573			0.837867	+	0.545875i	$0.183803\pi$
$440$		0			0
$441$	432.075	−	71.7502i	0.979761	−	0.162699i
$442$		0			0
$443$		−	333.798i		−	0.753494i	−0.926316	−	0.376747i	$-0.877043\pi$
$443$		−	333.798i		−	0.753494i	0.926316	−	0.376747i	$-0.122957\pi$
$444$		0			0
$445$	389.629			0.875571
$446$		0			0
$447$	−52.3498	+	4.31703i	−0.117114	+	0.00965779i
$448$		0			0
$449$		−	396.485i		−	0.883039i	−0.897252	−	0.441520i	$-0.854440\pi$
$449$		−	396.485i		−	0.883039i	0.897252	−	0.441520i	$-0.145560\pi$
$450$		0			0
$451$	−310.100			−0.687584
$452$		0			0
$453$	29.4901	+	357.607i	0.0650996	+	0.789419i
$454$		0			0
$455$			68.7836i			0.151173i
$456$		0			0
$457$	34.5362			0.0755715			0.0377857	−	0.999286i	$-0.487970\pi$
$457$	34.5362			0.0755715			0.0377857	+	0.999286i	$0.487970\pi$
$458$		0			0
$459$	−497.739	+	125.418i	−1.08440	+	0.273242i
$460$		0			0
$461$		−	324.050i		−	0.702929i	−0.936201	−	0.351464i	$-0.885684\pi$
$461$		−	324.050i		−	0.702929i	0.936201	−	0.351464i	$-0.114316\pi$
$462$		0			0
$463$	−6.33727			−0.0136874			−0.00684371	−	0.999977i	$-0.502178\pi$
$463$	−6.33727			−0.0136874			−0.00684371	+	0.999977i	$0.502178\pi$
$464$		0			0
$465$	520.539	−	42.9264i	1.11944	−	0.0923147i
$466$		0			0
$467$		−	450.706i		−	0.965109i	−0.875866	−	0.482554i	$-0.839709\pi$
$467$		−	450.706i		−	0.965109i	0.875866	−	0.482554i	$-0.160291\pi$
$468$		0			0
$469$	62.7658			0.133829
$470$		0			0
$471$	46.7200	+	566.542i	0.0991932	+	1.20285i
$472$		0			0
$473$			438.646i			0.927370i
$474$		0			0
$475$	−612.710			−1.28992
$476$		0			0
$477$	−131.064	−	789.258i	−0.274767	−	1.65463i
$478$		0			0
$479$			259.094i			0.540906i	0.962733	+	0.270453i	$0.0871735\pi$
$479$			259.094i			0.540906i	−0.962733	+	0.270453i	$0.912827\pi$
$480$		0			0
$481$	957.038			1.98968
$482$		0			0
$483$	35.4081	−	2.91993i	0.0733087	−	0.00604541i
$484$		0			0
$485$			619.538i			1.27740i
$486$		0			0
$487$	628.526			1.29061			0.645304	−	0.763926i	$-0.276731\pi$
$487$	628.526			1.29061			0.645304	+	0.763926i	$0.276731\pi$
$488$		0			0
$489$	25.8034	+	312.901i	0.0527677	+	0.639879i
$490$		0			0
$491$		−	352.334i		−	0.717585i	−0.933417	−	0.358792i	$-0.883189\pi$
$491$		−	352.334i		−	0.717585i	0.933417	−	0.358792i	$-0.116811\pi$
$492$		0			0
$493$	−418.831			−0.849557
$494$		0			0
$495$	511.896	−	85.0052i	1.03413	−	0.171728i
$496$		0			0
$497$		−	34.3195i		−	0.0690532i
$498$		0			0
$499$	240.576			0.482116			0.241058	−	0.970511i	$-0.422506\pi$
$499$	240.576			0.482116			0.241058	+	0.970511i	$0.422506\pi$
$500$		0			0
$501$	426.545	−	35.1751i	0.851388	−	0.0702098i
$502$		0			0
$503$		−	13.1648i		−	0.0261725i	−0.999914	−	0.0130862i	$-0.995834\pi$
$503$		−	13.1648i		−	0.0261725i	0.999914	−	0.0130862i	$-0.00416560\pi$
$504$		0			0
$505$	−757.621			−1.50024
$506$		0			0
$507$	37.5692	+	455.577i	0.0741010	+	0.898573i
$508$		0			0
$509$		−	468.752i		−	0.920927i	−0.887679	−	0.460464i	$-0.847683\pi$
$509$		−	468.752i		−	0.920927i	0.887679	−	0.460464i	$-0.152317\pi$
$510$		0			0
$511$	3.24044			0.00634138
$512$		0			0
$513$	−212.275	−	842.443i	−0.413791	−	1.64219i
$514$		0			0
$515$		−	464.784i		−	0.902494i
$516$		0			0
$517$	266.079			0.514660
