LMFDB - Embedded newform 768.4.c.m.767.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(767,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.767");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.c (of order $2$, degree $1$, not 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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-6}, \sqrt{-14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 10x^{2} + 4 $ x^4 + 10*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			767.1
Root			$-3.09557i$ of defining polynomial
Character	$\chi$	$=$	768.767
Dual form			768.4.c.m.767.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.58258 - 2.44949i) q^{3} +17.2813i q^{5} +0.269691i q^{7} +(15.0000 + 22.4499i) q^{9} +O(q^{10})$	\(q+(-4.58258 - 2.44949i) q^{3} +17.2813i q^{5} +0.269691i q^{7} +(15.0000 + 22.4499i) q^{9} -41.1652 q^{11} -64.6606 q^{13} +(42.3303 - 79.1927i) q^{15} -63.4170i q^{17} -154.026i q^{19} +(0.660606 - 1.23588i) q^{21} -43.3394 q^{23} -173.642 q^{25} +(-13.7477 - 139.621i) q^{27} +240.702i q^{29} +79.1927i q^{31} +(188.642 + 100.834i) q^{33} -4.66061 q^{35} +132.624 q^{37} +(296.312 + 158.385i) q^{39} -101.216i q^{41} +120.317i q^{43} +(-387.964 + 259.219i) q^{45} +293.285 q^{47} +342.927 q^{49} +(-155.339 + 290.613i) q^{51} -6.17940i q^{53} -711.386i q^{55} +(-377.285 + 705.835i) q^{57} +45.8258 q^{59} -651.230 q^{61} +(-6.05455 + 4.04537i) q^{63} -1117.42i q^{65} +685.273i q^{67} +(198.606 + 106.159i) q^{69} +836.515 q^{71} -285.927 q^{73} +(795.730 + 425.335i) q^{75} -11.1019i q^{77} -940.874i q^{79} +(-279.000 + 673.498i) q^{81} +1011.11 q^{83} +1095.93 q^{85} +(589.597 - 1103.03i) q^{87} -432.503i q^{89} -17.4384i q^{91} +(193.982 - 362.907i) q^{93} +2661.76 q^{95} +1007.28 q^{97} +(-617.477 - 924.155i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 60 q^{9}+O(q^{10}) $ 4 * q + 60 * q^9	$ 4 q + 60 q^{9} - 128 q^{11} - 112 q^{13} + 96 q^{15} - 144 q^{21} - 320 q^{23} - 108 q^{25} + 168 q^{33} + 128 q^{35} - 496 q^{37} + 672 q^{39} - 672 q^{45} - 388 q^{49} - 768 q^{51} - 336 q^{57} - 112 q^{61} - 1344 q^{63} - 672 q^{69} - 320 q^{71} + 616 q^{73} + 2688 q^{75} - 1116 q^{81} + 2688 q^{83} + 2624 q^{85} + 672 q^{87} + 336 q^{93} + 4928 q^{95} + 2856 q^{97} - 1920 q^{99}+O(q^{100}) $ 4 * q + 60 * q^9 - 128 * q^11 - 112 * q^13 + 96 * q^15 - 144 * q^21 - 320 * q^23 - 108 * q^25 + 168 * q^33 + 128 * q^35 - 496 * q^37 + 672 * q^39 - 672 * q^45 - 388 * q^49 - 768 * q^51 - 336 * q^57 - 112 * q^61 - 1344 * q^63 - 672 * q^69 - 320 * q^71 + 616 * q^73 + 2688 * q^75 - 1116 * q^81 + 2688 * q^83 + 2624 * q^85 + 672 * q^87 + 336 * q^93 + 4928 * q^95 + 2856 * q^97 - 1920 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−4.58258	−	2.44949i	−0.881917	−	0.471405i
$3$	−4.58258	−	2.44949i	−0.881917	−	0.471405i
$4$		0			0
$5$			17.2813i			1.54568i	0.634598	+	0.772842i	$0.281166\pi$
$5$			17.2813i			1.54568i	−0.634598	+	0.772842i	$0.718834\pi$
$6$		0			0
$7$			0.269691i			0.0145620i	0.999973	+	0.00728098i	$0.00231763\pi$
$7$			0.269691i			0.0145620i	−0.999973	+	0.00728098i	$0.997682\pi$
$8$		0			0
$9$	15.0000	+	22.4499i	0.555556	+	0.831479i
$10$		0			0
$11$	−41.1652			−1.12834			−0.564171	−	0.825658i	$-0.690804\pi$
$11$	−41.1652			−1.12834			−0.564171	+	0.825658i	$0.690804\pi$
$12$		0			0
$13$	−64.6606			−1.37951			−0.689755	−	0.724043i	$-0.742282\pi$
$13$	−64.6606			−1.37951			−0.689755	+	0.724043i	$0.742282\pi$
$14$		0			0
$15$	42.3303	−	79.1927i	0.728642	−	1.36317i
$16$		0			0
$17$		−	63.4170i		−	0.904758i	−0.891826	−	0.452379i	$-0.850575\pi$
$17$		−	63.4170i		−	0.904758i	0.891826	−	0.452379i	$-0.149425\pi$
$18$		0			0
$19$		−	154.026i		−	1.85979i	−0.367828	−	0.929894i	$-0.619899\pi$
$19$		−	154.026i		−	1.85979i	0.367828	−	0.929894i	$-0.380101\pi$
$20$		0			0
$21$	0.660606	−	1.23588i	0.00686457	−	0.0128424i
$22$		0			0
$23$	−43.3394			−0.392908			−0.196454	−	0.980513i	$-0.562943\pi$
$23$	−43.3394			−0.392908			−0.196454	+	0.980513i	$0.562943\pi$
$24$		0			0
$25$	−173.642			−1.38914
$26$		0			0
$27$	−13.7477	−	139.621i	−0.0979908	−	0.995187i
$28$		0			0
$29$			240.702i			1.54128i	0.637268	+	0.770642i	$0.280064\pi$
$29$			240.702i			1.54128i	−0.637268	+	0.770642i	$0.719936\pi$
$30$		0			0
$31$			79.1927i			0.458821i	0.973330	+	0.229410i	$0.0736797\pi$
$31$			79.1927i			0.458821i	−0.973330	+	0.229410i	$0.926320\pi$
$32$		0			0
$33$	188.642	+	100.834i	0.995104	+	0.531905i
$34$		0			0
$35$	−4.66061			−0.0225082
$36$		0			0
$37$	132.624			0.589278			0.294639	−	0.955609i	$-0.404801\pi$
$37$	132.624			0.589278			0.294639	+	0.955609i	$0.404801\pi$
$38$		0			0
$39$	296.312	+	158.385i	1.21661	+	0.650307i
$40$		0			0
$41$		−	101.216i		−	0.385543i	−0.981244	−	0.192772i	$-0.938252\pi$
$41$		−	101.216i		−	0.385543i	0.981244	−	0.192772i	$-0.0617476\pi$
$42$		0			0
$43$			120.317i			0.426701i	0.976976	+	0.213351i	$0.0684377\pi$
$43$			120.317i			0.426701i	−0.976976	+	0.213351i	$0.931562\pi$
$44$		0			0
$45$	−387.964	+	259.219i	−1.28520	+	0.858713i
$46$		0			0
$47$	293.285			0.910213			0.455106	−	0.890437i	$-0.349601\pi$
$47$	293.285			0.910213			0.455106	+	0.890437i	$0.349601\pi$
$48$		0			0
$49$	342.927			0.999788
$50$		0			0
$51$	−155.339	+	290.613i	−0.426507	+	0.797922i
$52$		0			0
$53$		−	6.17940i		−	0.0160152i	−0.999968	−	0.00800760i	$-0.997451\pi$
$53$		−	6.17940i		−	0.0160152i	0.999968	−	0.00800760i	$-0.00254893\pi$
$54$		0			0
$55$		−	711.386i		−	1.74406i
$56$		0			0
$57$	−377.285	+	705.835i	−0.876712	+	1.64018i
$58$		0			0
$59$	45.8258			0.101119			0.0505594	−	0.998721i	$-0.483900\pi$
$59$	45.8258			0.101119			0.0505594	+	0.998721i	$0.483900\pi$
$60$		0			0
$61$	−651.230			−1.36691			−0.683455	−	0.729993i	$-0.739523\pi$
$61$	−651.230			−1.36691			−0.683455	+	0.729993i	$0.739523\pi$
$62$		0			0
$63$	−6.05455	+	4.04537i	−0.0121080	+	0.00808997i
$64$		0			0
$65$		−	1117.42i		−	2.13229i
