LMFDB - Embedded newform 768.4.c.d.767.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,4,Mod(767,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,-2,0,0,0,0,0,-50,0,92,0,-88,0,104,0,0,0,0,0,-104,0,-176, 0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-26}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 26 $ x^2 + 26
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			767.1
Root			$5.09902i$ of defining polynomial
Character	$\chi$	$=$	768.767
Dual form			768.4.c.d.767.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 5.09902i) q^{3} +10.1980i q^{5} -10.1980i q^{7} +(-25.0000 + 10.1980i) q^{9} +46.0000 q^{11} -44.0000 q^{13} +(52.0000 - 10.1980i) q^{15} -20.3961i q^{17} +10.1980i q^{19} +(-52.0000 + 10.1980i) q^{21} -88.0000 q^{23} +21.0000 q^{25} +(77.0000 + 117.277i) q^{27} -254.951i q^{29} +214.159i q^{31} +(-46.0000 - 234.555i) q^{33} +104.000 q^{35} -332.000 q^{37} +(44.0000 + 224.357i) q^{39} +489.506i q^{41} -234.555i q^{43} +(-104.000 - 254.951i) q^{45} -384.000 q^{47} +239.000 q^{49} +(-104.000 + 20.3961i) q^{51} +458.912i q^{53} +469.110i q^{55} +(52.0000 - 10.1980i) q^{57} -630.000 q^{59} -236.000 q^{61} +(104.000 + 254.951i) q^{63} -448.714i q^{65} -50.9902i q^{67} +(88.0000 + 448.714i) q^{69} +680.000 q^{71} -422.000 q^{73} +(-21.0000 - 107.079i) q^{75} -469.110i q^{77} +744.457i q^{79} +(521.000 - 509.902i) q^{81} +186.000 q^{83} +208.000 q^{85} +(-1300.00 + 254.951i) q^{87} +958.616i q^{89} +448.714i q^{91} +(1092.00 - 214.159i) q^{93} -104.000 q^{95} -1062.00 q^{97} +(-1150.00 + 469.110i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 50 q^{9} + 92 q^{11} - 88 q^{13} + 104 q^{15} - 104 q^{21} - 176 q^{23} + 42 q^{25} + 154 q^{27} - 92 q^{33} + 208 q^{35} - 664 q^{37} + 88 q^{39} - 208 q^{45} - 768 q^{47} + 478 q^{49}+ \cdots - 2300 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 50 * q^9 + 92 * q^11 - 88 * q^13 + 104 * q^15 - 104 * q^21 - 176 * q^23 + 42 * q^25 + 154 * q^27 - 92 * q^33 + 208 * q^35 - 664 * q^37 + 88 * q^39 - 208 * q^45 - 768 * q^47 + 478 * q^49 - 208 * q^51 + 104 * q^57 - 1260 * q^59 - 472 * q^61 + 208 * q^63 + 176 * q^69 + 1360 * q^71 - 844 * q^73 - 42 * q^75 + 1042 * q^81 + 372 * q^83 + 416 * q^85 - 2600 * q^87 + 2184 * q^93 - 208 * q^95 - 2124 * q^97 - 2300 * q^99

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
$3$	−1.00000	−	5.09902i	−0.192450	−	0.981307i
$3$	−1.00000	−	5.09902i	−0.192450	−	0.981307i
$4$		0			0
$5$			10.1980i			0.912140i	0.889944	+	0.456070i	$0.150743\pi$
$5$			10.1980i			0.912140i	−0.889944	+	0.456070i	$0.849257\pi$
$6$		0			0
