LMFDB - Embedded newform 768.4.f.b.383.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(383,768)]

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, 1, 1]))

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

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

chi := DirichletCharacter("768.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.f (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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.8
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.4.f.b.383.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.73205 + 4.89898i) q^{3} +16.9706 q^{5} +17.3205i q^{7} +(-21.0000 + 16.9706i) q^{9} -29.3939i q^{11} +26.0000i q^{13} +(29.3939 + 83.1384i) q^{15} +67.8823i q^{17} -107.387 q^{19} +(-84.8528 + 30.0000i) q^{21} -176.363 q^{23} +163.000 q^{25} +(-119.512 - 73.4847i) q^{27} +16.9706 q^{29} -31.1769i q^{31} +(144.000 - 50.9117i) q^{33} +293.939i q^{35} +206.000i q^{37} +(-127.373 + 45.0333i) q^{39} +305.470i q^{41} +93.5307 q^{43} +(-356.382 + 288.000i) q^{45} -117.576 q^{47} +43.0000 q^{49} +(-332.554 + 117.576i) q^{51} -50.9117 q^{53} -498.831i q^{55} +(-186.000 - 526.087i) q^{57} +558.484i q^{59} -278.000i q^{61} +(-293.939 - 363.731i) q^{63} +441.235i q^{65} +890.274 q^{67} +(-305.470 - 864.000i) q^{69} -58.7878 q^{71} +422.000 q^{73} +(282.324 + 798.534i) q^{75} +509.117 q^{77} -668.572i q^{79} +(153.000 - 712.764i) q^{81} -29.3939i q^{83} +1152.00i q^{85} +(29.3939 + 83.1384i) q^{87} +373.352i q^{89} -450.333 q^{91} +(152.735 - 54.0000i) q^{93} -1822.42 q^{95} -1070.00 q^{97} +(498.831 + 617.271i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 168 q^{9} + 1304 q^{25} + 1152 q^{33} + 344 q^{49} - 1488 q^{57} + 3376 q^{73} + 1224 q^{81} - 8560 q^{97}+O(q^{100}) $ 8 * q - 168 * q^9 + 1304 * q^25 + 1152 * q^33 + 344 * q^49 - 1488 * q^57 + 3376 * q^73 + 1224 * q^81 - 8560 * q^97

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
$3$	1.73205	+	4.89898i	0.333333	+	0.942809i
$3$	1.73205	+	4.89898i	0.333333	+	0.942809i
$4$		0			0
$5$	16.9706			1.51789			0.758947	−	0.651153i	$-0.225714\pi$
$5$	16.9706			1.51789			0.758947	+	0.651153i	$0.225714\pi$
$6$		0			0
$7$			17.3205i			0.935220i	0.883935	+	0.467610i	$0.154885\pi$
$7$			17.3205i			0.935220i	−0.883935	+	0.467610i	$0.845115\pi$
$8$		0			0
$9$	−21.0000	+	16.9706i	−0.777778	+	0.628539i
$10$		0			0
$11$		−	29.3939i		−	0.805690i	−0.915268	−	0.402845i	$-0.868021\pi$
$11$		−	29.3939i		−	0.805690i	0.915268	−	0.402845i	$-0.131979\pi$
$12$		0			0
$13$			26.0000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			26.0000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$	29.3939	+	83.1384i	0.505964	+	1.43108i
$16$		0			0
$17$			67.8823i			0.968463i	0.874940	+	0.484231i	$0.160901\pi$
$17$			67.8823i			0.968463i	−0.874940	+	0.484231i	$0.839099\pi$
$18$		0			0
$19$	−107.387			−1.29665			−0.648324	−	0.761365i	$-0.724530\pi$
$19$	−107.387			−1.29665			−0.648324	+	0.761365i	$0.724530\pi$
$20$		0			0
$21$	−84.8528	+	30.0000i	−0.881733	+	0.311740i
$22$		0			0
$23$	−176.363			−1.59888			−0.799441	−	0.600745i	$-0.794871\pi$
$23$	−176.363			−1.59888			−0.799441	+	0.600745i	$0.794871\pi$
$24$		0			0
$25$	163.000			1.30400
$26$		0			0
$27$	−119.512	−	73.4847i	−0.851852	−	0.523783i
$28$		0			0
$29$	16.9706			0.108667			0.0543337	−	0.998523i	$-0.482697\pi$
$29$	16.9706			0.108667			0.0543337	+	0.998523i	$0.482697\pi$
$30$		0			0
$31$		−	31.1769i		−	0.180630i	−0.995913	−	0.0903151i	$-0.971213\pi$
$31$		−	31.1769i		−	0.180630i	0.995913	−	0.0903151i	$-0.0287874\pi$
$32$		0			0
$33$	144.000	−	50.9117i	0.759612	−	0.268563i
$34$		0			0
$35$			293.939i			1.41956i
$36$		0			0
$37$			206.000i			0.915302i	0.889132	+	0.457651i	$0.151309\pi$
$37$			206.000i			0.915302i	−0.889132	+	0.457651i	$0.848691\pi$
$38$		0			0
$39$	−127.373	+	45.0333i	−0.522976	+	0.184900i
$40$		0			0
$41$			305.470i			1.16357i	0.813342	+	0.581786i	$0.197646\pi$
$41$			305.470i			1.16357i	−0.813342	+	0.581786i	$0.802354\pi$
$42$		0			0
$43$	93.5307			0.331705			0.165852	−	0.986151i	$-0.446962\pi$
$43$	93.5307			0.331705			0.165852	+	0.986151i	$0.446962\pi$
$44$		0			0
$45$	−356.382	+	288.000i	−1.18058	+	0.954056i
$46$		0			0
$47$	−117.576			−0.364897			−0.182448	−	0.983215i	$-0.558402\pi$
$47$	−117.576			−0.364897			−0.182448	+	0.983215i	$0.558402\pi$
$48$		0			0
$49$	43.0000			0.125364
$50$		0			0
$51$	−332.554	+	117.576i	−0.913075	+	0.322821i
$52$		0			0
$53$	−50.9117			−0.131948			−0.0659741	−	0.997821i	$-0.521015\pi$
$53$	−50.9117			−0.131948			−0.0659741	+	0.997821i	$0.521015\pi$
$54$		0			0
$55$		−	498.831i		−	1.22295i
$56$		0			0
$57$	−186.000	−	526.087i	−0.432216	−	1.22249i
$58$		0			0
$59$			558.484i			1.23235i	0.787611	+	0.616173i	$0.211318\pi$
$59$			558.484i			1.23235i	−0.787611	+	0.616173i	$0.788682\pi$
$60$		0			0
$61$		−	278.000i		−	0.583512i	−0.956493	−	0.291756i	$-0.905760\pi$
$61$		−	278.000i		−	0.583512i	0.956493	−	0.291756i	$-0.0942396\pi$
$62$		0			0
$63$	−293.939	−	363.731i	−0.587822	−	0.727393i
$64$		0			0
$65$			441.235i			0.841976i
$66$		0			0
$67$	890.274			1.62335			0.811674	−	0.584111i	$-0.198557\pi$
$67$	890.274			1.62335			0.811674	+	0.584111i	$0.198557\pi$
$68$		0			0
$69$	−305.470	−	864.000i	−0.532961	−	1.50744i
$70$		0			0
$71$	−58.7878			−0.0982651			−0.0491326	−	0.998792i	$-0.515646\pi$
$71$	−58.7878			−0.0982651			−0.0491326	+	0.998792i	$0.515646\pi$
$72$		0			0
$73$	422.000			0.676594			0.338297	−	0.941039i	$-0.390149\pi$
$73$	422.000			0.676594			0.338297	+	0.941039i	$0.390149\pi$
$74$		0			0
$75$	282.324	+	798.534i	0.434667	+	1.22942i
$76$		0			0
$77$	509.117			0.753497
$78$		0			0