$66$		0			0
$67$			685.273i			1.24954i	0.780807	+	0.624772i	$0.214808\pi$
$67$			685.273i			1.24954i	−0.780807	+	0.624772i	$0.785192\pi$
$68$		0			0
$69$	198.606	+	106.159i	0.346512	+	0.185219i
$70$		0			0
$71$	836.515			1.39825			0.699127	−	0.714997i	$-0.253572\pi$
$71$	836.515			1.39825			0.699127	+	0.714997i	$0.253572\pi$
$72$		0			0
$73$	−285.927			−0.458428			−0.229214	−	0.973376i	$-0.573616\pi$
$73$	−285.927			−0.458428			−0.229214	+	0.973376i	$0.573616\pi$
$74$		0			0
$75$	795.730	+	425.335i	1.22511	+	0.654847i
$76$		0			0
$77$		−	11.1019i		−	0.0164309i
$78$		0			0
$79$		−	940.874i		−	1.33996i	−0.742381	−	0.669978i	$-0.766303\pi$
$79$		−	940.874i		−	1.33996i	0.742381	−	0.669978i	$-0.233697\pi$
$80$		0			0
$81$	−279.000	+	673.498i	−0.382716	+	0.923866i
$82$		0			0
$83$	1011.11			1.33715			0.668577	−	0.743643i	$-0.266904\pi$
$83$	1011.11			1.33715			0.668577	+	0.743643i	$0.266904\pi$
$84$		0			0
$85$	1095.93			1.39847
$86$		0			0
$87$	589.597	−	1103.03i	0.726568	−	1.35928i
$88$		0			0
$89$		−	432.503i		−	0.515115i	−0.966263	−	0.257558i	$-0.917082\pi$
$89$		−	432.503i		−	0.515115i	0.966263	−	0.257558i	$-0.0829177\pi$
$90$		0			0
$91$		−	17.4384i		−	0.0200884i
$92$		0			0
$93$	193.982	−	362.907i	0.216290	−	0.404642i
$94$		0			0
$95$	2661.76			2.87464
$96$		0			0
$97$	1007.28			1.05437			0.527187	−	0.849749i	$-0.323247\pi$
$97$	1007.28			1.05437			0.527187	+	0.849749i	$0.323247\pi$
$98$		0			0
$99$	−617.477	−	924.155i	−0.626857	−	0.938193i
$100$		0			0
$101$		−	1217.08i		−	1.19905i	−0.800355	−	0.599526i	$-0.795356\pi$
$101$		−	1217.08i		−	1.19905i	0.800355	−	0.599526i	$-0.204644\pi$
$102$		0			0
$103$			1357.60i			1.29872i	0.760479	+	0.649362i	$0.224964\pi$
$103$			1357.60i			1.29872i	−0.760479	+	0.649362i	$0.775036\pi$
$104$		0			0
$105$	21.3576	+	11.4161i	0.0198503	+	0.0106105i
$106$		0			0
$107$	−1490.17			−1.34636			−0.673180	−	0.739478i	$-0.735072\pi$
$107$	−1490.17			−1.34636			−0.673180	+	0.739478i	$0.735072\pi$
$108$		0			0
$109$	1865.91			1.63965			0.819824	−	0.572615i	$-0.194071\pi$
$109$	1865.91			1.63965			0.819824	+	0.572615i	$0.194071\pi$
$110$		0			0
$111$	−607.761	−	324.862i	−0.519694	−	0.277788i
$112$		0			0
$113$			1047.66i			0.872177i	0.899904	+	0.436088i	$0.143636\pi$
$113$			1047.66i			0.872177i	−0.899904	+	0.436088i	$0.856364\pi$
$114$		0			0
$115$		−	748.960i		−	0.607312i
$116$		0			0
$117$	−969.909	−	1451.63i	−0.766394	−	1.14703i
$118$		0			0
$119$	17.1030			0.0131750
$120$		0			0
$121$	363.570			0.273155
$122$		0			0
$123$	−247.927	+	463.829i	−0.181747	+	0.340017i
$124$		0			0
$125$		−	840.603i		−	0.601487i
$126$		0			0
$127$		−	1033.28i		−	0.721960i	−0.932574	−	0.360980i	$-0.882442\pi$
$127$		−	1033.28i		−	0.721960i	0.932574	−	0.360980i	$-0.117558\pi$
$128$		0			0
$129$	294.715	−	551.362i	0.201149	−	0.376315i
$130$		0			0
$131$	−359.626			−0.239852			−0.119926	−	0.992783i	$-0.538266\pi$
$131$	−359.626			−0.239852			−0.119926	+	0.992783i	$0.538266\pi$
$132$		0			0
$133$	41.5394			0.0270821
$134$		0			0
$135$	2412.83	−	237.578i	1.53825	−	0.151463i
$136$		0			0
$137$		−	379.245i		−	0.236504i	−0.992984	−	0.118252i	$-0.962271\pi$
$137$		−	379.245i		−	0.236504i	0.992984	−	0.118252i	$-0.0377291\pi$
$138$		0			0
$139$		−	1984.09i		−	1.21071i	−0.795957	−	0.605353i	$-0.793032\pi$
$139$		−	1984.09i		−	1.21071i	0.795957	−	0.605353i	$-0.206968\pi$
$140$		0			0
$141$	−1344.00	−	718.398i	−0.802732	−	0.429078i
$142$		0			0
$143$	2661.76			1.55656
$144$		0			0
$145$	−4159.64			−2.38234
$146$		0			0
$147$	−1571.49	−	839.997i	−0.881730	−	0.471305i
$148$		0			0
$149$			2323.13i			1.27730i	0.769497	+	0.638651i	$0.220507\pi$
$149$			2323.13i			1.27730i	−0.769497	+	0.638651i	$0.779493\pi$
$150$		0			0
$151$		−	2613.36i		−	1.40843i	−0.709989	−	0.704213i	$-0.751300\pi$
$151$		−	2613.36i		−	1.40843i	0.709989	−	0.704213i	$-0.248700\pi$
$152$		0			0
$153$	1423.71	−	951.256i	0.752288	−	0.502644i
$154$		0			0
$155$	−1368.55			−0.709192
$156$		0			0
$157$	2780.41			1.41338			0.706690	−	0.707524i	$-0.250188\pi$
$157$	2780.41			1.41338			0.706690	+	0.707524i	$0.250188\pi$
$158$		0			0
$159$	−15.1364	+	28.3176i	−0.00754964	+	0.0141241i
$160$		0			0
$161$		−	11.6882i		−	0.00572151i
$162$		0			0
$163$		−	635.294i		−	0.305276i	−0.988282	−	0.152638i	$-0.951223\pi$
$163$		−	635.294i		−	0.305276i	0.988282	−	0.152638i	$-0.0487769\pi$
$164$		0			0
$165$	−1742.53	+	3259.98i	−0.822158	+	1.53812i
$166$		0			0
$167$	−319.521			−0.148056			−0.0740278	−	0.997256i	$-0.523585\pi$
$167$	−319.521			−0.148056			−0.0740278	+	0.997256i	$0.523585\pi$
$168$		0			0
$169$	1983.99			0.903047
$170$		0			0
$171$	3457.87	−	2310.39i	1.54638	−	1.03322i
$172$		0			0
$173$		−	1862.71i		−	0.818609i	−0.912398	−	0.409305i	$-0.865771\pi$
$173$		−	1862.71i		−	0.818609i	0.912398	−	0.409305i	$-0.134229\pi$
$174$		0			0
$175$		−	46.8298i		−	0.0202286i
$176$		0			0
$177$	−210.000	−	112.250i	−0.0891783	−	0.0476678i
$178$		0			0
$179$	3946.43			1.64788			0.823940	−	0.566678i	$-0.191771\pi$
$179$	3946.43			1.64788			0.823940	+	0.566678i	$0.191771\pi$
$180$		0			0
$181$	−1187.01			−0.487458			−0.243729	−	0.969843i	$-0.578371\pi$
$181$	−1187.01			−0.487458			−0.243729	+	0.969843i	$0.578371\pi$
$182$		0			0
$183$	2984.31	+	1595.18i	1.20550	+	0.644367i
$184$		0			0
$185$			2291.92i			0.910838i
$186$		0			0
$187$			2610.57i			1.02088i
$188$		0			0
$189$	37.6545	−	3.70764i	0.0144919	−	0.00142694i
$190$		0			0
$191$	−1147.96			−0.434889			−0.217444	−	0.976073i	$-0.569772\pi$
$191$	−1147.96			−0.434889			−0.217444	+	0.976073i	$0.569772\pi$
$192$		0			0
$193$	304.933			0.113728			0.0568642	−	0.998382i	$-0.481890\pi$
$193$	304.933			0.113728			0.0568642	+	0.998382i	$0.481890\pi$
$194$		0			0
$195$	−2737.10	+	5120.65i	−1.00517	+	1.88050i
$196$		0			0
$197$		−	1252.94i		−	0.453140i	−0.973995	−	0.226570i	$-0.927249\pi$