$7$		−	10.1980i		−	0.550642i	−0.961352	−	0.275321i	$-0.911216\pi$
$7$		−	10.1980i		−	0.550642i	0.961352	−	0.275321i	$-0.0887842\pi$
$8$		0			0
$9$	−25.0000	+	10.1980i	−0.925926	+	0.377705i
$10$		0			0
$11$	46.0000			1.26087			0.630433	−	0.776244i	$-0.282877\pi$
$11$	46.0000			1.26087			0.630433	+	0.776244i	$0.282877\pi$
$12$		0			0
$13$	−44.0000			−0.938723			−0.469362	−	0.883006i	$-0.655516\pi$
$13$	−44.0000			−0.938723			−0.469362	+	0.883006i	$0.655516\pi$
$14$		0			0
$15$	52.0000	−	10.1980i	0.895089	−	0.175541i
$16$		0			0
$17$		−	20.3961i		−	0.290987i	−0.989359	−	0.145493i	$-0.953523\pi$
$17$		−	20.3961i		−	0.290987i	0.989359	−	0.145493i	$-0.0464770\pi$
$18$		0			0
$19$			10.1980i			0.123136i	0.998103	+	0.0615682i	$0.0196102\pi$
$19$			10.1980i			0.123136i	−0.998103	+	0.0615682i	$0.980390\pi$
$20$		0			0
$21$	−52.0000	+	10.1980i	−0.540349	+	0.105971i
$22$		0			0
$23$	−88.0000			−0.797794			−0.398897	−	0.916996i	$-0.630607\pi$
$23$	−88.0000			−0.797794			−0.398897	+	0.916996i	$0.630607\pi$
$24$		0			0
$25$	21.0000			0.168000
$26$		0			0
$27$	77.0000	+	117.277i	0.548839	+	0.835928i
$28$		0			0
$29$		−	254.951i		−	1.63252i	−0.577682	−	0.816262i	$-0.696042\pi$
$29$		−	254.951i		−	1.63252i	0.577682	−	0.816262i	$-0.303958\pi$
$30$		0			0
$31$			214.159i			1.24078i	0.784295	+	0.620388i	$0.213025\pi$
$31$			214.159i			1.24078i	−0.784295	+	0.620388i	$0.786975\pi$
$32$		0			0
$33$	−46.0000	−	234.555i	−0.242654	−	1.23730i
$34$		0			0
$35$	104.000			0.502263
$36$		0			0
$37$	−332.000			−1.47515			−0.737574	−	0.675266i	$-0.764029\pi$
$37$	−332.000			−1.47515			−0.737574	+	0.675266i	$0.764029\pi$
$38$		0			0
$39$	44.0000	+	224.357i	0.180657	+	0.921176i
$40$		0			0
$41$			489.506i			1.86458i	0.361706	+	0.932292i	$0.382195\pi$
$41$			489.506i			1.86458i	−0.361706	+	0.932292i	$0.617805\pi$
$42$		0			0
$43$		−	234.555i		−	0.831844i	−0.909400	−	0.415922i	$-0.863459\pi$
$43$		−	234.555i		−	0.831844i	0.909400	−	0.415922i	$-0.136541\pi$
$44$		0			0
$45$	−104.000	−	254.951i	−0.344520	−	0.844574i
$46$		0			0
$47$	−384.000			−1.19175			−0.595874	−	0.803078i	$-0.703194\pi$
$47$	−384.000			−1.19175			−0.595874	+	0.803078i	$0.703194\pi$
$48$		0			0
$49$	239.000			0.696793
$50$		0			0
$51$	−104.000	+	20.3961i	−0.285547	+	0.0560004i
$52$		0			0
$53$			458.912i			1.18937i	0.803960	+	0.594683i	$0.202722\pi$
$53$			458.912i			1.18937i	−0.803960	+	0.594683i	$0.797278\pi$
$54$		0			0
$55$			469.110i			1.15009i
$56$		0			0
$57$	52.0000	−	10.1980i	0.120835	−	0.0236976i
$58$		0			0
$59$	−630.000			−1.39015			−0.695076	−	0.718936i	$-0.744629\pi$
$59$	−630.000			−1.39015			−0.695076	+	0.718936i	$0.744629\pi$