$79$		−	668.572i		−	0.952154i	−0.879404	−	0.476077i	$-0.842058\pi$
$79$		−	668.572i		−	0.952154i	0.879404	−	0.476077i	$-0.157942\pi$
$80$		0			0
$81$	153.000	−	712.764i	0.209877	−	0.977728i
$82$		0			0
$83$		−	29.3939i		−	0.0388723i	−0.999811	−	0.0194361i	$-0.993813\pi$
$83$		−	29.3939i		−	0.0388723i	0.999811	−	0.0194361i	$-0.00618710\pi$
$84$		0			0
$85$			1152.00i			1.47002i
$86$		0			0
$87$	29.3939	+	83.1384i	0.0362225	+	0.102453i
$88$		0			0
$89$			373.352i			0.444666i	0.974971	+	0.222333i	$0.0713672\pi$
$89$			373.352i			0.444666i	−0.974971	+	0.222333i	$0.928633\pi$
$90$		0			0
$91$	−450.333			−0.518766
$92$		0			0
$93$	152.735	−	54.0000i	0.170300	−	0.0602101i
$94$		0			0
$95$	−1822.42			−1.96817
$96$		0			0
$97$	−1070.00			−1.12002			−0.560011	−	0.828486i	$-0.689203\pi$
$97$	−1070.00			−1.12002			−0.560011	+	0.828486i	$0.689203\pi$
$98$		0			0
$99$	498.831	+	617.271i	0.506408	+	0.626648i
$100$		0			0
$101$	1781.91			1.75551			0.877755	−	0.479109i	$-0.159040\pi$
$101$	1781.91			1.75551			0.877755	+	0.479109i	$0.159040\pi$
$102$		0			0
$103$			17.3205i			0.0165693i	0.999966	+	0.00828466i	$0.00263712\pi$
$103$			17.3205i			0.0165693i	−0.999966	+	0.00828466i	$0.997363\pi$
$104$		0			0
$105$	−1440.00	+	509.117i	−1.33838	+	0.473188i
$106$		0			0
$107$			1381.51i			1.24819i	0.781350	+	0.624093i	$0.214531\pi$
$107$			1381.51i			1.24819i	−0.781350	+	0.624093i	$0.785469\pi$
$108$		0			0
$109$		−	358.000i		−	0.314589i	−0.987552	−	0.157294i	$-0.949723\pi$
$109$		−	358.000i		−	0.314589i	0.987552	−	0.157294i	$-0.0502772\pi$
$110$		0			0
$111$	−1009.19	+	356.802i	−0.862955	+	0.305101i
$112$		0			0
$113$			67.8823i			0.0565117i	0.999601	+	0.0282559i	$0.00899532\pi$
$113$			67.8823i			0.0565117i	−0.999601	+	0.0282559i	$0.991005\pi$
$114$		0			0
$115$	−2992.98			−2.42693
$116$		0			0
$117$	−441.235	−	546.000i	−0.348651	−	0.431433i
$118$		0			0
$119$	−1175.76			−0.905725
$120$		0			0
$121$	467.000			0.350864
$122$		0			0
$123$	−1496.49	+	529.090i	−1.09703	+	0.387857i
$124$		0			0
$125$	644.881			0.461440
$126$		0			0
$127$		−	31.1769i		−	0.0217835i	−0.999941	−	0.0108917i	$-0.996533\pi$
$127$		−	31.1769i		−	0.0217835i	0.999941	−	0.0108917i	$-0.00346702\pi$
$128$		0			0
$129$	162.000	+	458.205i	0.110568	+	0.312734i
$130$		0			0
$131$			1734.24i			1.15665i	0.815806	+	0.578325i	$0.196293\pi$
$131$			1734.24i			1.15665i	−0.815806	+	0.578325i	$0.803707\pi$
$132$		0			0
$133$		−	1860.00i		−	1.21265i
$134$		0			0
$135$	−2028.18	−	1247.08i	−1.29302	−	0.795046i
$136$		0			0
$137$		−	2274.06i		−	1.41814i	−0.705136	−	0.709072i	$-0.749114\pi$
$137$		−	2274.06i		−	1.41814i	0.705136	−	0.709072i	$-0.250886\pi$
$138$		0			0
$139$	−1541.53			−0.940651			−0.470325	−	0.882493i	$-0.655863\pi$
$139$	−1541.53			−0.940651			−0.470325	+	0.882493i	$0.655863\pi$
$140$		0			0
$141$	−203.647	−	576.000i	−0.121632	−	0.344028i
$142$		0			0
$143$	764.241			0.446916
$144$		0			0
$145$	288.000			0.164946
$146$		0			0
$147$	74.4782	+	210.656i	0.0417881	+	0.118195i
$148$		0			0
$149$	−2358.91			−1.29698			−0.648488	−	0.761225i	$-0.724598\pi$
$149$	−2358.91			−1.29698			−0.648488	+	0.761225i	$0.724598\pi$
$150$		0			0
$151$			2040.36i			1.09961i	0.835292	+	0.549807i	$0.185299\pi$
$151$			2040.36i			1.09961i	−0.835292	+	0.549807i	$0.814701\pi$
$152$		0			0
$153$	−1152.00	−	1425.53i	−0.608717	−	0.753249i
$154$		0			0
$155$		−	529.090i		−	0.274178i
$156$		0			0
$157$			106.000i			0.0538836i	0.999637	+	0.0269418i	$0.00857687\pi$
$157$			106.000i			0.0538836i	−0.999637	+	0.0269418i	$0.991423\pi$
$158$		0			0
$159$	−88.1816	−	249.415i	−0.0439828	−	0.124402i
$160$		0			0
$161$		−	3054.70i		−	1.49531i
$162$		0			0
$163$	−79.6743			−0.0382857			−0.0191429	−	0.999817i	$-0.506094\pi$
$163$	−79.6743			−0.0382857			−0.0191429	+	0.999817i	$0.506094\pi$
$164$		0			0
$165$	2443.76	−	864.000i	1.15301	−	0.407650i
$166$		0			0
$167$	−3115.75			−1.44374			−0.721868	−	0.692030i	$-0.756716\pi$
$167$	−3115.75			−1.44374			−0.721868	+	0.692030i	$0.756716\pi$
$168$		0			0
$169$	1521.00			0.692308
$170$		0			0
$171$	2255.13	−	1822.42i	1.00850	−	0.814994i
$172$		0			0
$173$	−2223.14			−0.977009			−0.488504	−	0.872561i	$-0.662457\pi$
$173$	−2223.14			−0.977009			−0.488504	+	0.872561i	$0.662457\pi$
$174$		0			0
$175$			2823.24i			1.21953i
$176$		0			0
$177$	−2736.00	+	967.322i	−1.16187	+	0.410782i
$178$		0			0
$179$			2792.42i			1.16601i	0.812470	+	0.583003i	$0.198122\pi$
$179$			2792.42i			1.16601i	−0.812470	+	0.583003i	$0.801878\pi$
$180$		0			0
$181$			4510.00i			1.85208i	0.377431	+	0.926038i	$0.376808\pi$
$181$			4510.00i			1.85208i	−0.377431	+	0.926038i	$0.623192\pi$
$182$		0			0
$183$	1361.92	−	481.510i	0.550141	−	0.194504i
$184$		0			0
$185$			3495.94i			1.38933i
$186$		0			0
$187$	1995.32			0.780280
$188$		0			0
$189$	1272.79	−	2070.00i	0.489852	−	0.796668i
$190$		0			0
$191$	−1881.21			−0.712667			−0.356334	−	0.934359i	$-0.615973\pi$
$191$	−1881.21			−0.712667			−0.356334	+	0.934359i	$0.615973\pi$
$192$		0			0
$193$	4994.00			1.86257			0.931285	−	0.364292i	$-0.118689\pi$
$193$	4994.00			1.86257			0.931285	+	0.364292i	$0.118689\pi$
$194$		0			0
$195$	−2161.60	+	764.241i	−0.793822	+	0.280659i
$196$		0			0
$197$	−2155.26			−0.779472			−0.389736	−	0.920927i	$-0.627434\pi$
$197$	−2155.26			−0.779472			−0.389736	+	0.920927i	$0.627434\pi$
$198$		0			0
$199$		−	3973.32i		−	1.41538i	−0.706521	−	0.707692i	$-0.749736\pi$
$199$		−	3973.32i		−	1.41538i	0.706521	−	0.707692i	$-0.250264\pi$
$200$		0			0
$201$	1542.00	+	4361.43i	0.541116	+	1.53051i
$202$		0			0
$203$			293.939i			0.101628i
$204$		0			0
$205$			5184.00i			1.76618i
$206$		0			0
$207$	3703.63	−	2992.98i	1.24357	−	1.00496i
$208$		0			0