$518$		0			0
$519$	−34.8598	+	2.87472i	−0.0671672	+	0.00553895i
$520$		0			0
$521$		−	456.517i		−	0.876232i	−0.898918	−	0.438116i	$-0.855646\pi$
$521$		−	456.517i		−	0.876232i	0.898918	−	0.438116i	$-0.144354\pi$
$522$		0			0
$523$	−287.095			−0.548939			−0.274469	−	0.961596i	$-0.588502\pi$
$523$	−287.095			−0.548939			−0.274469	+	0.961596i	$0.588502\pi$
$524$		0			0
$525$	2.71442	+	32.9160i	0.00517033	+	0.0626971i
$526$		0			0
$527$			498.742i			0.946379i
$528$		0			0
$529$	109.421			0.206845
$530$		0			0
$531$	−93.1698	−	561.062i	−0.175461	−	1.05661i
$532$		0			0
$533$		−	639.875i		−	1.20052i
$534$		0			0
$535$	−337.841			−0.631479
$536$		0			0
$537$	−120.784	+	9.96046i	−0.224923	+	0.0185483i
$538$		0			0
$539$		−	422.801i		−	0.784418i
$540$		0			0
$541$	804.779			1.48758			0.743788	−	0.668416i	$-0.233027\pi$
$541$	804.779			1.48758			0.743788	+	0.668416i	$0.233027\pi$
$542$		0			0
$543$	−29.8766	−	362.294i	−0.0550214	−	0.667208i
$544$		0			0
$545$		−	87.7186i		−	0.160952i
$546$		0			0
$547$	−76.3034			−0.139494			−0.0697471	−	0.997565i	$-0.522219\pi$
$547$	−76.3034			−0.139494			−0.0697471	+	0.997565i	$0.522219\pi$
$548$		0			0
$549$	−297.708	+	49.4373i	−0.542274	+	0.0900498i
$550$		0			0
$551$		−	708.889i		−	1.28655i
$552$		0			0
$553$	−45.6718			−0.0825892
$554$		0			0
$555$	1059.27	−	87.3528i	1.90859	−	0.157392i
$556$		0			0
$557$		−	540.544i		−	0.970455i	−0.874388	−	0.485228i	$-0.838737\pi$
$557$		−	540.544i		−	0.970455i	0.874388	−	0.485228i	$-0.161263\pi$
$558$		0			0
$559$	−905.121			−1.61918
$560$		0			0
$561$	40.7228	+	493.819i	0.0725897	+	0.880247i
$562$		0			0
$563$			1047.65i			1.86084i	0.366491	+	0.930422i	$0.380559\pi$
$563$			1047.65i			1.86084i	−0.366491	+	0.930422i	$0.619441\pi$
$564$		0			0
$565$	417.742			0.739366
$566$		0			0
$567$	−44.3173	+	15.1360i	−0.0781610	+	0.0266949i
$568$		0			0
$569$			958.593i			1.68470i	0.538933	+	0.842348i	$0.318827\pi$
$569$			958.593i			1.68470i	−0.538933	+	0.842348i	$0.681173\pi$
$570$		0			0
$571$	118.951			0.208321			0.104161	−	0.994560i	$-0.466784\pi$
$571$	118.951			0.208321			0.104161	+	0.994560i	$0.466784\pi$
$572$		0			0
$573$	−1008.44	+	83.1611i	−1.75993	+	0.145133i
$574$		0			0
$575$		−	390.048i		−	0.678344i
$576$		0			0
$577$	−355.825			−0.616681			−0.308340	−	0.951276i	$-0.599774\pi$
$577$	−355.825			−0.616681			−0.308340	+	0.951276i	$0.599774\pi$
$578$		0			0
$579$	−56.1999	−	681.499i	−0.0970637	−	1.17703i
$580$		0			0
$581$		−	28.0570i		−	0.0482910i
$582$		0			0
$583$	−772.318			−1.32473
$584$		0			0
$585$	175.404	+	1056.27i	0.299835	+	1.80559i
$586$		0			0
$587$		−	571.017i		−	0.972771i	−0.873744	−	0.486386i	$-0.838315\pi$
$587$		−	571.017i		−	0.972771i	0.873744	−	0.486386i	$-0.161685\pi$
$588$		0			0
$589$	−844.140			−1.43318
$590$		0			0
$591$	515.078	−	42.4760i	0.871537	−	0.0718714i
$592$		0			0
$593$		−	370.411i		−	0.624640i	−0.949977	−	0.312320i	$-0.898894\pi$
$593$		−	370.411i		−	0.624640i	0.949977	−	0.312320i	$-0.101106\pi$
$594$		0			0
$595$	−72.9432			−0.122594
$596$		0			0
$597$	−0.147648	−	1.79043i	−0.000247317	−	0.00299905i
$598$		0			0
$599$			68.6579i			0.114621i	0.998356	+	0.0573104i	$0.0182525\pi$