$197$		−	1252.94i		−	0.453140i	0.973995	−	0.226570i	$-0.0727512\pi$
$198$		0			0
$199$		−	4664.28i		−	1.66152i	−0.556632	−	0.830759i	$-0.687907\pi$
$199$		−	4664.28i		−	1.66152i	0.556632	−	0.830759i	$-0.312093\pi$
$200$		0			0
$201$	1678.57	−	3140.32i	0.589041	−	1.10199i
$202$		0			0
$203$	−64.9152			−0.0224441
$204$		0			0
$205$	1749.14			0.595928
$206$		0			0
$207$	−650.091	−	972.967i	−0.218282	−	0.326695i
$208$		0			0
$209$			6340.50i			2.09848i
$210$		0			0
$211$			1845.39i			0.602095i	0.953609	+	0.301047i	$0.0973362\pi$
$211$			1845.39i			0.602095i	−0.953609	+	0.301047i	$0.902664\pi$
$212$		0			0
$213$	−3833.39	−	2049.04i	−1.23314	−	0.659144i
$214$		0			0
$215$	−2079.23			−0.659546
$216$		0			0
$217$	−21.3576			−0.00668132
$218$		0			0
$219$	1310.28	+	700.376i	0.404296	+	0.216105i
$220$		0			0
$221$			4100.58i			1.24812i
$222$		0			0
$223$			2965.36i			0.890473i	0.895413	+	0.445237i	$0.146880\pi$
$223$			2965.36i			0.890473i	−0.895413	+	0.445237i	$0.853120\pi$
$224$		0			0
$225$	−2604.64	−	3898.26i	−0.771744	−	1.15504i
$226$		0			0
$227$	−1538.99			−0.449983			−0.224991	−	0.974361i	$-0.572235\pi$
$227$	−1538.99			−0.449983			−0.224991	+	0.974361i	$0.572235\pi$
$228$		0			0
$229$	1658.05			0.478460			0.239230	−	0.970963i	$-0.423105\pi$
$229$	1658.05			0.478460			0.239230	+	0.970963i	$0.423105\pi$
$230$		0			0
$231$	−27.1939	+	50.8752i	−0.00774558	+	0.0144907i
$232$		0			0
$233$		−	1241.69i		−	0.349123i	−0.984646	−	0.174561i	$-0.944149\pi$
$233$		−	1241.69i		−	0.349123i	0.984646	−	0.174561i	$-0.0558507\pi$
$234$		0			0
$235$			5068.34i			1.40690i
$236$		0			0
$237$	−2304.66	+	4311.63i	−0.631662	+	1.18173i
$238$		0			0
$239$	2373.58			0.642401			0.321201	−	0.947011i	$-0.395914\pi$
$239$	2373.58			0.642401			0.321201	+	0.947011i	$0.395914\pi$
$240$		0			0
$241$	2592.35			0.692896			0.346448	−	0.938069i	$-0.387388\pi$
$241$	2592.35			0.692896			0.346448	+	0.938069i	$0.387388\pi$
$242$		0			0
$243$	2928.27	−	2402.95i	0.773038	−	0.634359i
$244$		0			0
$245$			5926.22i			1.54536i
$246$		0			0
$247$			9959.41i			2.56559i
$248$		0			0
$249$	−4633.49	−	2476.71i	−1.17926	−	0.630341i
$250$		0			0
$251$	−2750.30			−0.691624			−0.345812	−	0.938304i	$-0.612397\pi$
$251$	−2750.30			−0.691624			−0.345812	+	0.938304i	$0.612397\pi$
$252$		0			0
$253$	1784.07			0.443335
$254$		0			0
$255$	−5022.17	−	2684.46i	−1.23334	−	0.659245i
$256$		0			0
$257$		−	218.163i		−	0.0529519i	−0.999649	−	0.0264759i	$-0.991571\pi$
$257$		−	218.163i		−	0.0529519i	0.999649	−	0.0264759i	$-0.00842854\pi$
$258$		0			0
$259$			35.7676i			0.00858104i
$260$		0			0
$261$	−5403.75	+	3610.53i	−1.28155	+	0.856269i
$262$		0			0
$263$	−7341.25			−1.72122			−0.860611	−	0.509264i	$-0.829918\pi$
$263$	−7341.25			−1.72122			−0.860611	+	0.509264i	$0.829918\pi$
$264$		0			0
$265$	106.788			0.0247544
$266$		0			0
$267$	−1059.41	+	1981.98i	−0.242828	+	0.454289i
$268$		0			0
$269$			54.3577i			0.0123206i	0.999981	+	0.00616031i	$0.00196090\pi$
$269$			54.3577i			0.0123206i	−0.999981	+	0.00616031i	$0.998039\pi$
$270$		0			0
$271$			3633.43i			0.814446i	0.913329	+	0.407223i	$0.133503\pi$
$271$			3633.43i			0.814446i	−0.913329	+	0.407223i	$0.866497\pi$
$272$		0			0
$273$	−42.7152	+	79.9127i	−0.00946974	+	0.0177163i
$274$		0			0
$275$	7148.02			1.56742
$276$		0			0
$277$	6204.12			1.34574			0.672868	−	0.739762i	$-0.265062\pi$
$277$	6204.12			1.34574			0.672868	+	0.739762i	$0.265062\pi$
$278$		0			0
$279$	−1777.87	+	1187.89i	−0.381500	+	0.254900i
$280$		0			0
$281$			5651.49i			1.19979i	0.800081	+	0.599893i	$0.204790\pi$
$281$			5651.49i			1.19979i	−0.800081	+	0.599893i	$0.795210\pi$
$282$		0			0
$283$		−	5857.07i		−	1.23027i	−0.788421	−	0.615136i	$-0.789101\pi$
$283$		−	5857.07i		−	1.23027i	0.788421	−	0.615136i	$-0.210899\pi$
$284$		0			0
$285$	−12197.7	−	6519.96i	−2.53520	−	1.35512i
$286$		0			0
$287$	27.2970			0.00561426
$288$		0			0
$289$	891.279			0.181412
$290$		0			0
$291$	−4615.96	−	2467.33i	−0.929870	−	0.497037i
$292$		0			0
$293$		−	5395.42i		−	1.07578i	−0.843015	−	0.537891i	$-0.819221\pi$
$293$		−	5395.42i		−	1.07578i	0.843015	−	0.537891i	$-0.180779\pi$
$294$		0			0
$295$			791.927i			0.156298i
$296$		0			0
$297$	565.927	+	5747.52i	0.110567	+	1.12291i
$298$		0			0
$299$	2802.35			0.542021
$300$		0			0
$301$	−32.4484			−0.00621361
$302$		0			0
$303$	−2981.23	+	5577.38i	−0.565239	+	1.05747i
$304$		0			0
$305$		−	11254.1i		−	2.11281i
$306$		0			0
$307$		−	8188.99i		−	1.52238i	−0.648530	−	0.761189i	$-0.724616\pi$
$307$		−	8188.99i		−	1.52238i	0.648530	−	0.761189i	$-0.275384\pi$
$308$		0			0
$309$	3325.44	−	6221.32i	0.612225	−	1.14537i
$310$		0			0
$311$	1838.42			0.335200			0.167600	−	0.985855i	$-0.446398\pi$
$311$	1838.42			0.335200			0.167600	+	0.985855i	$0.446398\pi$
$312$		0			0
$313$	293.079			0.0529259			0.0264629	−	0.999650i	$-0.491576\pi$
$313$	293.079			0.0529259			0.0264629	+	0.999650i	$0.491576\pi$
$314$		0			0
$315$	−69.9091	−	104.630i	−0.0125045	−	0.0187151i
$316$		0			0
$317$			1343.02i			0.237954i	0.992897	+	0.118977i	$0.0379614\pi$
$317$			1343.02i			0.237954i	−0.992897	+	0.118977i	$0.962039\pi$
$318$		0			0
$319$		−	9908.53i		−	1.73909i
$320$		0			0
$321$	6828.84	+	3650.17i	1.18738	+	0.634680i
$322$		0			0
$323$	−9767.87			−1.68266
$324$		0			0
$325$	11227.8			1.91633
$326$		0			0
$327$	−8550.67	−	4570.53i	−1.44603	−	0.772938i
$328$		0			0
$329$			79.0963i			0.0132545i
$330$		0			0
$331$		−	8825.95i		−	1.46561i	−0.680437	−	0.732807i	$-0.738210\pi$
$331$		−	8825.95i		−	1.46561i	0.680437	−	0.732807i	$-0.261790\pi$
$332$		0			0
$333$	1989.36	+	2977.41i	0.327377	+	0.489973i
$334$		0			0
$335$	−11842.4			−1.93140
$336$		0			0
$337$	3842.48			0.621108			0.310554	−	0.950556i	$-0.399485\pi$
$337$	3842.48			0.621108			0.310554	+	0.950556i	$0.399485\pi$
$338$		0			0
$339$	2566.24	−	4801.00i	0.411148	−	0.769187i
$340$		0			0
$341$		−	3259.98i		−	0.517706i
$342$		0			0