$209$			3156.52i			1.04470i
$210$		0			0
$211$	4908.63			1.60154			0.800768	−	0.598974i	$-0.204425\pi$
$211$	4908.63			1.60154			0.800768	+	0.598974i	$0.204425\pi$
$212$		0			0
$213$	−101.823	−	288.000i	−0.0327550	−	0.0926452i
$214$		0			0
$215$	1587.27			0.503492
$216$		0			0
$217$	540.000			0.168929
$218$		0			0
$219$	730.925	+	2067.37i	0.225531	+	0.637899i
$220$		0			0
$221$	−1764.94			−0.537206
$222$		0			0
$223$		−	4021.82i		−	1.20772i	−0.797091	−	0.603859i	$-0.793629\pi$
$223$		−	4021.82i		−	1.20772i	0.797091	−	0.603859i	$-0.206371\pi$
$224$		0			0
$225$	−3423.00	+	2766.20i	−1.01422	+	0.819615i
$226$		0			0
$227$		−	2968.78i		−	0.868039i	−0.900903	−	0.434020i	$-0.857095\pi$
$227$		−	2968.78i		−	0.868039i	0.900903	−	0.434020i	$-0.142905\pi$
$228$		0			0
$229$			430.000i			0.124084i	0.998074	+	0.0620419i	$0.0197612\pi$
$229$			430.000i			0.124084i	−0.998074	+	0.0620419i	$0.980239\pi$
$230$		0			0
$231$	881.816	+	2494.15i	0.251166	+	0.710404i
$232$		0			0
$233$		−	6346.99i		−	1.78457i	−0.451471	−	0.892286i	$-0.649101\pi$
$233$		−	6346.99i		−	1.78457i	0.451471	−	0.892286i	$-0.350899\pi$
$234$		0			0
$235$	−1995.32			−0.553874
$236$		0			0
$237$	3275.32	−	1158.00i	0.897700	−	0.317385i
$238$		0			0
$239$	4115.14			1.11375			0.556875	−	0.830596i	$-0.312000\pi$
$239$	4115.14			1.11375			0.556875	+	0.830596i	$0.312000\pi$
$240$		0			0
$241$	4690.00			1.25357			0.626783	−	0.779194i	$-0.284371\pi$
$241$	4690.00			1.25357			0.626783	+	0.779194i	$0.284371\pi$
$242$		0			0
$243$	3756.82	−	484.999i	0.991770	−	0.128036i
$244$		0			0
$245$	729.734			0.190290
$246$		0			0
$247$		−	2792.07i		−	0.719251i
$248$		0			0
$249$	144.000	−	50.9117i	0.0366491	−	0.0129574i
$250$		0			0
$251$			1263.94i			0.317845i	0.987291	+	0.158922i	$0.0508019\pi$
$251$			1263.94i			0.317845i	−0.987291	+	0.158922i	$0.949198\pi$
$252$		0			0
$253$			5184.00i			1.28820i
$254$		0			0
$255$	−5643.62	+	1995.32i	−1.38595	+	0.490008i
$256$		0			0
$257$		−	271.529i		−	0.0659047i	−0.999457	−	0.0329524i	$-0.989509\pi$
$257$		−	271.529i		−	0.0659047i	0.999457	−	0.0329524i	$-0.0104910\pi$
$258$		0			0
$259$	−3568.02			−0.856009
$260$		0			0
$261$	−356.382	+	288.000i	−0.0845191	+	0.0683017i
$262$		0			0
$263$	5114.53			1.19915			0.599574	−	0.800320i	$-0.295337\pi$
$263$	5114.53			1.19915			0.599574	+	0.800320i	$0.295337\pi$
$264$		0			0
$265$	−864.000			−0.200283
$266$		0			0
$267$	−1829.05	+	646.665i	−0.419235	+	0.148222i
$268$		0			0
$269$	−3173.50			−0.719299			−0.359649	−	0.933087i	$-0.617104\pi$
$269$	−3173.50			−0.719299			−0.359649	+	0.933087i	$0.617104\pi$
$270$		0			0
$271$		−	668.572i		−	0.149863i	−0.997189	−	0.0749314i	$-0.976126\pi$
$271$		−	668.572i		−	0.149863i	0.997189	−	0.0749314i	$-0.0238738\pi$
$272$		0			0
$273$	−780.000	−	2206.17i	−0.172922	−	0.489098i
$274$		0			0
$275$		−	4791.20i		−	1.05062i
$276$		0			0
$277$		−	2018.00i		−	0.437725i	−0.975756	−	0.218863i	$-0.929765\pi$
$277$		−	2018.00i		−	0.437725i	0.975756	−	0.218863i	$-0.0702346\pi$
$278$		0			0
$279$	529.090	+	654.715i	0.113533	+	0.140490i
$280$		0			0
$281$			4717.82i			1.00157i	0.865572	+	0.500785i	$0.166955\pi$
$281$			4717.82i			1.00157i	−0.865572	+	0.500785i	$0.833045\pi$
$282$		0			0
$283$	5081.84			1.06743			0.533717	−	0.845663i	$-0.320795\pi$
$283$	5081.84			1.06743			0.533717	+	0.845663i	$0.320795\pi$
$284$		0			0
$285$	−3156.52	−	8928.00i	−0.656057	−	1.85561i
$286$		0			0
$287$	−5290.90			−1.08819
$288$		0			0
$289$	305.000			0.0620802
$290$		0			0
$291$	−1853.29	−	5241.91i	−0.373340	−	1.05597i
$292$		0			0
$293$	424.264			0.0845931			0.0422965	−	0.999105i	$-0.486533\pi$
$293$	424.264			0.0845931			0.0422965	+	0.999105i	$0.486533\pi$
$294$		0			0
$295$			9477.78i			1.87057i
$296$		0			0
$297$	−2160.00	+	3512.91i	−0.422006	+	0.686328i
$298$		0			0
$299$		−	4585.44i		−	0.886900i
$300$		0			0
$301$			1620.00i			0.310217i
$302$		0			0
$303$	3086.36	+	8729.54i	0.585170	+	1.65511i
$304$		0			0
$305$		−	4717.82i		−	0.885709i
$306$		0			0
$307$	3522.99			0.654944			0.327472	−	0.944861i	$-0.393803\pi$
$307$	3522.99			0.654944			0.327472	+	0.944861i	$0.393803\pi$
$308$		0			0
$309$	−84.8528	+	30.0000i	−0.0156217	+	0.00552311i
$310$		0			0
$311$	9464.83			1.72573			0.862864	−	0.505437i	$-0.168669\pi$
$311$	9464.83			1.72573			0.862864	+	0.505437i	$0.168669\pi$
$312$		0			0
$313$	3718.00			0.671418			0.335709	−	0.941966i	$-0.391024\pi$
$313$	3718.00			0.671418			0.335709	+	0.941966i	$0.391024\pi$
$314$		0			0
$315$	−4988.31	−	6172.71i	−0.892251	−	1.10410i
$316$		0			0
$317$	7212.49			1.27790			0.638949	−	0.769249i	$-0.279369\pi$
$317$	7212.49			1.27790			0.638949	+	0.769249i	$0.279369\pi$
$318$		0			0
$319$		−	498.831i		−	0.0875522i
$320$		0			0
$321$	−6768.00	+	2392.85i	−1.17680	+	0.416062i
$322$		0			0
$323$		−	7289.68i		−	1.25575i
$324$		0			0
$325$			4238.00i			0.723329i
$326$		0			0
$327$	1753.83	−	620.074i	0.296597	−	0.104863i
$328$		0			0
$329$		−	2036.47i		−	0.341259i
$330$		0			0
$331$	−1541.53			−0.255982			−0.127991	−	0.991775i	$-0.540853\pi$
$331$	−1541.53			−0.255982			−0.127991	+	0.991775i	$0.540853\pi$
$332$		0			0
$333$	−3495.94	−	4326.00i	−0.575304	−	0.711902i
$334$		0			0
$335$	15108.5			2.46407
$336$		0			0
$337$	2530.00			0.408955			0.204478	−	0.978871i	$-0.434450\pi$
$337$	2530.00			0.408955			0.204478	+	0.978871i	$0.434450\pi$
$338$		0			0
$339$	−332.554	+	117.576i	−0.0532798	+	0.0188372i
$340$		0			0
$341$	−916.410			−0.145532
$342$		0			0
$343$			6685.72i			1.05246i
$344$		0			0
$345$	−5184.00	−	14662.6i	−0.808977	−	2.28813i
$346$		0			0
$347$		−	5555.44i		−	0.859458i	−0.902958	−	0.429729i	$-0.858609\pi$
$347$		−	5555.44i		−	0.859458i	0.902958	−	0.429729i	$-0.141391\pi$