$599$			68.6579i			0.114621i	−0.998356	+	0.0573104i	$0.981748\pi$
$600$		0			0
$601$	9.39898			0.0156389			0.00781945	−	0.999969i	$-0.497511\pi$
$601$	9.39898			0.0156389			0.00781945	+	0.999969i	$0.497511\pi$
$602$		0			0
$603$	963.857	−	160.058i	1.59844	−	0.265436i
$604$		0			0
$605$			302.096i			0.499333i
$606$		0			0
$607$	1064.78			1.75417			0.877085	−	0.480336i	$-0.159485\pi$
$607$	1064.78			1.75417			0.877085	+	0.480336i	$0.159485\pi$
$608$		0			0
$609$	−38.0829	+	3.14051i	−0.0625335	+	0.00515684i
$610$		0			0
$611$			549.039i			0.898591i
$612$		0			0
$613$	−651.884			−1.06343			−0.531716	−	0.846922i	$-0.678453\pi$
$613$	−651.884			−1.06343			−0.531716	+	0.846922i	$0.678453\pi$
$614$		0			0
$615$	−58.4040	−	708.227i	−0.0949659	−	1.15159i
$616$		0			0
$617$		−	146.821i		−	0.237959i	−0.992897	−	0.118980i	$-0.962038\pi$
$617$		−	146.821i		−	0.237959i	0.992897	−	0.118980i	$-0.0379623\pi$
$618$		0			0
$619$	−596.307			−0.963340			−0.481670	−	0.876353i	$-0.659970\pi$
$619$	−596.307			−0.963340			−0.481670	+	0.876353i	$0.659970\pi$
$620$		0			0
$621$	536.295	−	135.133i	0.863598	−	0.217606i
$622$		0			0
$623$			33.9441i			0.0544850i
$624$		0			0
$625$	−738.453			−1.18152
$626$		0			0
$627$	−835.808	+	68.9250i	−1.33303	+	0.109928i
$628$		0			0
$629$			1014.91i			1.61353i
$630$		0			0
$631$	77.6556			0.123067			0.0615337	−	0.998105i	$-0.480401\pi$
$631$	77.6556			0.123067			0.0615337	+	0.998105i	$0.480401\pi$
$632$		0			0
$633$	−59.9235	−	726.652i	−0.0946658	−	1.14795i
$634$		0			0
$635$			1570.56i			2.47333i
$636$		0			0
$637$	872.427			1.36959
$638$		0			0
$639$	−87.5172	−	527.023i	−0.136960	−	0.824762i
$640$		0			0
$641$			574.386i			0.896078i	0.894014	+	0.448039i	$0.147877\pi$
$641$			574.386i			0.896078i	−0.894014	+	0.448039i	$0.852123\pi$
$642$		0			0
$643$	1217.20			1.89299			0.946497	−	0.322712i	$-0.104594\pi$
$643$	1217.20			1.89299			0.946497	+	0.322712i	$0.104594\pi$
$644$		0			0
$645$	−1001.81	+	82.6141i	−1.55319	+	0.128084i
$646$		0			0
$647$			802.675i			1.24061i	0.784360	+	0.620305i	$0.212991\pi$
$647$			802.675i			1.24061i	−0.784360	+	0.620305i	$0.787009\pi$
$648$		0			0
$649$	−549.020			−0.845948
$650$		0			0
$651$	3.73970	+	45.3489i	0.00574455	+	0.0696604i
$652$		0			0
$653$			216.282i			0.331212i	0.986192	+	0.165606i	$0.0529581\pi$
$653$			216.282i			0.331212i	−0.986192	+	0.165606i	$0.947042\pi$
$654$		0			0
$655$	−804.410			−1.22811
$656$		0			0
$657$	49.7615	−	8.26338i	0.0757405	−	0.0125774i
$658$		0			0
$659$		−	611.696i		−	0.928218i	−0.885778	−	0.464109i	$-0.846375\pi$
$659$		−	611.696i		−	0.928218i	0.885778	−	0.464109i	$-0.153625\pi$
$660$		0			0
$661$	−214.961			−0.325206			−0.162603	−	0.986692i	$-0.551989\pi$
$661$	−214.961			−0.325206			−0.162603	+	0.986692i	$0.551989\pi$
$662$		0			0
$663$	−1018.97	+	84.0293i	−1.53690	+	0.126741i
$664$		0			0
$665$		−	123.459i		−	0.185653i
$666$		0			0
$667$	451.275			0.676574
$668$		0			0
$669$	−47.1357	−	571.583i	−0.0704569	−	0.854384i
$670$		0			0
$671$			291.319i			0.434156i
$672$		0			0
$673$	−900.809			−1.33850			−0.669249	−	0.743038i	$-0.733384\pi$
$673$	−900.809			−1.33850			−0.669249	+	0.743038i	$0.733384\pi$
$674$		0			0