$343$			184.988i			0.0291208i
$344$		0			0
$345$	−1834.57	+	3432.17i	−0.286290	+	0.535599i
$346$		0			0
$347$	−9565.43			−1.47983			−0.739913	−	0.672703i	$-0.765133\pi$
$347$	−9565.43			−1.47983			−0.739913	+	0.672703i	$0.765133\pi$
$348$		0			0
$349$	7514.35			1.15253			0.576266	−	0.817262i	$-0.304509\pi$
$349$	7514.35			1.15253			0.576266	+	0.817262i	$0.304509\pi$
$350$		0			0
$351$	888.936	+	9027.97i	0.135179	+	1.37287i
$352$		0			0
$353$			4302.30i			0.648693i	0.945938	+	0.324346i	$0.105144\pi$
$353$			4302.30i			0.648693i	−0.945938	+	0.324346i	$0.894856\pi$
$354$		0			0
$355$			14456.0i			2.16126i
$356$		0			0
$357$	−78.3758	−	41.8937i	−0.0116193	−	0.00621078i
$358$		0			0
$359$	−2543.75			−0.373967			−0.186983	−	0.982363i	$-0.559871\pi$
$359$	−2543.75			−0.373967			−0.186983	+	0.982363i	$0.559871\pi$
$360$		0			0
$361$	−16865.0			−2.45881
$362$		0			0
$363$	−1666.09	−	890.560i	−0.240900	−	0.128767i
$364$		0			0
$365$		−	4941.19i		−	0.708585i
$366$		0			0
$367$			5291.35i			0.752606i	0.926497	+	0.376303i	$0.122805\pi$
$367$			5291.35i			0.752606i	−0.926497	+	0.376303i	$0.877195\pi$
$368$		0			0
$369$	2272.29	−	1518.24i	0.320571	−	0.214191i
$370$		0			0
$371$	1.66653			0.000233213
$372$		0			0
$373$	7860.77			1.09119			0.545597	−	0.838048i	$-0.316303\pi$
$373$	7860.77			1.09119			0.545597	+	0.838048i	$0.316303\pi$
$374$		0			0
$375$	−2059.05	+	3852.13i	−0.283543	+	0.530461i
$376$		0			0
$377$		−	15563.9i		−	2.12622i
$378$		0			0
$379$			3577.47i			0.484861i	0.970169	+	0.242431i	$0.0779447\pi$
$379$			3577.47i			0.484861i	−0.970169	+	0.242431i	$0.922055\pi$
$380$		0			0
$381$	−2531.01	+	4735.09i	−0.340335	+	0.636709i
$382$		0			0
$383$	−9144.30			−1.21998			−0.609990	−	0.792409i	$-0.708826\pi$
$383$	−9144.30			−1.21998			−0.609990	+	0.792409i	$0.708826\pi$
$384$		0			0
$385$	191.855			0.0253969
$386$		0			0
$387$	−2701.11	+	1804.75i	−0.354794	+	0.237056i
$388$		0			0
$389$		−	12996.7i		−	1.69398i	−0.531611	−	0.846989i	$-0.678413\pi$
$389$		−	12996.7i		−	1.69398i	0.531611	−	0.846989i	$-0.321587\pi$
$390$		0			0
$391$			2748.46i			0.355487i
$392$		0			0
$393$	1648.01	+	880.900i	0.211530	+	0.113067i
$394$		0			0
$395$	16259.5			2.07115
$396$		0			0
$397$	−2309.36			−0.291949			−0.145974	−	0.989288i	$-0.546632\pi$
$397$	−2309.36			−0.291949			−0.145974	+	0.989288i	$0.546632\pi$
$398$		0			0
$399$	−190.357	−	101.750i	−0.0238842	−	0.0127666i
$400$		0			0
$401$		−	4913.59i		−	0.611903i	−0.952047	−	0.305951i	$-0.901026\pi$
$401$		−	4913.59i		−	0.611903i	0.952047	−	0.305951i	$-0.0989745\pi$
$402$		0			0
$403$		−	5120.65i		−	0.632947i
$404$		0			0
$405$	−11638.9	−	4821.48i	−1.42801	−	0.591558i
$406$		0			0
$407$	−5459.50			−0.664907
$408$		0			0
$409$	3643.58			0.440497			0.220248	−	0.975444i	$-0.429313\pi$
$409$	3643.58			0.440497			0.220248	+	0.975444i	$0.429313\pi$
$410$		0			0
$411$	−928.958	+	1737.92i	−0.111489	+	0.208577i
$412$		0			0
$413$			12.3588i			0.00147249i
$414$		0			0
$415$			17473.3i			2.06682i
$416$		0			0
$417$	−4860.00	+	9092.23i	−0.570732	+	1.06774i
$418$		0			0
$419$	−7200.87			−0.839583			−0.419792	−	0.907621i	$-0.637897\pi$
$419$	−7200.87			−0.839583			−0.419792	+	0.907621i	$0.637897\pi$
$420$		0			0
$421$	6441.62			0.745713			0.372857	−	0.927889i	$-0.378378\pi$
$421$	6441.62			0.745713			0.372857	+	0.927889i	$0.378378\pi$
$422$		0			0
$423$	4399.27	+	6584.23i	0.505674	+	0.756823i
$424$		0			0
$425$			11011.9i			1.25684i
$426$		0			0
$427$		−	175.631i		−	0.0199049i
$428$		0			0
$429$	−12197.7	−	6519.96i	−1.37276	−	0.733769i
$430$		0			0
$431$	−5306.42			−0.593043			−0.296521	−	0.955026i	$-0.595827\pi$
$431$	−5306.42			−0.593043			−0.296521	+	0.955026i	$0.595827\pi$
$432$		0			0
$433$	2336.16			0.259281			0.129640	−	0.991561i	$-0.458618\pi$
$433$	2336.16			0.259281			0.129640	+	0.991561i	$0.458618\pi$
$434$		0			0
$435$	19061.8	+	10189.0i	2.10102	+	1.12304i
$436$		0			0
$437$			6675.39i			0.730726i
$438$		0			0
$439$			286.123i			0.0311068i	0.999879	+	0.0155534i	$0.00495100\pi$
$439$			286.123i			0.0311068i	−0.999879	+	0.0155534i	$0.995049\pi$
$440$		0			0
$441$	5143.91	+	7698.70i	0.555438	+	0.831303i
$442$		0			0
$443$	8021.59			0.860309			0.430155	−	0.902755i	$-0.358459\pi$
$443$	8021.59			0.860309			0.430155	+	0.902755i	$0.358459\pi$
$444$		0			0
$445$	7474.21			0.796205
$446$		0			0
$447$	5690.48	−	10645.9i	0.602126	−	1.12647i
$448$		0			0
$449$		−	12605.3i		−	1.32491i	−0.749103	−	0.662453i	$-0.769515\pi$
$449$		−	12605.3i		−	1.32491i	0.749103	−	0.662453i	$-0.230485\pi$
$450$		0			0
$451$			4166.57i			0.435024i
$452$		0			0
$453$	−6401.40	+	11975.9i	−0.663938	+	1.24211i
$454$		0			0
$455$	301.358			0.0310502
$456$		0			0
$457$	5652.98			0.578633			0.289317	−	0.957233i	$-0.406572\pi$
$457$	5652.98			0.578633			0.289317	+	0.957233i	$0.406572\pi$
$458$		0			0
$459$	−8854.35	+	871.840i	−0.900404	+	0.0886580i
$460$		0			0
$461$			2094.76i			0.211633i	0.994386	+	0.105817i	$0.0337456\pi$
$461$			2094.76i			0.211633i	−0.994386	+	0.105817i	$0.966254\pi$
$462$		0			0
$463$		−	7783.64i		−	0.781288i	−0.920542	−	0.390644i	$-0.872252\pi$
$463$		−	7783.64i		−	0.781288i	0.920542	−	0.390644i	$-0.127748\pi$
$464$		0			0
$465$	6271.49	+	3352.25i	0.625448	+	0.334316i
$466$		0			0
$467$	12864.3			1.27470			0.637352	−	0.770573i	$-0.280030\pi$
$467$	12864.3			1.27470			0.637352	+	0.770573i	$0.280030\pi$
$468$		0			0
$469$	−184.812			−0.0181958
$470$		0			0
$471$	−12741.4	−	6810.58i	−1.24648	−	0.666273i
$472$		0			0
$473$		−	4952.87i		−	0.481465i
$474$		0			0
$475$			26745.4i			2.58350i
$476$		0			0
$477$	138.727	−	92.6910i	0.0133163	−	0.00889733i
$478$		0			0
$479$	−8397.65			−0.801040			−0.400520	−	0.916288i	$-0.631171\pi$
$479$	−8397.65			−0.801040			−0.400520	+	0.916288i	$0.631171\pi$
$480$		0			0
$481$	−8575.56			−0.812915
$482$		0			0
$483$	−28.6302	+	53.5623i	−0.00269715	+	0.00504590i
$484$		0			0