$348$		0			0
$349$			7786.00i			1.19420i	0.802168	+	0.597099i	$0.203680\pi$
$349$			7786.00i			1.19420i	−0.802168	+	0.597099i	$0.796320\pi$
$350$		0			0
$351$	1910.60	−	3107.30i	0.290542	−	0.472522i
$352$		0			0
$353$		−	407.294i		−	0.0614109i	−0.999528	−	0.0307054i	$-0.990225\pi$
$353$		−	407.294i		−	0.0614109i	0.999528	−	0.0307054i	$-0.00977538\pi$
$354$		0			0
$355$	−997.661			−0.149156
$356$		0			0
$357$	−2036.47	−	5760.00i	−0.301908	−	0.853926i
$358$		0			0
$359$	−8759.38			−1.28775			−0.643875	−	0.765131i	$-0.722674\pi$
$359$	−8759.38			−1.28775			−0.643875	+	0.765131i	$0.722674\pi$
$360$		0			0
$361$	4673.00			0.681295
$362$		0			0
$363$	808.868	+	2287.82i	0.116955	+	0.330798i
$364$		0			0
$365$	7161.58			1.02700
$366$		0			0
$367$			12578.2i			1.78903i	0.447037	+	0.894515i	$0.352479\pi$
$367$			12578.2i			1.78903i	−0.447037	+	0.894515i	$0.647521\pi$
$368$		0			0
$369$	−5184.00	−	6414.87i	−0.731350	−	0.905000i
$370$		0			0
$371$		−	881.816i		−	0.123401i
$372$		0			0
$373$		−	4258.00i		−	0.591075i	−0.955331	−	0.295537i	$-0.904501\pi$
$373$		−	4258.00i		−	0.591075i	0.955331	−	0.295537i	$-0.0954987\pi$
$374$		0			0
$375$	1116.97	+	3159.26i	0.153813	+	0.435049i
$376$		0			0
$377$			441.235i			0.0602778i
$378$		0			0
$379$	2726.25			0.369493			0.184747	−	0.982786i	$-0.440854\pi$
$379$	2726.25			0.369493			0.184747	+	0.982786i	$0.440854\pi$
$380$		0			0
$381$	152.735	−	54.0000i	0.0205377	−	0.00726116i
$382$		0			0
$383$	−4232.72			−0.564704			−0.282352	−	0.959311i	$-0.591115\pi$
$383$	−4232.72			−0.564704			−0.282352	+	0.959311i	$0.591115\pi$
$384$		0			0
$385$	8640.00			1.14373
$386$		0			0
$387$	−1964.15	+	1587.27i	−0.257993	+	0.208489i
$388$		0			0
$389$	5447.55			0.710030			0.355015	−	0.934861i	$-0.384476\pi$
$389$	5447.55			0.710030			0.355015	+	0.934861i	$0.384476\pi$
$390$		0			0
$391$		−	11971.9i		−	1.54846i
$392$		0			0
$393$	−8496.00	+	3003.79i	−1.09050	+	0.385550i
$394$		0			0
$395$		−	11346.0i		−	1.44527i
$396$		0			0
$397$		−	13574.0i		−	1.71602i	−0.513634	−	0.858009i	$-0.671701\pi$
$397$		−	13574.0i		−	1.71602i	0.513634	−	0.858009i	$-0.328299\pi$
$398$		0			0
$399$	9112.10	−	3221.61i	1.14330	−	0.404217i
$400$		0			0
$401$			11743.6i			1.46247i	0.682128	+	0.731233i	$0.261055\pi$
$401$			11743.6i			1.46247i	−0.682128	+	0.731233i	$0.738945\pi$
$402$		0			0
$403$	810.600			0.100196
$404$		0			0
$405$	2596.50	−	12096.0i	0.318570	−	1.48409i
$406$		0			0
$407$	6055.14			0.737450
$408$		0			0
$409$	−2890.00			−0.349392			−0.174696	−	0.984622i	$-0.555894\pi$
$409$	−2890.00			−0.349392			−0.174696	+	0.984622i	$0.555894\pi$
$410$		0			0
$411$	11140.6	−	3938.78i	1.33704	−	0.472715i
$412$		0			0
$413$	−9673.22			−1.15251
$414$		0			0
$415$		−	498.831i		−	0.0590039i
$416$		0			0
$417$	−2670.00	−	7551.90i	−0.313550	−	0.886854i
$418$		0			0
$419$			8318.47i			0.969890i	0.874545	+	0.484945i	$0.161160\pi$
$419$			8318.47i			0.969890i	−0.874545	+	0.484945i	$0.838840\pi$
$420$		0			0
$421$			10414.0i			1.20558i	0.797902	+	0.602788i	$0.205943\pi$
$421$			10414.0i			1.20558i	−0.797902	+	0.602788i	$0.794057\pi$
$422$		0			0
$423$	2469.09	−	1995.32i	0.283809	−	0.229352i
$424$		0			0
$425$			11064.8i			1.26288i
$426$		0			0
$427$	4815.10			0.545712
$428$		0			0
$429$	1323.70	+	3744.00i	0.148972	+	0.421357i
$430$		0			0
$431$	13050.9			1.45856			0.729279	−	0.684216i	$-0.239855\pi$
$431$	13050.9			1.45856			0.729279	+	0.684216i	$0.239855\pi$
$432$		0			0
$433$	−9214.00			−1.02262			−0.511312	−	0.859395i	$-0.670841\pi$
$433$	−9214.00			−1.02262			−0.511312	+	0.859395i	$0.670841\pi$
$434$		0			0
$435$	498.831	+	1410.91i	0.0549818	+	0.155512i
$436$		0			0
$437$	18939.1			2.07319
$438$		0			0
$439$		−	7326.57i		−	0.796534i	−0.917270	−	0.398267i	$-0.869612\pi$
$439$		−	7326.57i		−	0.796534i	0.917270	−	0.398267i	$-0.130388\pi$
$440$		0			0
$441$	−903.000	+	729.734i	−0.0975057	+	0.0787965i
$442$		0			0
$443$			2204.54i			0.236435i	0.992988	+	0.118218i	$0.0377181\pi$
$443$			2204.54i			0.236435i	−0.992988	+	0.118218i	$0.962282\pi$
$444$		0			0
$445$			6336.00i			0.674956i
$446$		0			0
$447$	−4085.75	−	11556.2i	−0.432325	−	1.22280i
$448$		0			0
$449$			2851.05i			0.299665i	0.988711	+	0.149832i	$0.0478734\pi$
$449$			2851.05i			0.299665i	−0.988711	+	0.149832i	$0.952127\pi$
$450$		0			0
$451$	8978.95			0.937477
$452$		0			0
$453$	−9995.66	+	3534.00i	−1.03673	+	0.366538i
$454$		0			0
$455$	−7642.41			−0.787432
$456$		0			0
$457$	38.0000			0.00388964			0.00194482	−	0.999998i	$-0.499381\pi$
$457$	38.0000			0.00388964			0.00194482	+	0.999998i	$0.499381\pi$
$458$		0			0
$459$	4988.31	−	8112.71i	0.507264	−	0.824987i
$460$		0			0
$461$	15833.5			1.59966			0.799828	−	0.600230i	$-0.204924\pi$
$461$	15833.5			1.59966			0.799828	+	0.600230i	$0.204924\pi$
$462$		0			0
$463$		−	11365.7i		−	1.14084i	−0.821353	−	0.570421i	$-0.806780\pi$
$463$		−	11365.7i		−	1.14084i	0.821353	−	0.570421i	$-0.193220\pi$
$464$		0			0
$465$	2592.00	−	916.410i	0.258497	−	0.0913925i
$466$		0			0
$467$		−	499.696i		−	0.0495143i	−0.999693	−	0.0247571i	$-0.992119\pi$
$467$		−	499.696i		−	0.0495143i	0.999693	−	0.0247571i	$-0.00788125\pi$
$468$		0			0
$469$			15420.0i			1.51819i
$470$		0			0
$471$	−519.292	+	183.597i	−0.0508019	+	0.0179612i
$472$		0			0
$473$		−	2749.23i		−	0.267251i
$474$		0			0
$475$	−17504.1			−1.69083
$476$		0			0
$477$	1069.15	−	864.000i	0.102626	−	0.0829347i
$478$		0			0
$479$	−8700.59			−0.829937			−0.414969	−	0.909836i	$-0.636207\pi$
$479$	−8700.59			−0.829937			−0.414969	+	0.909836i	$0.636207\pi$
$480$		0			0
$481$	−5356.00			−0.507718
$482$		0			0
$483$	14964.9	−	5290.90i	1.40979	−	0.498435i
$484$		0			0
$485$	−18158.5			−1.70007
$486$		0			0