$675$	125.622	+	498.549i	0.186107	+	0.738591i
$676$		0			0
$677$			322.312i			0.476088i	0.971254	+	0.238044i	$0.0765063\pi$
$677$			322.312i			0.476088i	−0.971254	+	0.238044i	$0.923494\pi$
$678$		0			0
$679$	−53.9736			−0.0794898
$680$		0			0
$681$	−26.2160	+	2.16191i	−0.0384964	+	0.00317461i
$682$		0			0
$683$			451.564i			0.661148i	0.943780	+	0.330574i	$0.107242\pi$
$683$			451.564i			0.661148i	−0.943780	+	0.330574i	$0.892758\pi$
$684$		0			0
$685$	−907.047			−1.32416
$686$		0			0
$687$	32.4816	+	393.882i	0.0472803	+	0.573337i
$688$		0			0
$689$		−	1593.64i		−	2.31297i
$690$		0			0
$691$	−259.954			−0.376199			−0.188100	−	0.982150i	$-0.560233\pi$
$691$	−259.954			−0.376199			−0.188100	+	0.982150i	$0.560233\pi$
$692$		0			0
$693$	7.40558	+	44.5959i	0.0106863	+	0.0643520i
$694$		0			0
$695$		−	196.183i		−	0.282277i
$696$		0			0
$697$	678.570			0.973558
$698$		0			0
$699$	287.070	−	23.6733i	0.410687	−	0.0338674i
$700$		0			0
$701$			434.792i			0.620246i	0.950696	+	0.310123i	$0.100370\pi$
$701$			434.792i			0.620246i	−0.950696	+	0.310123i	$0.899630\pi$
$702$		0			0
$703$	−1717.78			−2.44350
$704$		0			0
$705$	50.1131	+	607.688i	0.0710823	+	0.861968i
$706$		0			0
$707$		−	66.0033i		−	0.0933568i
$708$		0			0
$709$	−1136.05			−1.60232			−0.801162	−	0.598448i	$-0.795784\pi$
$709$	−1136.05			−1.60232			−0.801162	+	0.598448i	$0.795784\pi$
$710$		0			0
$711$	−701.355	+	116.467i	−0.986434	+	0.163807i
$712$		0			0
$713$		−	537.375i		−	0.753682i
$714$		0			0
$715$	1033.60			1.44559
$716$		0			0
$717$	950.705	−	78.4001i	1.32595	−	0.109345i
$718$		0			0
$719$			1284.33i			1.78627i	0.449792	+	0.893133i	$0.351498\pi$
$719$			1284.33i			1.78627i	−0.449792	+	0.893133i	$0.648502\pi$
$720$		0			0
$721$	40.4916			0.0561603
$722$		0			0
$723$	−28.1542	−	341.407i	−0.0389408	−	0.472209i
$724$		0			0
$725$			419.513i			0.578638i
$726$		0			0
$727$	−31.6562			−0.0435437			−0.0217718	−	0.999763i	$-0.506931\pi$
$727$	−31.6562			−0.0435437			−0.0217718	+	0.999763i	$0.506931\pi$
$728$		0			0
$729$	−641.956	+	345.447i	−0.880598	+	0.473864i
$730$		0			0
$731$		−	959.856i		−	1.31307i
$732$		0			0
$733$	−596.421			−0.813671			−0.406836	−	0.913501i	$-0.633368\pi$
$733$	−596.421			−0.813671			−0.406836	+	0.913501i	$0.633368\pi$
$734$		0			0
$735$	965.619	−	79.6300i	1.31377	−	0.108340i
$736$		0			0
$737$		−	943.170i		−	1.27974i
$738$		0			0
$739$	1146.14			1.55093			0.775464	−	0.631391i	$-0.217516\pi$
$739$	1146.14			1.55093			0.775464	+	0.631391i	$0.217516\pi$
$740$		0			0
$741$	−142.223	−	1724.64i	−0.191934	−	2.32745i
$742$		0			0
$743$			842.850i			1.13439i	0.823584	+	0.567194i	$0.191971\pi$
$743$			842.850i			1.13439i	−0.823584	+	0.567194i	$0.808029\pi$
$744$		0			0
$745$	−116.198			−0.155970
$746$		0			0
$747$	−71.5476	−	430.855i	−0.0957799	−	0.576780i
$748$		0			0
$749$		−	29.4324i		−	0.0392956i
$750$		0			0
$751$	797.793			1.06231			0.531154	−	0.847275i	$-0.321759\pi$
$751$	797.793			1.06231			0.531154	+	0.847275i	$0.321759\pi$
$752$		0			0
$753$	−1126.73	+	92.9162i	−1.49633	+	0.123395i
$754$		0			0
$755$			793.761i			1.05134i
$756$		0			0
$757$	278.402			0.367770			0.183885	−	0.982948i	$-0.441133\pi$