$485$			17407.2i			1.62973i
$486$		0			0
$487$			9622.06i			0.895313i	0.894206	+	0.447656i	$0.147741\pi$
$487$			9622.06i			0.895313i	−0.894206	+	0.447656i	$0.852259\pi$
$488$		0			0
$489$	−1556.15	+	2911.28i	−0.143909	+	0.269228i
$490$		0			0
$491$	3382.57			0.310902			0.155451	−	0.987844i	$-0.450317\pi$
$491$	3382.57			0.310902			0.155451	+	0.987844i	$0.450317\pi$
$492$		0			0
$493$	15264.6			1.39449
$494$		0			0
$495$	15970.6	−	10670.8i	1.45015	−	0.968922i
$496$		0			0
$497$			225.601i			0.0203613i
$498$		0			0
$499$		−	12894.7i		−	1.15681i	−0.815751	−	0.578403i	$-0.803676\pi$
$499$		−	12894.7i		−	1.15681i	0.815751	−	0.578403i	$-0.196324\pi$
$500$		0			0
$501$	1464.23	+	782.664i	0.130573	+	0.0697941i
$502$		0			0
$503$	1741.55			0.154377			0.0771885	−	0.997017i	$-0.475406\pi$
$503$	1741.55			0.154377			0.0771885	+	0.997017i	$0.475406\pi$
$504$		0			0
$505$	21032.8			1.85336
$506$		0			0
$507$	−9091.80	−	4859.77i	−0.796412	−	0.425700i
$508$		0			0
$509$		−	5025.21i		−	0.437600i	−0.975770	−	0.218800i	$-0.929786\pi$
$509$		−	5025.21i		−	0.437600i	0.975770	−	0.218800i	$-0.0702142\pi$
$510$		0			0
$511$		−	77.1120i		−	0.00667561i
$512$		0			0
$513$	−21505.2	+	2117.51i	−1.85084	+	0.182242i
$514$		0			0
$515$	−23461.1			−2.00742
$516$		0			0
$517$	−12073.1			−1.02703
$518$		0			0
$519$	−4562.69	+	8536.02i	−0.385896	+	0.721945i
$520$		0			0
$521$		−	19105.2i		−	1.60656i	−0.595605	−	0.803278i	$-0.703087\pi$
$521$		−	19105.2i		−	1.60656i	0.595605	−	0.803278i	$-0.296913\pi$
$522$		0			0
$523$		−	2025.34i		−	0.169334i	−0.996409	−	0.0846671i	$-0.973017\pi$
$523$		−	2025.34i		−	0.169334i	0.996409	−	0.0846671i	$-0.0269827\pi$
$524$		0			0
$525$	−114.709	+	214.601i	−0.00953584	+	0.0178399i
$526$		0			0
$527$	5022.17			0.415122
$528$		0			0
$529$	−10288.7			−0.845623
$530$		0			0
$531$	687.386	+	1028.79i	0.0561771	+	0.0840781i
$532$		0			0
$533$			6544.68i			0.531860i
$534$		0			0
$535$		−	25752.1i		−	2.08105i
$536$		0			0
$537$	−18084.8	−	9666.75i	−1.45329	−	0.776818i
$538$		0			0
$539$	−14116.7			−1.12810
$540$		0			0
$541$	−1585.39			−0.125991			−0.0629955	−	0.998014i	$-0.520065\pi$
$541$	−1585.39			−0.125991			−0.0629955	+	0.998014i	$0.520065\pi$
$542$		0			0
$543$	5439.57	+	2907.57i	0.429898	+	0.229790i
$544$		0			0
$545$			32245.3i			2.53438i
$546$		0			0
$547$		−	4480.61i		−	0.350232i	−0.984548	−	0.175116i	$-0.943970\pi$
$547$		−	4480.61i		−	0.350232i	0.984548	−	0.175116i	$-0.0560300\pi$
$548$		0			0
$549$	−9768.45	−	14620.1i	−0.759394	−	1.13656i
$550$		0			0
$551$	37074.3			2.86646
$552$		0			0
$553$	253.745			0.0195124
$554$		0			0
$555$	5614.02	−	10502.9i	0.429373	−	0.803283i
$556$		0			0
$557$			12069.6i			0.918139i	0.888400	+	0.459070i	$0.151817\pi$
$557$			12069.6i			0.918139i	−0.888400	+	0.459070i	$0.848183\pi$
$558$		0			0
$559$		−	7779.77i		−	0.588639i
$560$		0			0
$561$	6394.57	−	11963.1i	0.481246	−	0.900329i
$562$		0			0
$563$	6218.25			0.465485			0.232742	−	0.972538i	$-0.425230\pi$
$563$	6218.25			0.465485			0.232742	+	0.972538i	$0.425230\pi$
$564$		0			0
$565$	−18105.0			−1.34811
$566$		0			0
$567$	−181.636	−	75.2438i	−0.0134533	−	0.00557309i
$568$		0			0
$569$			7681.87i			0.565977i	0.959123	+	0.282988i	$0.0913258\pi$
$569$			7681.87i			0.565977i	−0.959123	+	0.282988i	$0.908674\pi$
$570$		0			0
$571$		−	21206.3i		−	1.55421i	−0.629371	−	0.777105i	$-0.716687\pi$
$571$		−	21206.3i		−	1.55421i	0.629371	−	0.777105i	$-0.283313\pi$
$572$		0			0
$573$	5260.63	+	2811.93i	0.383536	+	0.205008i
$574$		0			0
$575$	7525.56			0.545804
$576$		0			0
$577$	9075.07			0.654766			0.327383	−	0.944892i	$-0.393833\pi$
$577$	9075.07			0.654766			0.327383	+	0.944892i	$0.393833\pi$
$578$		0			0
$579$	−1397.38	−	746.931i	−0.100299	−	0.0536121i
$580$		0			0
$581$			272.688i			0.0194716i
$582$		0			0
$583$			254.376i			0.0180706i
$584$		0			0
$585$	25086.0	−	16761.3i	1.77295	−	1.18460i
$586$		0			0
$587$	15653.2			1.10064			0.550320	−	0.834954i	$-0.314506\pi$
$587$	15653.2			1.10064			0.550320	+	0.834954i	$0.314506\pi$
$588$		0			0
$589$	12197.7			0.853309
$590$		0			0
$591$	−3069.08	+	5741.72i	−0.213612	+	0.399632i
$592$		0			0
$593$		−	1646.23i		−	0.114001i	−0.998374	−	0.0570003i	$-0.981846\pi$
$593$		−	1646.23i		−	0.114001i	0.998374	−	0.0570003i	$-0.0181536\pi$
$594$		0			0
$595$			295.562i			0.0203645i
$596$		0			0
$597$	−11425.1	+	21374.4i	−0.783247	+	1.46532i
$598$		0			0
$599$	6400.85			0.436614			0.218307	−	0.975880i	$-0.429947\pi$
$599$	6400.85			0.436614			0.218307	+	0.975880i	$0.429947\pi$
$600$		0			0
$601$	−1282.48			−0.0870443			−0.0435222	−	0.999052i	$-0.513858\pi$
$601$	−1282.48			−0.0870443			−0.0435222	+	0.999052i	$0.513858\pi$
$602$		0			0
$603$	−15384.3	+	10279.1i	−1.03897	+	0.694191i
$604$		0			0
$605$			6282.95i			0.422212i
$606$		0			0
$607$		−	5242.27i		−	0.350539i	−0.984521	−	0.175269i	$-0.943920\pi$
$607$		−	5242.27i		−	0.350539i	0.984521	−	0.175269i	$-0.0560796\pi$
$608$		0			0
$609$	297.479	+	159.009i	0.0197938	+	0.0105802i
$610$		0			0
$611$	−18964.0			−1.25565
$612$		0			0
$613$	−9985.56			−0.657933			−0.328966	−	0.944342i	$-0.606700\pi$
$613$	−9985.56			−0.657933			−0.328966	+	0.944342i	$0.606700\pi$
$614$		0			0
$615$	−8015.56	−	4284.50i	−0.525559	−	0.280923i
$616$		0			0
$617$			26504.8i			1.72941i	0.502283	+	0.864703i	$0.332493\pi$
$617$			26504.8i			1.72941i	−0.502283	+	0.864703i	$0.667507\pi$
$618$		0			0
$619$		−	22327.0i		−	1.44976i	−0.688877	−	0.724878i	$-0.741896\pi$
$619$		−	22327.0i		−	1.44976i	0.688877	−	0.724878i	$-0.258104\pi$
$620$		0			0
$621$	595.818	+	6051.09i	0.0385014	+	0.391017i
$622$		0			0
$623$	116.642			0.00750108
$624$		0			0
$625$	−7178.61			−0.459431
$626$		0			0
$627$	15531.0	−	29055.8i	0.989231	−	1.85068i
$628$		0			0
$629$		−	8410.64i		−	0.533154i
$630$		0			0
$631$		−	5040.02i		−	0.317971i	−0.987281	−	0.158986i	$-0.949178\pi$