$487$			4007.97i			0.372933i	0.982461	+	0.186466i	$0.0597035\pi$
$487$			4007.97i			0.372933i	−0.982461	+	0.186466i	$0.940296\pi$
$488$		0			0
$489$	−138.000	−	390.323i	−0.0127619	−	0.0360961i
$490$		0			0
$491$			16901.5i			1.55347i	0.629828	+	0.776734i	$0.283125\pi$
$491$			16901.5i			1.55347i	−0.629828	+	0.776734i	$0.716875\pi$
$492$		0			0
$493$			1152.00i			0.105240i
$494$		0			0
$495$	8465.44	+	10475.4i	0.768673	+	0.951184i
$496$		0			0
$497$		−	1018.23i		−	0.0918994i
$498$		0			0
$499$	−12439.6			−1.11598			−0.557988	−	0.829849i	$-0.688427\pi$
$499$	−12439.6			−1.11598			−0.557988	+	0.829849i	$0.688427\pi$
$500$		0			0
$501$	−5396.64	−	15264.0i	−0.481246	−	1.36117i
$502$		0			0
$503$	12756.9			1.13082			0.565411	−	0.824809i	$-0.308717\pi$
$503$	12756.9			1.13082			0.565411	+	0.824809i	$0.308717\pi$
$504$		0			0
$505$	30240.0			2.66468
$506$		0			0
$507$	2634.45	+	7451.35i	0.230769	+	0.652714i
$508$		0			0
$509$	−5549.37			−0.483245			−0.241622	−	0.970370i	$-0.577680\pi$
$509$	−5549.37			−0.483245			−0.241622	+	0.970370i	$0.577680\pi$
$510$		0			0
$511$			7309.25i			0.632764i
$512$		0			0
$513$	12834.0	+	7891.31i	1.10455	+	0.679162i
$514$		0			0
$515$			293.939i			0.0251505i
$516$		0			0
$517$			3456.00i			0.293994i
$518$		0			0
$519$	−3850.60	−	10891.1i	−0.325670	−	0.921133i
$520$		0			0
$521$		−	11506.0i		−	0.967541i	−0.875195	−	0.483770i	$-0.839267\pi$
$521$		−	11506.0i		−	0.967541i	0.875195	−	0.483770i	$-0.160733\pi$
$522$		0			0
$523$	17441.8			1.45827			0.729134	−	0.684371i	$-0.239923\pi$
$523$	17441.8			1.45827			0.729134	+	0.684371i	$0.239923\pi$
$524$		0			0
$525$	−13831.0	+	4890.00i	−1.14978	+	0.406509i
$526$		0			0
$527$	2116.36			0.174934
$528$		0			0
$529$	18937.0			1.55642
$530$		0			0
$531$	−9477.78	−	11728.2i	−0.774578	−	0.958491i
$532$		0			0
$533$	−7942.22			−0.645433
$534$		0			0
$535$			23445.0i			1.89461i
$536$		0			0
$537$	−13680.0	+	4836.61i	−1.09932	+	0.388669i
$538$		0			0
$539$		−	1263.94i		−	0.101005i
$540$		0			0
$541$			6794.00i			0.539920i	0.962871	+	0.269960i	$0.0870105\pi$
$541$			6794.00i			0.539920i	−0.962871	+	0.269960i	$0.912989\pi$
$542$		0			0
$543$	−22094.4	+	7811.55i	−1.74615	+	0.617358i
$544$		0			0
$545$		−	6075.46i		−	0.477512i
$546$		0			0
$547$	3910.97			0.305706			0.152853	−	0.988249i	$-0.451154\pi$
$547$	3910.97			0.305706			0.152853	+	0.988249i	$0.451154\pi$
$548$		0			0
$549$	4717.82	+	5838.00i	0.366760	+	0.453843i
$550$		0			0
$551$	−1822.42			−0.140903
$552$		0			0
$553$	11580.0			0.890473
$554$		0			0
$555$	−17126.5	+	6055.14i	−1.30987	+	0.463110i
$556$		0			0
$557$	−22587.8			−1.71827			−0.859135	−	0.511749i	$-0.828998\pi$
$557$	−22587.8			−1.71827			−0.859135	+	0.511749i	$0.828998\pi$
$558$		0			0
$559$			2431.80i			0.183997i
$560$		0			0
$561$	3456.00	+	9775.04i	0.260093	+	0.735655i
$562$		0			0
$563$		−	8729.98i		−	0.653508i	−0.945109	−	0.326754i	$-0.894045\pi$
$563$		−	8729.98i		−	0.653508i	0.945109	−	0.326754i	$-0.105955\pi$
$564$		0			0
$565$			1152.00i			0.0857788i
$566$		0			0
$567$	12345.4	+	2650.04i	0.914390	+	0.196281i
$568$		0			0
$569$			16122.0i			1.18782i	0.804531	+	0.593911i	$0.202417\pi$
$569$			16122.0i			1.18782i	−0.804531	+	0.593911i	$0.797583\pi$
$570$		0			0
$571$	−9495.10			−0.695898			−0.347949	−	0.937513i	$-0.613122\pi$
$571$	−9495.10			−0.695898			−0.347949	+	0.937513i	$0.613122\pi$
$572$		0			0
$573$	−3258.35	−	9216.00i	−0.237556	−	0.671909i
$574$		0			0
$575$	−28747.2			−2.08494
$576$		0			0
$577$	−8974.00			−0.647474			−0.323737	−	0.946147i	$-0.604939\pi$
$577$	−8974.00			−0.647474			−0.323737	+	0.946147i	$0.604939\pi$
$578$		0			0
$579$	8649.86	+	24465.5i	0.620857	+	1.75605i
$580$		0			0
$581$	509.117			0.0363541
$582$		0			0
$583$			1496.49i			0.106309i
$584$		0			0
$585$	−7488.00	−	9265.93i	−0.529215	−	0.654870i
$586$		0			0
$587$			4908.78i			0.345157i	0.984996	+	0.172578i	$0.0552098\pi$
$587$			4908.78i			0.345157i	−0.984996	+	0.172578i	$0.944790\pi$
$588$		0			0
$589$			3348.00i			0.234214i
$590$		0			0
$591$	−3733.02	−	10558.6i	−0.259824	−	0.734893i
$592$		0			0
$593$		−	14730.4i		−	1.02008i	−0.860151	−	0.510040i	$-0.829631\pi$
$593$		−	14730.4i		−	1.02008i	0.860151	−	0.510040i	$-0.170369\pi$
$594$		0			0
$595$	−19953.2			−1.37479
$596$		0			0
$597$	19465.2	−	6882.00i	1.33444	−	0.471795i
$598$		0			0
$599$	−10052.7			−0.685714			−0.342857	−	0.939388i	$-0.611395\pi$
$599$	−10052.7			−0.685714			−0.342857	+	0.939388i	$0.611395\pi$
$600$		0			0
$601$	−23690.0			−1.60788			−0.803939	−	0.594711i	$-0.797266\pi$
$601$	−23690.0			−1.60788			−0.803939	+	0.594711i	$0.797266\pi$
$602$		0			0
$603$	−18695.8	+	15108.5i	−1.26260	+	1.02034i
$604$		0			0
$605$	7925.25			0.532574
$606$		0			0
$607$		−	22589.4i		−	1.51050i	−0.655435	−	0.755252i	$-0.727515\pi$
$607$		−	22589.4i		−	1.51050i	0.655435	−	0.755252i	$-0.272485\pi$
$608$		0			0
$609$	−1440.00	+	509.117i	−0.0958157	+	0.0338760i
$610$		0			0
$611$		−	3056.96i		−	0.202408i
$612$		0			0
$613$			9422.00i			0.620801i	0.950606	+	0.310400i	$0.100463\pi$
$613$			9422.00i			0.620801i	−0.950606	+	0.310400i	$0.899537\pi$
$614$		0			0
$615$	−25396.3	+	8978.95i	−1.66517	+	0.588726i
$616$		0			0
$617$			984.293i			0.0642239i	0.999484	+	0.0321119i	$0.0102233\pi$
$617$			984.293i			0.0642239i	−0.999484	+	0.0321119i	$0.989777\pi$
$618$		0			0
$619$	−12238.7			−0.794691			−0.397345	−	0.917669i	$-0.630069\pi$
$619$	−12238.7			−0.794691			−0.397345	+	0.917669i	$0.630069\pi$
$620$		0			0
$621$	21077.4	+	12960.0i	1.36201	+	0.837467i
$622$		0			0
$623$	−6466.65			−0.415860
$624$		0			0
$625$	−9431.00			−0.603584
$626$		0			0
$627$	−15463.7	+	5467.26i	−0.984948	+	0.348232i
$628$		0			0
$629$	−13983.7			−0.886436