$757$	278.402			0.367770			0.183885	+	0.982948i	$0.441133\pi$
$758$		0			0
$759$	−43.8773	−	532.071i	−0.0578093	−	0.701015i
$760$		0			0
$761$		−	986.126i		−	1.29583i	−0.761713	−	0.647914i	$-0.775641\pi$
$761$		−	986.126i		−	1.29583i	0.761713	−	0.647914i	$-0.224359\pi$
$762$		0			0
$763$	7.64197			0.0100157
$764$		0			0
$765$	−1120.14	+	186.011i	−1.46424	+	0.243151i
$766$		0			0
$767$		−	1132.87i		−	1.47702i
$768$		0			0
$769$	242.053			0.314764			0.157382	−	0.987538i	$-0.449695\pi$
$769$	242.053			0.314764			0.157382	+	0.987538i	$0.449695\pi$
$770$		0			0
$771$	−454.580	+	37.4870i	−0.589598	+	0.0486213i
$772$		0			0
$773$		−	590.233i		−	0.763561i	−0.924253	−	0.381780i	$-0.875311\pi$
$773$		−	590.233i		−	0.763561i	0.924253	−	0.381780i	$-0.124689\pi$
$774$		0			0
$775$	499.553			0.644585
$776$		0			0
$777$	7.61009	+	92.2825i	0.00979420	+	0.118768i
$778$		0			0
$779$			1148.51i			1.47433i
$780$		0			0
$781$	−515.712			−0.660322
$782$		0			0
$783$	−576.808	+	145.341i	−0.736664	+	0.185621i
$784$		0			0
$785$			1257.52i			1.60194i
$786$		0			0
$787$	−565.777			−0.718903			−0.359451	−	0.933164i	$-0.617036\pi$
$787$	−565.777			−0.718903			−0.359451	+	0.933164i	$0.617036\pi$
$788$		0			0
$789$	1480.02	−	122.050i	1.87581	−	0.154689i
$790$		0			0
$791$			36.3933i			0.0460092i
$792$		0			0
$793$	−601.120			−0.758033
$794$		0			0
$795$	−145.458	−	1763.87i	−0.182966	−	2.21870i
$796$		0			0
$797$			1128.02i			1.41533i	0.706549	+	0.707664i	$0.250251\pi$
$797$			1128.02i			1.41533i	−0.706549	+	0.707664i	$0.749749\pi$
$798$		0			0
$799$	−582.241			−0.728712
$800$		0			0
$801$	86.5601	+	521.259i	0.108065	+	0.650761i
$802$		0			0
$803$		−	48.6935i		−	0.0606395i
$804$		0			0
$805$	78.5934			0.0976316
$806$		0			0
$807$	−518.907	+	42.7917i	−0.643007	+	0.0530257i
$808$		0			0
$809$		−	794.472i		−	0.982042i	−0.871148	−	0.491021i	$-0.836624\pi$
$809$		−	794.472i		−	0.982042i	0.871148	−	0.491021i	$-0.163376\pi$
$810$		0			0
$811$	−1430.50			−1.76387			−0.881937	−	0.471367i	$-0.843761\pi$
$811$	−1430.50			−1.76387			−0.881937	+	0.471367i	$0.843761\pi$
$812$		0			0
$813$	30.9220	+	374.971i	0.0380345	+	0.461219i
$814$		0			0
$815$			694.529i			0.852183i
$816$		0			0
$817$	1624.59			1.98849
$818$		0			0
$819$	−92.0211	+	15.2810i	−0.112358	+	0.0186581i
$820$		0			0
$821$		−	1289.62i		−	1.57079i	−0.618992	−	0.785397i	$-0.712459\pi$
$821$		−	1289.62i		−	1.57079i	0.618992	−	0.785397i	$-0.287541\pi$
$822$		0			0
$823$	1064.14			1.29300			0.646502	−	0.762913i	$-0.276231\pi$
$823$	1064.14			1.29300			0.646502	+	0.762913i	$0.276231\pi$
$824$		0			0
$825$	494.622	−	40.7891i	0.599542	−	0.0494413i
$826$		0			0
$827$			1258.48i			1.52174i	0.648907	+	0.760868i	$0.275227\pi$
$827$			1258.48i			1.52174i	−0.648907	+	0.760868i	$0.724773\pi$
$828$		0			0
$829$	1197.65			1.44469			0.722347	−	0.691531i	$-0.243064\pi$
$829$	1197.65			1.44469			0.722347	+	0.691531i	$0.243064\pi$
$830$		0			0
$831$	−55.7961	−	676.603i	−0.0671434	−	0.814203i
$832$		0			0
$833$			925.184i			1.11067i
$834$		0			0
$835$	946.779			1.13387
$836$		0			0
$837$	173.071	+	686.859i	0.206776	+	0.820620i
$838$		0			0
$839$			423.903i			0.505248i	0.967565	+	0.252624i	$0.0812936\pi$