$631$		−	5040.02i		−	0.317971i	0.987281	−	0.158986i	$-0.0508224\pi$
$632$		0			0
$633$	4520.27	−	8456.64i	0.283830	−	0.530998i
$634$		0			0
$635$	17856.4			1.11592
$636$		0			0
$637$	−22173.9			−1.37922
$638$		0			0
$639$	12547.7	+	18779.7i	0.776808	+	1.16262i
$640$		0			0
$641$			8994.13i			0.554207i	0.960840	+	0.277104i	$0.0893745\pi$
$641$			8994.13i			0.554207i	−0.960840	+	0.277104i	$0.910626\pi$
$642$		0			0
$643$			8789.99i			0.539103i	0.962986	+	0.269551i	$0.0868754\pi$
$643$			8789.99i			0.539103i	−0.962986	+	0.269551i	$0.913125\pi$
$644$		0			0
$645$	9528.23	+	5093.05i	0.581665	+	0.310913i
$646$		0			0
$647$	14441.6			0.877523			0.438762	−	0.898603i	$-0.355417\pi$
$647$	14441.6			0.877523			0.438762	+	0.898603i	$0.355417\pi$
$648$		0			0
$649$	−1886.42			−0.114096
$650$		0			0
$651$	97.8727	+	52.3152i	0.00589237	+	0.00314961i
$652$		0			0
$653$		−	16798.8i		−	1.00672i	−0.864078	−	0.503359i	$-0.832097\pi$
$653$		−	16798.8i		−	1.00672i	0.864078	−	0.503359i	$-0.167903\pi$
$654$		0			0
$655$		−	6214.79i		−	0.370736i
$656$		0			0
$657$	−4288.91	−	6419.05i	−0.254682	−	0.381174i
$658$		0			0
$659$	−10301.3			−0.608928			−0.304464	−	0.952524i	$-0.598477\pi$
$659$	−10301.3			−0.608928			−0.304464	+	0.952524i	$0.598477\pi$
$660$		0			0
$661$	31766.9			1.86927			0.934635	−	0.355609i	$-0.115726\pi$
$661$	31766.9			1.86927			0.934635	+	0.355609i	$0.115726\pi$
$662$		0			0
$663$	10044.3	−	18791.2i	0.588371	−	1.10074i
$664$		0			0
$665$			717.854i			0.0418604i
$666$		0			0
$667$		−	10431.9i		−	0.605583i
$668$		0			0
$669$	7263.63	−	13589.0i	0.419773	−	0.785323i
$670$		0			0
$671$	26808.0			1.54234
$672$		0			0
$673$	−12707.8			−0.727861			−0.363930	−	0.931426i	$-0.618565\pi$
$673$	−12707.8			−0.727861			−0.363930	+	0.931426i	$0.618565\pi$
$674$		0			0
$675$	2387.19	+	24244.1i	0.136123	+	1.38245i
$676$		0			0
$677$			19127.6i			1.08587i	0.839774	+	0.542936i	$0.182687\pi$
$677$			19127.6i			1.08587i	−0.839774	+	0.542936i	$0.817313\pi$
$678$		0			0
$679$			271.656i			0.0153537i
$680$		0			0
$681$	7052.52	+	3769.73i	0.396848	+	0.212124i
$682$		0			0
$683$	−14069.0			−0.788191			−0.394096	−	0.919069i	$-0.628942\pi$
$683$	−14069.0			−0.788191			−0.394096	+	0.919069i	$0.628942\pi$
$684$		0			0
$685$	6553.84			0.365561
$686$		0			0
$687$	−7598.16	−	4061.39i	−0.421962	−	0.225548i
$688$		0			0
$689$			399.564i			0.0220931i
$690$		0			0
$691$			19159.5i			1.05479i	0.849620	+	0.527396i	$0.176831\pi$
$691$			19159.5i			1.05479i	−0.849620	+	0.527396i	$0.823169\pi$
$692$		0			0
$693$	249.236	−	166.528i	0.0136619	−	0.00912825i
$694$		0			0
$695$	34287.5			1.87137
$696$		0			0
$697$	−6418.81			−0.348823
$698$		0			0
$699$	−3041.50	+	5690.12i	−0.164578	+	0.307897i
$700$		0			0
$701$		−	24736.8i		−	1.33281i	−0.745592	−	0.666403i	$-0.767833\pi$
$701$		−	24736.8i		−	1.33281i	0.745592	−	0.666403i	$-0.232167\pi$
$702$		0			0
$703$		−	20427.6i		−	1.09593i
$704$		0			0
$705$	12414.8	−	23226.0i	0.663220	−	1.24077i
$706$		0			0
$707$	328.237			0.0174605
$708$		0			0
$709$	10627.5			0.562937			0.281469	−	0.959570i	$-0.409178\pi$
$709$	10627.5			0.562937			0.281469	+	0.959570i	$0.409178\pi$
$710$		0			0
$711$	21122.6	−	14113.1i	1.11415	−	0.744420i
$712$		0			0
$713$		−	3432.17i		−	0.180274i
$714$		0			0
$715$			45998.7i			2.40595i
$716$		0			0
$717$	−10877.1	−	5814.05i	−0.566544	−	0.302831i
$718$		0			0
$719$	8256.38			0.428249			0.214124	−	0.976806i	$-0.431310\pi$
$719$	8256.38			0.428249			0.214124	+	0.976806i	$0.431310\pi$
$720$		0			0
$721$	−366.134			−0.0189120
$722$		0			0
$723$	−11879.6	−	6349.94i	−0.611077	−	0.326635i
$724$		0			0
$725$		−	41796.1i		−	2.14106i
$726$		0			0
$727$			21457.7i			1.09466i	0.836916	+	0.547332i	$0.184356\pi$
$727$			21457.7i			1.09466i	−0.836916	+	0.547332i	$0.815644\pi$
$728$		0			0
$729$	−19305.0	+	3838.94i	−0.980796	+	0.195038i
$730$		0			0
$731$	7630.15			0.386062
$732$		0			0
$733$	12687.2			0.639307			0.319654	−	0.947534i	$-0.396433\pi$
$733$	12687.2			0.639307			0.319654	+	0.947534i	$0.396433\pi$
$734$		0			0
$735$	14516.2	−	27157.4i	0.728488	−	1.36288i
$736$		0			0
$737$		−	28209.4i		−	1.40991i
$738$		0			0
$739$			7599.74i			0.378296i	0.981949	+	0.189148i	$0.0605727\pi$
$739$			7599.74i			0.378296i	−0.981949	+	0.189148i	$0.939427\pi$
$740$		0			0
$741$	24395.5	−	45639.7i	1.20943	−	2.26264i
$742$		0			0
$743$	−34061.2			−1.68181			−0.840904	−	0.541185i	$-0.817976\pi$
$743$	−34061.2			−1.68181			−0.840904	+	0.541185i	$0.817976\pi$
$744$		0			0
$745$	−40146.6			−1.97431
$746$		0			0
$747$	15166.7	+	22699.4i	0.742864	+	1.11182i
$748$		0			0
$749$		−	401.887i		−	0.0196056i
$750$		0			0
$751$			7091.67i			0.344579i	0.985046	+	0.172289i	$0.0551164\pi$
$751$			7091.67i			0.344579i	−0.985046	+	0.172289i	$0.944884\pi$
$752$		0			0
$753$	12603.5	+	6736.84i	0.609955	+	0.326035i
$754$		0			0
$755$	45162.2			2.17698
$756$		0			0
$757$	−10128.2			−0.486282			−0.243141	−	0.969991i	$-0.578178\pi$
$757$	−10128.2			−0.486282			−0.243141	+	0.969991i	$0.578178\pi$
$758$		0			0
$759$	−8175.65	−	4370.07i	−0.390984	−	0.208990i
$760$		0			0
$761$			22747.8i			1.08358i	0.840513	+	0.541792i	$0.182254\pi$
$761$			22747.8i			1.08358i	−0.840513	+	0.541792i	$0.817746\pi$
$762$		0			0
$763$			503.219i			0.0238765i
$764$		0			0
$765$	16438.9	+	24603.5i	0.776928	+	1.16280i
$766$		0			0
$767$	−2963.12			−0.139494
$768$		0			0
$769$	30168.9			1.41472			0.707359	−	0.706855i	$-0.249887\pi$
$769$	30168.9			1.41472			0.707359	+	0.706855i	$0.249887\pi$
$770$		0			0
$771$	−534.388	+	999.748i	−0.0249617	+	0.0466992i
$772$		0			0
$773$		−	12592.8i		−	0.585940i	−0.956122	−	0.292970i	$-0.905356\pi$
$773$		−	12592.8i		−	0.585940i	0.956122	−	0.292970i	$-0.0946437\pi$
$774$		0			0
$775$		−	13751.2i		−	0.637366i
$776$		0			0
$777$	87.6123	−	163.908i	0.00404514	−	0.00756776i
$778$		0			0
$779$	−15589.9			−0.717028
$780$		0			0
$781$	−34435.3			−1.57771
$782$		0			0