$630$		0			0
$631$		−	13922.2i		−	0.878344i	−0.898403	−	0.439172i	$-0.855272\pi$
$631$		−	13922.2i		−	0.878344i	0.898403	−	0.439172i	$-0.144728\pi$
$632$		0			0
$633$	8502.00	+	24047.3i	0.533845	+	1.50994i
$634$		0			0
$635$		−	529.090i		−	0.0330650i
$636$		0			0
$637$			1118.00i			0.0695397i
$638$		0			0
$639$	1234.54	−	997.661i	0.0764284	−	0.0617635i
$640$		0			0
$641$			16427.5i			1.01224i	0.862462	+	0.506121i	$0.168921\pi$
$641$			16427.5i			1.01224i	−0.862462	+	0.506121i	$0.831079\pi$
$642$		0			0
$643$	−1465.31			−0.0898700			−0.0449350	−	0.998990i	$-0.514308\pi$
$643$	−1465.31			−0.0898700			−0.0449350	+	0.998990i	$0.514308\pi$
$644$		0			0
$645$	2749.23	+	7776.00i	0.167831	+	0.474697i
$646$		0			0
$647$	−12286.6			−0.746581			−0.373290	−	0.927715i	$-0.621770\pi$
$647$	−12286.6			−0.746581			−0.373290	+	0.927715i	$0.621770\pi$
$648$		0			0
$649$	16416.0			0.992888
$650$		0			0
$651$	935.307	+	2645.45i	0.0563097	+	0.159268i
$652$		0			0
$653$	8502.25			0.509523			0.254762	−	0.967004i	$-0.418003\pi$
$653$	8502.25			0.509523			0.254762	+	0.967004i	$0.418003\pi$
$654$		0			0
$655$			29431.0i			1.75567i
$656$		0			0
$657$	−8862.00	+	7161.58i	−0.526240	+	0.425266i
$658$		0			0
$659$			17136.6i			1.01297i	0.862248	+	0.506486i	$0.169056\pi$
$659$			17136.6i			1.01297i	−0.862248	+	0.506486i	$0.830944\pi$
$660$		0			0
$661$		−	10018.0i		−	0.589493i	−0.955575	−	0.294747i	$-0.904765\pi$
$661$		−	10018.0i		−	0.589493i	0.955575	−	0.294747i	$-0.0952352\pi$
$662$		0			0
$663$	−3056.96	−	8646.40i	−0.179069	−	0.506483i
$664$		0			0
$665$		−	31565.2i		−	1.84067i
$666$		0			0
$667$	−2992.98			−0.173746
$668$		0			0
$669$	19702.8	−	6966.00i	1.13865	−	0.402573i
$670$		0			0
$671$	−8171.50			−0.470130
$672$		0			0
$673$	1682.00			0.0963393			0.0481696	−	0.998839i	$-0.484661\pi$
$673$	1682.00			0.0963393			0.0481696	+	0.998839i	$0.484661\pi$
$674$		0			0
$675$	−19480.4	−	11978.0i	−1.11081	−	0.683013i
$676$		0			0
$677$	9113.19			0.517354			0.258677	−	0.965964i	$-0.416714\pi$
$677$	9113.19			0.517354			0.258677	+	0.965964i	$0.416714\pi$
$678$		0			0
$679$		−	18532.9i		−	1.04747i
$680$		0			0
$681$	14544.0	−	5142.08i	0.818395	−	0.289346i
$682$		0			0
$683$			23250.6i			1.30257i	0.758832	+	0.651287i	$0.225771\pi$
$683$			23250.6i			1.30257i	−0.758832	+	0.651287i	$0.774229\pi$
$684$		0			0
$685$		−	38592.0i		−	2.15259i
$686$		0			0
$687$	−2106.56	+	744.782i	−0.116987	+	0.0413613i
$688$		0			0
$689$		−	1323.70i		−	0.0731917i
$690$		0			0
$691$	25222.1			1.38856			0.694280	−	0.719705i	$-0.255723\pi$
$691$	25222.1			1.38856			0.694280	+	0.719705i	$0.255723\pi$
$692$		0			0
$693$	−10691.5	+	8640.00i	−0.586053	+	0.473602i
$694$		0			0
$695$	−26160.6			−1.42781
$696$		0			0
$697$	−20736.0			−1.12688
$698$		0			0
$699$	31093.8	−	10993.3i	1.68251	−	0.594857i
$700$		0			0
$701$	−16410.5			−0.884190			−0.442095	−	0.896968i	$-0.645765\pi$
$701$	−16410.5			−0.884190			−0.442095	+	0.896968i	$0.645765\pi$
$702$		0			0
$703$		−	22121.8i		−	1.18682i
$704$		0			0
$705$	−3456.00	−	9775.04i	−0.184625	−	0.522198i
$706$		0			0
$707$			30863.6i			1.64179i
$708$		0			0
$709$		−	27602.0i		−	1.46208i	−0.682334	−	0.731040i	$-0.739035\pi$
$709$		−	27602.0i		−	1.46208i	0.682334	−	0.731040i	$-0.260965\pi$
$710$		0			0
$711$	11346.0	+	14040.0i	0.598466	+	0.740564i
$712$		0			0
$713$			5498.46i			0.288806i
$714$		0			0
$715$	12969.6			0.678371
$716$		0			0
$717$	7127.64	+	20160.0i	0.371250	+	1.05005i
$718$		0			0
$719$	31627.8			1.64050			0.820249	−	0.572006i	$-0.193835\pi$
$719$	31627.8			1.64050			0.820249	+	0.572006i	$0.193835\pi$
$720$		0			0
$721$	−300.000			−0.0154960
$722$		0			0
$723$	8123.32	+	22976.2i	0.417855	+	1.18187i
$724$		0			0
$725$	2766.20			0.141702
$726$		0			0
$727$		−	27279.8i		−	1.39168i	−0.718197	−	0.695840i	$-0.755032\pi$
$727$		−	27279.8i		−	1.39168i	0.718197	−	0.695840i	$-0.244968\pi$
$728$		0			0
$729$	8883.00	+	17564.5i	0.451303	+	0.892371i
$730$		0			0
$731$			6349.08i			0.321244i
$732$		0			0
$733$			15914.0i			0.801906i	0.916099	+	0.400953i	$0.131321\pi$
$733$			15914.0i			0.801906i	−0.916099	+	0.400953i	$0.868679\pi$
$734$		0			0
$735$	1263.94	+	3574.95i	0.0634299	+	0.179407i
$736$		0			0
$737$		−	26168.6i		−	1.30791i
$738$		0			0
$739$	2275.91			0.113289			0.0566447	−	0.998394i	$-0.481960\pi$
$739$	2275.91			0.113289			0.0566447	+	0.998394i	$0.481960\pi$
$740$		0			0
$741$	13678.3	−	4836.00i	0.678116	−	0.239750i
$742$		0			0
$743$	8641.80			0.426698			0.213349	−	0.976976i	$-0.431563\pi$
$743$	8641.80			0.426698			0.213349	+	0.976976i	$0.431563\pi$
$744$		0			0
$745$	−40032.0			−1.96867
$746$		0			0
$747$	498.831	+	617.271i	0.0244327	+	0.0302340i
$748$		0			0
$749$	−23928.5			−1.16733
$750$		0			0
$751$		−	16631.2i		−	0.808095i	−0.914738	−	0.404048i	$-0.867603\pi$
$751$		−	16631.2i		−	0.808095i	0.914738	−	0.404048i	$-0.132397\pi$
$752$		0			0
$753$	−6192.00	+	2189.20i	−0.299667	+	0.105948i
$754$		0			0
$755$			34626.0i			1.66910i
$756$		0			0
$757$			11422.0i			0.548401i	0.961673	+	0.274201i	$0.0884132\pi$
$757$			11422.0i			0.548401i	−0.961673	+	0.274201i	$0.911587\pi$
$758$		0			0
$759$	−25396.3	+	8978.95i	−1.21453	+	0.429401i
$760$		0			0
$761$		−	12660.0i		−	0.603057i	−0.953457	−	0.301528i	$-0.902503\pi$
$761$		−	12660.0i		−	0.603057i	0.953457	−	0.301528i	$-0.0974968\pi$
$762$		0			0
$763$	6200.74			0.294210
$764$		0			0
$765$	−19550.1	−	24192.0i	−0.923967	−	1.14335i
$766$		0			0
$767$	−14520.6			−0.683582
$768$		0			0
$769$	18818.0			0.882437			0.441219	−	0.897400i	$-0.354546\pi$
$769$	18818.0			0.882437			0.441219	+	0.897400i	$0.354546\pi$
$770$		0			0
$771$	1330.22	−	470.302i	0.0621356	−	0.0219682i
$772$		0			0