$839$			423.903i			0.505248i	−0.967565	+	0.252624i	$0.918706\pi$
$840$		0			0
$841$	355.635			0.422872
$842$		0			0
$843$	41.5094	−	3.42308i	0.0492401	−	0.00406059i
$844$		0			0
$845$			1011.22i			1.19671i
$846$		0			0
$847$	−26.3183			−0.0310724
$848$		0			0
$849$	75.5605	+	916.272i	0.0889995	+	1.07924i
$850$		0			0
$851$		−	1093.53i		−	1.28499i
$852$		0			0
$853$	−33.0080			−0.0386964			−0.0193482	−	0.999813i	$-0.506159\pi$
$853$	−33.0080			−0.0386964			−0.0193482	+	0.999813i	$0.506159\pi$
$854$		0			0
$855$	−314.830	−	1895.89i	−0.368223	−	2.21741i
$856$		0			0
$857$			315.995i			0.368722i	0.982859	+	0.184361i	$0.0590215\pi$
$857$			315.995i			0.368722i	−0.982859	+	0.184361i	$0.940978\pi$
$858$		0			0
$859$	−41.3157			−0.0480975			−0.0240487	−	0.999711i	$-0.507656\pi$
$859$	−41.3157			−0.0480975			−0.0240487	+	0.999711i	$0.507656\pi$
$860$		0			0
$861$	61.7000	−	5.08810i	0.0716609	−	0.00590953i
$862$		0			0
$863$		−	1173.36i		−	1.35963i	−0.733382	−	0.679817i	$-0.762059\pi$
$863$		−	1173.36i		−	1.35963i	0.733382	−	0.679817i	$-0.237941\pi$
$864$		0			0
$865$	−77.3763			−0.0894524
$866$		0			0
$867$	−17.8553	−	216.520i	−0.0205944	−	0.249735i
$868$		0			0
$869$			686.302i			0.789760i
$870$		0			0
$871$	1946.18			2.23442
$872$		0			0
$873$	−828.839	+	137.637i	−0.949415	+	0.157659i
$874$		0			0
$875$		−	22.8605i		−	0.0261262i
$876$		0			0
$877$	−917.504			−1.04618			−0.523092	−	0.852276i	$-0.675222\pi$
$877$	−917.504			−1.04618			−0.523092	+	0.852276i	$0.675222\pi$
$878$		0			0
$879$	364.370	−	30.0478i	0.414528	−	0.0341841i
$880$		0			0
$881$			1662.48i			1.88704i	0.331322	+	0.943518i	$0.392505\pi$
$881$			1662.48i			1.88704i	−0.331322	+	0.943518i	$0.607495\pi$
$882$		0			0
$883$	1258.25			1.42497			0.712486	−	0.701686i	$-0.247569\pi$
$883$	1258.25			1.42497			0.712486	+	0.701686i	$0.247569\pi$
$884$		0			0
$885$	−103.402	−	1253.89i	−0.116838	−	1.41682i
$886$		0			0
$887$			237.781i			0.268073i	0.990976	+	0.134037i	$0.0427940\pi$
$887$			237.781i			0.268073i	−0.990976	+	0.134037i	$0.957206\pi$
$888$		0			0
$889$	−136.826			−0.153910
$890$		0			0
$891$	227.446	+	665.947i	0.255270	+	0.747416i
$892$		0			0
$893$		−	985.466i		−	1.10354i
$894$		0			0
$895$	−268.097			−0.299550
$896$		0			0
$897$	1097.90	−	90.5383i	1.22397	−	0.100935i
$898$		0			0
$899$			577.970i			0.642903i
$900$		0			0
$901$	1690.01			1.87570
$902$		0			0
$903$	−7.19727	−	87.2764i	−0.00797039	−	0.0966517i
$904$		0			0
$905$		−	804.164i		−	0.888579i
$906$		0			0
$907$	−1118.51			−1.23320			−0.616599	−	0.787277i	$-0.711490\pi$
$907$	−1118.51			−1.23320			−0.616599	+	0.787277i	$0.711490\pi$
$908$		0			0
$909$	−168.313	−	1013.57i	−0.185163	−	1.11504i
$910$		0			0
$911$		−	307.830i		−	0.337903i	−0.985624	−	0.168952i	$-0.945962\pi$
$911$		−	307.830i		−	0.337903i	0.985624	−	0.168952i	$-0.0540382\pi$
$912$		0			0
$913$	−421.608			−0.461783
$914$		0			0
$915$	−665.332	+	54.8667i	−0.727138	+	0.0599636i
$916$		0			0
$917$		−	70.0794i		−	0.0764225i
$918$		0			0
$919$	195.828			0.213088			0.106544	−	0.994308i	$-0.466022\pi$
$919$	195.828			0.213088			0.106544	+	0.994308i	$0.466022\pi$
$920$		0			0
$921$	−4.69525	−	56.9362i	−0.00509799	−	0.0618199i