$783$	33607.0	−	3309.10i	1.53387	−	0.151032i
$784$		0			0
$785$			48049.0i			2.18464i
$786$		0			0
$787$		−	14594.4i		−	0.661036i	−0.943800	−	0.330518i	$-0.892777\pi$
$787$		−	14594.4i		−	0.661036i	0.943800	−	0.330518i	$-0.107223\pi$
$788$		0			0
$789$	33641.9	+	17982.3i	1.51797	+	0.811391i
$790$		0			0
$791$	−282.546			−0.0127006
$792$		0			0
$793$	42108.9			1.88567
$794$		0			0
$795$	−489.364	−	261.576i	−0.0218314	−	0.0116694i
$796$		0			0
$797$			36750.1i			1.63332i	0.577118	+	0.816661i	$0.304177\pi$
$797$			36750.1i			1.63332i	−0.577118	+	0.816661i	$0.695823\pi$
$798$		0			0
$799$		−	18599.3i		−	0.823522i
$800$		0			0
$801$	9709.67	−	6487.55i	0.428308	−	0.286175i
$802$		0			0
$803$	11770.2			0.517264
$804$		0			0
$805$	201.988			0.00884365
$806$		0			0
$807$	133.149	−	249.098i	0.00580799	−	0.0108658i
$808$		0			0
$809$		−	15914.1i		−	0.691605i	−0.938307	−	0.345803i	$-0.887607\pi$
$809$		−	15914.1i		−	0.691605i	0.938307	−	0.345803i	$-0.112393\pi$
$810$		0			0
$811$		−	1811.41i		−	0.0784304i	−0.999231	−	0.0392152i	$-0.987514\pi$
$811$		−	1811.41i		−	0.0784304i	0.999231	−	0.0392152i	$-0.0124858\pi$
$812$		0			0
$813$	8900.04	−	16650.5i	0.383934	−	0.718274i
$814$		0			0
$815$	10978.7			0.471861
$816$		0			0
$817$	18531.9			0.793574
$818$		0			0
$819$	391.491	−	261.576i	0.0167031	−	0.0111602i
$820$		0			0
$821$		−	15523.7i		−	0.659904i	−0.943998	−	0.329952i	$-0.892967\pi$
$821$		−	15523.7i		−	0.659904i	0.943998	−	0.329952i	$-0.107033\pi$
$822$		0			0
$823$		−	2879.06i		−	0.121941i	−0.998140	−	0.0609707i	$-0.980580\pi$
$823$		−	2879.06i		−	0.121941i	0.998140	−	0.0609707i	$-0.0194196\pi$
$824$		0			0
$825$	−32756.3	−	17509.0i	−1.38234	−	0.738891i
$826$		0			0
$827$	2163.66			0.0909766			0.0454883	−	0.998965i	$-0.485516\pi$
$827$	2163.66			0.0909766			0.0454883	+	0.998965i	$0.485516\pi$
$828$		0			0
$829$	−22363.4			−0.936928			−0.468464	−	0.883483i	$-0.655192\pi$
$829$	−22363.4			−0.936928			−0.468464	+	0.883483i	$0.655192\pi$
$830$		0			0
$831$	−28430.8	−	15196.9i	−1.18683	−	0.634387i
$832$		0			0
$833$		−	21747.4i		−	0.904566i
$834$		0			0
$835$		−	5521.73i		−	0.228847i
$836$		0			0
$837$	11057.0	−	1088.72i	0.456612	−	0.0449602i
$838$		0			0
$839$	10859.0			0.446833			0.223417	−	0.974723i	$-0.428279\pi$
$839$	10859.0			0.446833			0.223417	+	0.974723i	$0.428279\pi$
$840$		0			0
$841$	−33548.4			−1.37556
$842$		0			0
$843$	13843.3	−	25898.4i	0.565584	−	1.05811i
$844$		0			0
$845$			34285.9i			1.39583i
$846$		0			0
$847$			98.0515i			0.00397767i
$848$		0			0
$849$	−14346.8	+	26840.5i	−0.579955	+	1.08500i
$850$		0			0
$851$	−5747.85			−0.231532
$852$		0			0
$853$	−23518.0			−0.944012			−0.472006	−	0.881595i	$-0.656470\pi$
$853$	−23518.0			−0.944012			−0.472006	+	0.881595i	$0.656470\pi$
$854$		0			0
$855$	39926.5	+	59756.4i	1.59702	+	2.39021i
$856$		0			0
$857$			22859.6i			0.911164i	0.890194	+	0.455582i	$0.150569\pi$
$857$			22859.6i			0.911164i	−0.890194	+	0.455582i	$0.849431\pi$
$858$		0			0
$859$			25943.4i			1.03048i	0.857047	+	0.515238i	$0.172296\pi$
$859$			25943.4i			1.03048i	−0.857047	+	0.515238i	$0.827704\pi$
$860$		0			0
$861$	−125.091	−	66.8638i	−0.00495131	−	0.00264659i
$862$		0			0
$863$	−10444.3			−0.411967			−0.205983	−	0.978555i	$-0.566039\pi$
$863$	−10444.3			−0.411967			−0.205983	+	0.978555i	$0.566039\pi$
$864$		0			0
$865$	32190.0			1.26531
$866$		0			0
$867$	−4084.35	−	2183.18i	−0.159991	−	0.0855186i
$868$		0			0
$869$			38731.2i			1.51193i
$870$		0			0
$871$		−	44310.2i		−	1.72376i
$872$		0			0
$873$	15109.3	+	22613.5i	0.585763	+	0.876690i
$874$		0			0
$875$	226.703			0.00875882
$876$		0			0
$877$	−42209.3			−1.62521			−0.812603	−	0.582818i	$-0.801950\pi$
$877$	−42209.3			−1.62521			−0.812603	+	0.582818i	$0.801950\pi$
$878$		0			0
$879$	−13216.0	+	24724.9i	−0.507128	+	0.948750i
$880$		0			0
$881$		−	42919.3i		−	1.64130i	−0.571429	−	0.820651i	$-0.693611\pi$
$881$		−	42919.3i		−	1.64130i	0.571429	−	0.820651i	$-0.306389\pi$
$882$		0			0
$883$		−	9983.97i		−	0.380507i	−0.981735	−	0.190253i	$-0.939069\pi$
$883$		−	9983.97i		−	0.380507i	0.981735	−	0.190253i	$-0.0609309\pi$
$884$		0			0
$885$	1939.82	−	3629.07i	0.0736794	−	0.137842i
$886$		0			0
$887$	−42646.7			−1.61436			−0.807180	−	0.590305i	$-0.799007\pi$
$887$	−42646.7			−1.61436			−0.807180	+	0.590305i	$0.799007\pi$
$888$		0			0
$889$	278.667			0.0105131
$890$		0			0
$891$	11485.1	−	27724.7i	0.431835	−	1.04244i
$892$		0			0
$893$		−	45173.5i		−	1.69280i
$894$		0			0
$895$			68199.4i			2.54710i
$896$		0			0
$897$	−12842.0	−	6864.33i	−0.478017	−	0.255511i
$898$		0			0
$899$	−19061.8			−0.707173
$900$		0			0
$901$	−391.879			−0.0144899
$902$		0			0
$903$	148.697	+	79.4821i	0.00547989	+	0.00292912i
$904$		0			0
$905$		−	20513.1i		−	0.753456i
$906$		0			0
$907$		−	37924.9i		−	1.38840i	−0.719784	−	0.694198i	$-0.755759\pi$
$907$		−	37924.9i		−	1.38840i	0.719784	−	0.694198i	$-0.244241\pi$
$908$		0			0
$909$	27323.5	−	18256.3i	0.996988	−	0.666140i
$910$		0			0
$911$	18358.2			0.667655			0.333827	−	0.942634i	$-0.391660\pi$
$911$	18358.2			0.667655			0.333827	+	0.942634i	$0.391660\pi$
$912$		0			0
$913$	−41622.5			−1.50877
$914$		0			0
$915$	−27566.8	+	51572.7i	−0.995989	+	1.86332i
$916$		0			0
$917$		−	96.9879i		−	0.00349272i
$918$		0			0
$919$			30049.2i			1.07860i	0.842114	+	0.539300i	$0.181311\pi$
$919$			30049.2i			1.07860i	−0.842114	+	0.539300i	$0.818689\pi$
$920$		0			0
$921$	−20058.8	+	37526.6i	−0.717656	+	1.34261i
$922$		0			0
$923$	−54089.6			−1.92891
$924$		0			0
$925$	−23029.2			−0.818589
$926$		0			0
$927$	−30478.1	+	20364.1i	−1.07986	+	0.721514i
$928$		0			0
$929$		−	34115.5i		−	1.20484i	−0.798181	−	0.602418i	$-0.794204\pi$
$929$		−	34115.5i		−	1.20484i	0.798181	−	0.602418i	$-0.205796\pi$
$930$		0			0
$931$		−	52819.7i		−	1.85939i
$932$		0			0
$933$	−8424.69	−	4503.19i	−0.295618	−	0.158015i
$934$		0			0
$935$	−45114.0			−1.57795