$773$	1917.67			0.0892289			0.0446144	−	0.999004i	$-0.485794\pi$
$773$	1917.67			0.0892289			0.0446144	+	0.999004i	$0.485794\pi$
$774$		0			0
$775$		−	5081.84i		−	0.235542i
$776$		0			0
$777$	−6180.00	−	17479.7i	−0.285336	−	0.807053i
$778$		0			0
$779$		−	32803.6i		−	1.50874i
$780$		0			0
$781$			1728.00i			0.0791712i
$782$		0			0
$783$	−2028.18	−	1247.08i	−0.0925685	−	0.0569181i
$784$		0			0
$785$			1798.88i			0.0817895i
$786$		0			0
$787$	19845.8			0.898892			0.449446	−	0.893308i	$-0.351621\pi$
$787$	19845.8			0.898892			0.449446	+	0.893308i	$0.351621\pi$
$788$		0			0
$789$	8858.63	+	25056.0i	0.399716	+	1.13057i
$790$		0			0
$791$	−1175.76			−0.0528509
$792$		0			0
$793$	7228.00			0.323674
$794$		0			0
$795$	−1496.49	−	4232.72i	−0.0667611	−	0.188829i
$796$		0			0
$797$	30971.3			1.37649			0.688243	−	0.725480i	$-0.258382\pi$
$797$	30971.3			1.37649			0.688243	+	0.725480i	$0.258382\pi$
$798$		0			0
$799$		−	7981.29i		−	0.353389i
$800$		0			0
$801$	−6336.00	−	7840.40i	−0.279490	−	0.345851i
$802$		0			0
$803$		−	12404.2i		−	0.545125i
$804$		0			0
$805$		−	51840.0i		−	2.26971i
$806$		0			0
$807$	−5496.65	−	15546.9i	−0.239766	−	0.678162i
$808$		0			0
$809$			9537.46i			0.414486i	0.978290	+	0.207243i	$0.0664491\pi$
$809$			9537.46i			0.414486i	−0.978290	+	0.207243i	$0.933551\pi$
$810$		0			0
$811$	−3897.11			−0.168738			−0.0843688	−	0.996435i	$-0.526887\pi$
$811$	−3897.11			−0.168738			−0.0843688	+	0.996435i	$0.526887\pi$
$812$		0			0
$813$	3275.32	−	1158.00i	0.141292	−	0.0499543i
$814$		0			0
$815$	−1352.12			−0.0581137
$816$		0			0
$817$	−10044.0			−0.430104
$818$		0			0
$819$	9457.00	−	7642.41i	0.403485	−	0.326065i
$820$		0			0
$821$	13254.0			0.563420			0.281710	−	0.959500i	$-0.409098\pi$
$821$	13254.0			0.563420			0.281710	+	0.959500i	$0.409098\pi$
$822$		0			0
$823$			10021.6i			0.424462i	0.977219	+	0.212231i	$0.0680730\pi$
$823$			10021.6i			0.424462i	−0.977219	+	0.212231i	$0.931927\pi$
$824$		0			0
$825$	23472.0	−	8298.61i	0.990534	−	0.350206i
$826$		0			0
$827$			31128.1i			1.30886i	0.756121	+	0.654432i	$0.227092\pi$
$827$			31128.1i			1.30886i	−0.756121	+	0.654432i	$0.772908\pi$
$828$		0			0
$829$			12554.0i			0.525957i	0.964802	+	0.262979i	$0.0847048\pi$
$829$			12554.0i			0.525957i	−0.964802	+	0.262979i	$0.915295\pi$
$830$		0			0
$831$	9886.14	−	3495.28i	0.412691	−	0.145908i
$832$		0			0
$833$			2918.94i			0.121411i
$834$		0			0
$835$	−52876.0			−2.19144
$836$		0			0
$837$	−2291.03	+	3726.00i	−0.0946110	+	0.153870i
$838$		0			0
$839$	−21692.7			−0.892628			−0.446314	−	0.894877i	$-0.647263\pi$
$839$	−21692.7			−0.892628			−0.446314	+	0.894877i	$0.647263\pi$
$840$		0			0
$841$	−24101.0			−0.988191
$842$		0			0
$843$	−23112.5	+	8171.50i	−0.944290	+	0.333857i
$844$		0			0
$845$	25812.2			1.05085
$846$		0			0
$847$			8088.68i			0.328135i
$848$		0			0
$849$	8802.00	+	24895.8i	0.355811	+	1.00639i
$850$		0			0
$851$		−	36330.8i		−	1.46346i
$852$		0			0
$853$		−	4546.00i		−	0.182476i	−0.995829	−	0.0912380i	$-0.970918\pi$
$853$		−	4546.00i		−	0.182476i	0.995829	−	0.0912380i	$-0.0290824\pi$
$854$		0			0
$855$	38270.8	−	30927.5i	1.53080	−	1.23707i
$856$		0			0
$857$		−	32617.4i		−	1.30010i	−0.759889	−	0.650052i	$-0.774747\pi$
$857$		−	32617.4i		−	1.30010i	0.759889	−	0.650052i	$-0.225253\pi$
$858$		0			0
$859$	21404.7			0.850196			0.425098	−	0.905147i	$-0.360240\pi$
$859$	21404.7			0.850196			0.425098	+	0.905147i	$0.360240\pi$
$860$		0			0
$861$	−9164.10	−	25920.0i	−0.362732	−	1.02596i
$862$		0			0
$863$	−10111.5			−0.398840			−0.199420	−	0.979914i	$-0.563906\pi$
$863$	−10111.5			−0.398840			−0.199420	+	0.979914i	$0.563906\pi$
$864$		0			0
$865$	−37728.0			−1.48299
$866$		0			0
$867$	528.275	+	1494.19i	0.0206934	+	0.0585298i
$868$		0			0
$869$	−19651.9			−0.767141
$870$		0			0
$871$			23147.1i			0.900471i
$872$		0			0
$873$	22470.0	−	18158.5i	0.871127	−	0.703977i
$874$		0			0
$875$			11169.7i			0.431547i
$876$		0			0
$877$		−	20870.0i		−	0.803569i	−0.915734	−	0.401784i	$-0.868390\pi$
$877$		−	20870.0i		−	0.803569i	0.915734	−	0.401784i	$-0.131610\pi$
$878$		0			0
$879$	734.847	+	2078.46i	0.0281977	+	0.0797551i
$880$		0			0
$881$		−	28714.2i		−	1.09808i	−0.835797	−	0.549038i	$-0.814994\pi$
$881$		−	28714.2i		−	1.09808i	0.835797	−	0.549038i	$-0.185006\pi$
$882$		0			0
$883$	−2823.24			−0.107599			−0.0537994	−	0.998552i	$-0.517133\pi$
$883$	−2823.24			−0.107599			−0.0537994	+	0.998552i	$0.517133\pi$
$884$		0			0
$885$	−46431.5	+	16416.0i	−1.76359	+	0.623523i
$886$		0			0
$887$	2410.30			0.0912400			0.0456200	−	0.998959i	$-0.485474\pi$
$887$	2410.30			0.0912400			0.0456200	+	0.998959i	$0.485474\pi$
$888$		0			0
$889$	540.000			0.0203723
$890$		0			0
$891$	−20950.9	−	4497.26i	−0.787745	−	0.169095i
$892$		0			0
$893$	12626.1			0.473143
$894$		0			0
$895$			47388.9i			1.76987i
$896$		0			0
$897$	22464.0	−	7942.22i	0.836177	−	0.295633i
$898$		0			0
$899$		−	529.090i		−	0.0196286i
$900$		0			0
$901$		−	3456.00i		−	0.127787i
$902$		0			0
$903$	−7936.35	+	2805.92i	−0.292475	+	0.103406i
$904$		0			0
$905$			76537.2i			2.81125i
$906$		0			0
$907$	−18474.1			−0.676319			−0.338159	−	0.941089i	$-0.609804\pi$
$907$	−18474.1			−0.676319			−0.338159	+	0.941089i	$0.609804\pi$
$908$		0			0
$909$	−37420.1	+	30240.0i	−1.36540	+	1.10341i
$910$		0			0
$911$	−44561.1			−1.62061			−0.810305	−	0.586008i	$-0.800699\pi$
$911$	−44561.1			−1.62061			−0.810305	+	0.586008i	$0.800699\pi$
$912$		0			0
$913$	−864.000			−0.0313190
$914$		0			0
$915$	23112.5	−	8171.50i	0.835055	−	0.295236i
$916$		0			0
$917$	−30037.9			−1.08172
$918$		0			0
$919$			40561.2i			1.45592i	0.685620	+	0.727959i	$0.259531\pi$
$919$			40561.2i			1.45592i	−0.685620	+	0.727959i	$0.740469\pi$