$922$		0			0
$923$		−	1064.14i		−	1.15292i
$924$		0			0
$925$	1016.56			1.09899
$926$		0			0
$927$	621.804	−	103.257i	0.670771	−	0.111388i
$928$		0			0
$929$		−	80.3569i		−	0.0864982i	−0.999064	−	0.0432491i	$-0.986229\pi$
$929$		−	80.3569i		−	0.0864982i	0.999064	−	0.0432491i	$-0.0137709\pi$
$930$		0			0
$931$	−1565.91			−1.68197
$932$		0			0
$933$	−667.211	+	55.0217i	−0.715124	+	0.0589728i
$934$		0			0
$935$			1096.10i			1.17230i
$936$		0			0
$937$	502.272			0.536043			0.268021	−	0.963413i	$-0.413630\pi$
$937$	502.272			0.536043			0.268021	+	0.963413i	$0.413630\pi$
$938$		0			0
$939$	−48.0069	−	582.147i	−0.0511255	−	0.619965i
$940$		0			0
$941$			1752.05i			1.86190i	0.365144	+	0.930951i	$0.381020\pi$
$941$			1752.05i			1.86190i	−0.365144	+	0.930951i	$0.618980\pi$
$942$		0			0
$943$	−731.133			−0.775326
$944$		0			0
$945$	−100.456	+	25.3125i	−0.106303	+	0.0267857i
$946$		0			0
$947$			634.493i			0.670003i	0.942218	+	0.335001i	$0.108737\pi$
$947$			634.493i			0.670003i	−0.942218	+	0.335001i	$0.891263\pi$
$948$		0			0
$949$	100.476			0.105876
$950$		0			0
$951$	1591.18	−	131.217i	1.67316	−	0.137977i
$952$		0			0
$953$		−	590.937i		−	0.620081i	−0.950723	−	0.310040i	$-0.899657\pi$
$953$		−	590.937i		−	0.620081i	0.950723	−	0.310040i	$-0.100343\pi$
$954$		0			0
$955$	−2238.38			−2.34385
$956$		0			0
$957$	47.1919	+	572.264i	0.0493123	+	0.597977i
$958$		0			0
$959$		−	79.0211i		−	0.0823994i
$960$		0			0
$961$	−272.757			−0.283827
$962$		0			0
$963$	−75.0549	−	451.976i	−0.0779386	−	0.469341i
$964$		0			0
$965$		−	1512.69i		−	1.56755i
$966$		0			0
$967$	−1194.01			−1.23476			−0.617381	−	0.786665i	$-0.711806\pi$
$967$	−1194.01			−1.23476			−0.617381	+	0.786665i	$0.711806\pi$
$968$		0			0
$969$	1828.94	−	150.823i	1.88745	−	0.155649i
$970$		0			0
$971$		−	827.456i		−	0.852169i	−0.904683	−	0.426084i	$-0.859893\pi$
$971$		−	827.456i		−	0.852169i	0.904683	−	0.426084i	$-0.140107\pi$
$972$		0			0
$973$	17.0913			0.0175655
$974$		0			0
$975$	84.1660	+	1020.63i	0.0863241	+	1.04679i
$976$		0			0
$977$		−	1770.96i		−	1.81266i	−0.422576	−	0.906328i	$-0.638874\pi$
$977$		−	1770.96i		−	1.81266i	0.422576	−	0.906328i	$-0.361126\pi$
$978$		0			0
$979$	510.072			0.521013
$980$		0			0
$981$	117.353	−	19.4876i	0.119626	−	0.0198650i
$982$		0			0
$983$		−	406.666i		−	0.413699i	−0.978373	−	0.206850i	$-0.933679\pi$
$983$		−	406.666i		−	0.413699i	0.978373	−	0.206850i	$-0.0663211\pi$
$984$		0			0
$985$	1143.29			1.16070
$986$		0			0
$987$	−52.9412	+	4.36580i	−0.0536385	+	0.00442331i
$988$		0			0
$989$			1034.21i			1.04571i
$990$		0			0
$991$	1364.81			1.37720			0.688602	−	0.725140i	$-0.258225\pi$
$991$	1364.81			1.37720			0.688602	+	0.725140i	$0.258225\pi$
$992$		0			0
$993$	22.6161	+	274.251i	0.0227756	+	0.276184i
$994$		0			0
$995$		−	3.97413i		−	0.00399410i
$996$		0			0
$997$	1201.02			1.20463			0.602317	−	0.798257i	$-0.294244\pi$
$997$	1201.02			1.20463			0.602317	+	0.798257i	$0.294244\pi$
$998$		0			0
$999$	352.191	+	1397.72i	0.352544	+	1.39912i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.e.c.257.4	yes	8
3.2	odd	2	inner	384.3.e.c.257.3	yes	8
4.3	odd	2		384.3.e.b.257.5	yes	8