$936$		0			0
$937$	−1709.73			−0.0596100			−0.0298050	−	0.999556i	$-0.509489\pi$
$937$	−1709.73			−0.0596100			−0.0298050	+	0.999556i	$0.509489\pi$
$938$		0			0
$939$	−1343.06	−	717.894i	−0.0466762	−	0.0249495i
$940$		0			0
$941$			6856.94i			0.237545i	0.992921	+	0.118772i	$0.0378959\pi$
$941$			6856.94i			0.237545i	−0.992921	+	0.118772i	$0.962104\pi$
$942$		0			0
$943$			4386.64i			0.151483i
$944$		0			0
$945$	64.0727	+	650.718i	0.00220559	+	0.0223999i
$946$		0			0
$947$	−34558.0			−1.18583			−0.592917	−	0.805264i	$-0.702024\pi$
$947$	−34558.0			−1.18583			−0.592917	+	0.805264i	$0.702024\pi$
$948$		0			0
$949$	18488.2			0.632406
$950$		0			0
$951$	3289.71	−	6154.48i	0.112172	−	0.209855i
$952$		0			0
$953$		−	39786.8i		−	1.35238i	−0.736725	−	0.676192i	$-0.763629\pi$
$953$		−	39786.8i		−	1.35238i	0.736725	−	0.676192i	$-0.236371\pi$
$954$		0			0
$955$		−	19838.3i		−	0.672200i
$956$		0			0
$957$	−24270.8	+	45406.6i	−0.819817	+	1.53374i
$958$		0			0
$959$	102.279			0.00344397
$960$		0			0
$961$	23519.5			0.789484
$962$		0			0
$963$	−22352.6	−	33454.3i	−0.747978	−	1.11947i
$964$		0			0
$965$			5269.64i			0.175788i
$966$		0			0
$967$		−	11168.4i		−	0.371406i	−0.982606	−	0.185703i	$-0.940544\pi$
$967$		−	11168.4i		−	0.371406i	0.982606	−	0.185703i	$-0.0594563\pi$
$968$		0			0
$969$	44762.0	+	23926.3i	1.48397	+	0.793213i
$970$		0			0
$971$	−13885.1			−0.458901			−0.229450	−	0.973320i	$-0.573693\pi$
$971$	−13885.1			−0.458901			−0.229450	+	0.973320i	$0.573693\pi$
$972$		0			0
$973$	535.091			0.0176302
$974$		0			0
$975$	−51452.4	−	27502.4i	−1.69005	−	0.903367i
$976$		0			0
$977$			5137.09i			0.168219i	0.996457	+	0.0841096i	$0.0268046\pi$
$977$			5137.09i			0.168219i	−0.996457	+	0.0841096i	$0.973195\pi$
$978$		0			0
$979$			17804.1i			0.581226i
$980$		0			0
$981$	27988.6	+	41889.6i	0.910916	+	1.36333i
$982$		0			0
$983$	28815.3			0.934959			0.467480	−	0.884004i	$-0.345162\pi$
$983$	28815.3			0.934959			0.467480	+	0.884004i	$0.345162\pi$
$984$		0			0
$985$	21652.5			0.700412
$986$		0			0
$987$	193.746	−	362.465i	0.00624822	−	0.0116893i
$988$		0			0
$989$		−	5214.46i		−	0.167655i
$990$		0			0
$991$		−	52902.8i		−	1.69578i	−0.530175	−	0.847888i	$-0.677874\pi$
$991$		−	52902.8i		−	1.69578i	0.530175	−	0.847888i	$-0.322126\pi$
$992$		0			0
$993$	−21619.1	+	40445.6i	−0.690897	+	1.29255i
$994$		0			0
$995$	80604.7			2.56818
$996$		0			0
$997$	−29517.0			−0.937625			−0.468812	−	0.883298i	$-0.655318\pi$
$997$	−29517.0			−0.937625			−0.468812	+	0.883298i	$0.655318\pi$
$998$		0			0
$999$	−1823.28	−	18517.1i	−0.0577438	−	0.586442i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.c.m.767.1		4
3.2	odd	2		768.4.c.o.767.3		4
4.3	odd	2		768.4.c.o.767.4		4
8.3	odd	2		768.4.c.n.767.1		4
8.5	even	2		768.4.c.p.767.4		4
12.11	even	2	inner	768.4.c.m.767.2		4
16.3	odd	4		384.4.f.g.191.4	yes	8
16.5	even	4		384.4.f.h.191.3	yes	8
16.11	odd	4		384.4.f.g.191.5	yes	8
16.13	even	4		384.4.f.h.191.6	yes	8
24.5	odd	2		768.4.c.n.767.2		4
24.11	even	2		768.4.c.p.767.3		4
48.5	odd	4		384.4.f.g.191.2	✓	8
48.11	even	4		384.4.f.h.191.8	yes	8
48.29	odd	4		384.4.f.g.191.7	yes	8
48.35	even	4		384.4.f.h.191.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.f.g.191.2	✓	8	48.5	odd	4
384.4.f.g.191.4	yes	8	16.3	odd	4
384.4.f.g.191.5	yes	8	16.11	odd	4
384.4.f.g.191.7	yes	8	48.29	odd	4
384.4.f.h.191.1	yes	8	48.35	even	4
384.4.f.h.191.3	yes	8	16.5	even	4
384.4.f.h.191.6	yes	8	16.13	even	4
384.4.f.h.191.8	yes	8	48.11	even	4
768.4.c.m.767.1		4	1.1	even	1	trivial
768.4.c.m.767.2		4	12.11	even	2	inner
768.4.c.n.767.1		4	8.3	odd	2
768.4.c.n.767.2		4	24.5	odd	2
768.4.c.o.767.3		4	3.2	odd	2
768.4.c.o.767.4		4	4.3	odd	2
768.4.c.p.767.3		4	24.11	even	2
768.4.c.p.767.4		4	8.5	even	2

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.58258 - 2.44949i) q^{3} +17.2813i q^{5} +0.269691i q^{7} +(15.0000 + 22.4499i) q^{9} +O(q^{10})\)	\(q+(-4.58258 - 2.44949i) q^{3} +17.2813i q^{5} +0.269691i q^{7} +(15.0000 + 22.4499i) q^{9} -41.1652 q^{11} -64.6606 q^{13} +(42.3303 - 79.1927i) q^{15} -63.4170i q^{17} -154.026i q^{19} +(0.660606 - 1.23588i) q^{21} -43.3394 q^{23} -173.642 q^{25} +(-13.7477 - 139.621i) q^{27} +240.702i q^{29} +79.1927i q^{31} +(188.642 + 100.834i) q^{33} -4.66061 q^{35} +132.624 q^{37} +(296.312 + 158.385i) q^{39} -101.216i q^{41} +120.317i q^{43} +(-387.964 + 259.219i) q^{45} +293.285 q^{47} +342.927 q^{49} +(-155.339 + 290.613i) q^{51} -6.17940i q^{53} -711.386i q^{55} +(-377.285 + 705.835i) q^{57} +45.8258 q^{59} -651.230 q^{61} +(-6.05455 + 4.04537i) q^{63} -1117.42i q^{65} +685.273i q^{67} +(198.606 + 106.159i) q^{69} +836.515 q^{71} -285.927 q^{73} +(795.730 + 425.335i) q^{75} -11.1019i q^{77} -940.874i q^{79} +(-279.000 + 673.498i) q^{81} +1011.11 q^{83} +1095.93 q^{85} +(589.597 - 1103.03i) q^{87} -432.503i q^{89} -17.4384i q^{91} +(193.982 - 362.907i) q^{93} +2661.76 q^{95} +1007.28 q^{97} +(-617.477 - 924.155i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 60 q^{9}+O(q^{10}) \) 4 * q + 60 * q^9	\( 4 q + 60 q^{9} - 128 q^{11} - 112 q^{13} + 96 q^{15} - 144 q^{21} - 320 q^{23} - 108 q^{25} + 168 q^{33} + 128 q^{35} - 496 q^{37} + 672 q^{39} - 672 q^{45} - 388 q^{49} - 768 q^{51} - 336 q^{57} - 112 q^{61} - 1344 q^{63} - 672 q^{69} - 320 q^{71} + 616 q^{73} + 2688 q^{75} - 1116 q^{81} + 2688 q^{83} + 2624 q^{85} + 672 q^{87} + 336 q^{93} + 4928 q^{95} + 2856 q^{97} - 1920 q^{99}+O(q^{100}) \) 4 * q + 60 * q^9 - 128 * q^11 - 112 * q^13 + 96 * q^15 - 144 * q^21 - 320 * q^23 - 108 * q^25 + 168 * q^33 + 128 * q^35 - 496 * q^37 + 672 * q^39 - 672 * q^45 - 388 * q^49 - 768 * q^51 - 336 * q^57 - 112 * q^61 - 1344 * q^63 - 672 * q^69 - 320 * q^71 + 616 * q^73 + 2688 * q^75 - 1116 * q^81 + 2688 * q^83 + 2624 * q^85 + 672 * q^87 + 336 * q^93 + 4928 * q^95 + 2856 * q^97 - 1920 * q^99

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