$920$		0			0
$921$	6102.00	+	17259.1i	0.218315	+	0.617487i
$922$		0			0
$923$		−	1528.48i		−	0.0545077i
$924$		0			0
$925$			33578.0i			1.19355i
$926$		0			0
$927$	−293.939	−	363.731i	−0.0104145	−	0.0128873i
$928$		0			0
$929$			56478.0i			1.99460i	0.0734296	+	0.997300i	$0.476606\pi$
$929$			56478.0i			1.99460i	−0.0734296	+	0.997300i	$0.523394\pi$
$930$		0			0
$931$	−4617.65			−0.162553
$932$		0			0
$933$	16393.6	+	46368.0i	0.575242	+	1.62703i
$934$		0			0
$935$	33861.7			1.18438
$936$		0			0
$937$	7030.00			0.245101			0.122551	−	0.992462i	$-0.460893\pi$
$937$	7030.00			0.245101			0.122551	+	0.992462i	$0.460893\pi$
$938$		0			0
$939$	6439.76	+	18214.4i	0.223806	+	0.633019i
$940$		0			0
$941$	−27475.3			−0.951828			−0.475914	−	0.879492i	$-0.657883\pi$
$941$	−27475.3			−0.951828			−0.475914	+	0.879492i	$0.657883\pi$
$942$		0			0
$943$		−	53873.7i		−	1.86041i
$944$		0			0
$945$	21600.0	−	35129.1i	0.743543	−	1.20926i
$946$		0			0
$947$			3262.72i			0.111958i	0.998432	+	0.0559790i	$0.0178280\pi$
$947$			3262.72i			0.111958i	−0.998432	+	0.0559790i	$0.982172\pi$
$948$		0			0
$949$			10972.0i			0.375307i
$950$		0			0
$951$	12492.4	+	35333.8i	0.425966	+	1.20481i
$952$		0			0
$953$			8519.22i			0.289575i	0.989463	+	0.144787i	$0.0462498\pi$
$953$			8519.22i			0.289575i	−0.989463	+	0.144787i	$0.953750\pi$
$954$		0			0
$955$	−31925.2			−1.08175
$956$		0			0
$957$	2443.76	−	864.000i	0.0825450	−	0.0291841i
$958$		0			0
$959$	39387.8			1.32628
$960$		0			0
$961$	28819.0			0.967373
$962$		0			0
$963$	−23445.0	−	29011.8i	−0.784534	−	0.970811i
$964$		0			0
$965$	84751.0			2.82718
$966$		0			0
$967$			29226.6i			0.971939i	0.873976	+	0.485969i	$0.161533\pi$
$967$			29226.6i			0.971939i	−0.873976	+	0.485969i	$0.838467\pi$
$968$		0			0
$969$	35712.0	−	12626.1i	1.18394	−	0.418585i
$970$		0			0
$971$		−	47059.6i		−	1.55532i	−0.628686	−	0.777659i	$-0.716407\pi$
$971$		−	47059.6i		−	1.55532i	0.628686	−	0.777659i	$-0.283593\pi$
$972$		0			0
$973$		−	26700.0i		−	0.879715i
$974$		0			0
$975$	−20761.9	+	7340.43i	−0.681961	+	0.241110i
$976$		0			0
$977$			27492.3i			0.900263i	0.892962	+	0.450132i	$0.148623\pi$
$977$			27492.3i			0.900263i	−0.892962	+	0.450132i	$0.851377\pi$
$978$		0			0
$979$	10974.3			0.358263
$980$		0			0
$981$	6075.46	+	7518.00i	0.197731	+	0.244680i
$982$		0			0
$983$	37447.8			1.21506			0.607528	−	0.794299i	$-0.292161\pi$
$983$	37447.8			1.21506			0.607528	+	0.794299i	$0.292161\pi$
$984$		0			0
$985$	−36576.0			−1.18316
$986$		0			0
$987$	9976.61	−	3527.27i	0.321742	−	0.113753i
$988$		0			0
$989$	−16495.4			−0.530357
$990$		0			0
$991$			37270.3i			1.19468i	0.801988	+	0.597341i	$0.203776\pi$
$991$			37270.3i			1.19468i	−0.801988	+	0.597341i	$0.796224\pi$
$992$		0			0
$993$	−2670.00	−	7551.90i	−0.0853272	−	0.241342i
$994$		0			0
$995$		−	67429.6i		−	2.14840i
$996$		0			0
$997$		−	45106.0i		−	1.43282i	−0.697680	−	0.716410i	$-0.745784\pi$
$997$		−	45106.0i		−	1.43282i	0.697680	−	0.716410i	$-0.254216\pi$
$998$		0			0
$999$	15137.8	−	24619.4i	0.479420	−	0.779702i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.f.b.383.8		8
3.2	odd	2	inner	768.4.f.b.383.5		8
4.3	odd	2	inner	768.4.f.b.383.2		8
8.3	odd	2	inner	768.4.f.b.383.7		8
8.5	even	2	inner	768.4.f.b.383.1		8
12.11	even	2	inner	768.4.f.b.383.3		8
16.3	odd	4		48.4.c.b.47.3	yes	4
16.5	even	4		192.4.c.c.191.3		4
16.11	odd	4		192.4.c.c.191.2		4
16.13	even	4		48.4.c.b.47.2	yes	4
24.5	odd	2	inner	768.4.f.b.383.4		8
24.11	even	2	inner	768.4.f.b.383.6		8
48.5	odd	4		192.4.c.c.191.1		4
48.11	even	4		192.4.c.c.191.4		4
48.29	odd	4		48.4.c.b.47.4	yes	4
48.35	even	4		48.4.c.b.47.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.c.b.47.1	✓	4	48.35	even	4
48.4.c.b.47.2	yes	4	16.13	even	4
48.4.c.b.47.3	yes	4	16.3	odd	4
48.4.c.b.47.4	yes	4	48.29	odd	4
192.4.c.c.191.1		4	48.5	odd	4
192.4.c.c.191.2		4	16.11	odd	4
192.4.c.c.191.3		4	16.5	even	4
192.4.c.c.191.4		4	48.11	even	4
768.4.f.b.383.1		8	8.5	even	2	inner
768.4.f.b.383.2		8	4.3	odd	2	inner
768.4.f.b.383.3		8	12.11	even	2	inner
768.4.f.b.383.4		8	24.5	odd	2	inner
768.4.f.b.383.5		8	3.2	odd	2	inner
768.4.f.b.383.6		8	24.11	even	2	inner
768.4.f.b.383.7		8	8.3	odd	2	inner
768.4.f.b.383.8		8	1.1	even	1	trivial

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 + 4.89898i) q^{3} +16.9706 q^{5} +17.3205i q^{7} +(-21.0000 + 16.9706i) q^{9} -29.3939i q^{11} +26.0000i q^{13} +(29.3939 + 83.1384i) q^{15} +67.8823i q^{17} -107.387 q^{19} +(-84.8528 + 30.0000i) q^{21} -176.363 q^{23} +163.000 q^{25} +(-119.512 - 73.4847i) q^{27} +16.9706 q^{29} -31.1769i q^{31} +(144.000 - 50.9117i) q^{33} +293.939i q^{35} +206.000i q^{37} +(-127.373 + 45.0333i) q^{39} +305.470i q^{41} +93.5307 q^{43} +(-356.382 + 288.000i) q^{45} -117.576 q^{47} +43.0000 q^{49} +(-332.554 + 117.576i) q^{51} -50.9117 q^{53} -498.831i q^{55} +(-186.000 - 526.087i) q^{57} +558.484i q^{59} -278.000i q^{61} +(-293.939 - 363.731i) q^{63} +441.235i q^{65} +890.274 q^{67} +(-305.470 - 864.000i) q^{69} -58.7878 q^{71} +422.000 q^{73} +(282.324 + 798.534i) q^{75} +509.117 q^{77} -668.572i q^{79} +(153.000 - 712.764i) q^{81} -29.3939i q^{83} +1152.00i q^{85} +(29.3939 + 83.1384i) q^{87} +373.352i q^{89} -450.333 q^{91} +(152.735 - 54.0000i) q^{93} -1822.42 q^{95} -1070.00 q^{97} +(498.831 + 617.271i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 168 q^{9} + 1304 q^{25} + 1152 q^{33} + 344 q^{49} - 1488 q^{57} + 3376 q^{73} + 1224 q^{81} - 8560 q^{97}+O(q^{100}) \) 8 * q - 168 * q^9 + 1304 * q^25 + 1152 * q^33 + 344 * q^49 - 1488 * q^57 + 3376 * q^73 + 1224 * q^81 - 8560 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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