8.3	odd	2		384.3.e.d.257.4	yes	8
8.5	even	2		384.3.e.a.257.5	✓	8
12.11	even	2		384.3.e.b.257.6	yes	8
16.3	odd	4		768.3.h.g.641.15		16
16.5	even	4		768.3.h.h.641.15		16
16.11	odd	4		768.3.h.g.641.2		16
16.13	even	4		768.3.h.h.641.2		16
24.5	odd	2		384.3.e.a.257.6	yes	8
24.11	even	2		384.3.e.d.257.3	yes	8
48.5	odd	4		768.3.h.h.641.1		16
48.11	even	4		768.3.h.g.641.16		16
48.29	odd	4		768.3.h.h.641.16		16
48.35	even	4		768.3.h.g.641.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.e.a.257.5	✓	8	8.5	even	2
384.3.e.a.257.6	yes	8	24.5	odd	2
384.3.e.b.257.5	yes	8	4.3	odd	2
384.3.e.b.257.6	yes	8	12.11	even	2
384.3.e.c.257.3	yes	8	3.2	odd	2	inner
384.3.e.c.257.4	yes	8	1.1	even	1	trivial
384.3.e.d.257.3	yes	8	24.11	even	2
384.3.e.d.257.4	yes	8	8.3	odd	2
768.3.h.g.641.1		16	48.35	even	4
768.3.h.g.641.2		16	16.11	odd	4
768.3.h.g.641.15		16	16.3	odd	4
768.3.h.g.641.16		16	48.11	even	4
768.3.h.h.641.1		16	48.5	odd	4
768.3.h.h.641.2		16	16.13	even	4
768.3.h.h.641.15		16	16.5	even	4
768.3.h.h.641.16		16	48.29	odd	4

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.246559 + 2.98985i) q^{3} +6.63641i q^{5} -0.578158 q^{7} +(-8.87842 + 1.47435i) q^{9} +O(q^{10})\)	\(q+(0.246559 + 2.98985i) q^{3} +6.63641i q^{5} -0.578158 q^{7} +(-8.87842 + 1.47435i) q^{9} +8.68786i q^{11} -17.9269 q^{13} +(-19.8419 + 1.63626i) q^{15} -19.0110i q^{17} +32.1769 q^{19} +(-0.142550 - 1.72861i) q^{21} +20.4836i q^{23} -19.0419 q^{25} +(-6.59713 - 26.1816i) q^{27} -22.0310i q^{29} -26.2344 q^{31} +(-25.9754 + 2.14207i) q^{33} -3.83689i q^{35} -53.3855 q^{37} +(-4.42003 - 53.5988i) q^{39} +35.6935i q^{41} +50.4895 q^{43} +(-9.78437 - 58.9208i) q^{45} -30.6265i q^{47} -48.6657 q^{49} +(56.8401 - 4.68732i) q^{51} +88.8962i q^{53} -57.6562 q^{55} +(7.93348 + 96.2040i) q^{57} +63.1939i q^{59} +33.5317 q^{61} +(5.13313 - 0.852405i) q^{63} -118.970i q^{65} -108.562 q^{67} +(-61.2430 + 5.05041i) q^{69} +59.3600i q^{71} -5.60477 q^{73} +(-4.69495 - 56.9325i) q^{75} -5.02296i q^{77} +78.9955 q^{79} +(76.6526 - 26.1797i) q^{81} +48.5283i q^{83} +126.165 q^{85} +(65.8694 - 5.43193i) q^{87} -58.7109i q^{89} +10.3646 q^{91} +(-6.46831 - 78.4369i) q^{93} +213.539i q^{95} +93.3544 q^{97} +(-12.8089 - 77.1345i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{7}+O(q^{10}) \) 8 * q + 4 * q^3 - 8 * q^7	\( 8 q + 4 q^{3} - 8 q^{7} - 16 q^{15} - 24 q^{19} - 16 q^{21} - 40 q^{25} - 44 q^{27} + 56 q^{31} + 8 q^{33} - 32 q^{37} + 104 q^{39} + 136 q^{43} + 80 q^{45} + 72 q^{49} + 176 q^{51} - 192 q^{55} - 40 q^{57} + 160 q^{61} - 264 q^{63} - 280 q^{67} - 80 q^{69} - 80 q^{73} - 348 q^{75} + 408 q^{79} + 72 q^{81} - 192 q^{85} + 368 q^{87} + 336 q^{91} - 160 q^{93} + 96 q^{97} + 432 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 8 * q^7 - 16 * q^15 - 24 * q^19 - 16 * q^21 - 40 * q^25 - 44 * q^27 + 56 * q^31 + 8 * q^33 - 32 * q^37 + 104 * q^39 + 136 * q^43 + 80 * q^45 + 72 * q^49 + 176 * q^51 - 192 * q^55 - 40 * q^57 + 160 * q^61 - 264 * q^63 - 280 * q^67 - 80 * q^69 - 80 * q^73 - 348 * q^75 + 408 * q^79 + 72 * q^81 - 192 * q^85 + 368 * q^87 + 336 * q^91 - 160 * q^93 + 96 * q^97 + 432 * q^99

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