$60$		0			0
$61$	−236.000			−0.495356			−0.247678	−	0.968842i	$-0.579667\pi$
$61$	−236.000			−0.495356			−0.247678	+	0.968842i	$0.579667\pi$
$62$		0			0
$63$	104.000	+	254.951i	0.207980	+	0.509854i
$64$		0			0
$65$		−	448.714i		−	0.856247i
$66$		0			0
$67$		−	50.9902i		−	0.0929768i	−0.998919	−	0.0464884i	$-0.985197\pi$
$67$		−	50.9902i		−	0.0929768i	0.998919	−	0.0464884i	$-0.0148030\pi$
$68$		0			0
$69$	88.0000	+	448.714i	0.153536	+	0.782881i
$70$		0			0
$71$	680.000			1.13664			0.568318	−	0.822809i	$-0.307594\pi$
$71$	680.000			1.13664			0.568318	+	0.822809i	$0.307594\pi$
$72$		0			0
$73$	−422.000			−0.676594			−0.338297	−	0.941039i	$-0.609851\pi$
$73$	−422.000			−0.676594			−0.338297	+	0.941039i	$0.609851\pi$
$74$		0			0
$75$	−21.0000	−	107.079i	−0.0323316	−	0.164860i
$76$		0			0
$77$		−	469.110i		−	0.694286i
$78$		0			0
$79$			744.457i			1.06023i	0.847927	+	0.530114i	$0.177851\pi$
$79$			744.457i			1.06023i	−0.847927	+	0.530114i	$0.822149\pi$
$80$		0			0
$81$	521.000	−	509.902i	0.714678	−	0.699454i
$82$		0			0
$83$	186.000			0.245978			0.122989	−	0.992408i	$-0.460752\pi$
$83$	186.000			0.245978			0.122989	+	0.992408i	$0.460752\pi$
$84$		0			0
$85$	208.000			0.265421
$86$		0			0
$87$	−1300.00	+	254.951i	−1.60201	+	0.314179i
$88$		0			0
$89$			958.616i			1.14172i	0.821048	+	0.570860i	$0.193390\pi$
$89$			958.616i			1.14172i	−0.821048	+	0.570860i	$0.806610\pi$
$90$		0			0
$91$			448.714i			0.516901i
$92$		0			0
$93$	1092.00	−	214.159i	1.21758	−	0.238787i
$94$		0			0
$95$	−104.000			−0.112318
$96$		0			0
$97$	−1062.00			−1.11165			−0.555824	−	0.831300i	$-0.687597\pi$
$97$	−1062.00			−1.11165			−0.555824	+	0.831300i	$0.687597\pi$
$98$		0			0
$99$	−1150.00	+	469.110i	−1.16747	+	0.476235i
$100$		0			0
$101$			10.1980i			0.0100470i	0.999987	+	0.00502348i	$0.00159903\pi$
$101$			10.1980i			0.0100470i	−0.999987	+	0.00502348i	$0.998401\pi$
$102$		0			0
$103$			1988.62i			1.90237i	0.308618	+	0.951186i	$0.400133\pi$
$103$			1988.62i			1.90237i	−0.308618	+	0.951186i	$0.599867\pi$
$104$		0			0
$105$	−104.000	−	530.298i	−0.0966606	−	0.492874i
$106$		0			0
$107$	234.000			0.211417			0.105709	−	0.994397i	$-0.466289\pi$
$107$	234.000			0.211417			0.105709	+	0.994397i	$0.466289\pi$
$108$		0			0
$109$	724.000			0.636208			0.318104	−	0.948056i	$-0.396954\pi$
$109$	724.000			0.636208			0.318104	+	0.948056i	$0.396954\pi$
$110$		0			0
$111$	332.000	+	1692.87i	0.283892	+	1.44757i
$112$		0			0
$113$			1019.80i			0.848983i	0.905432	+	0.424492i	$0.139547\pi$
$113$			1019.80i			0.848983i	−0.905432	+	0.424492i	$0.860453\pi$
$114$		0			0
$115$		−	897.427i		−	0.727700i
$116$		0			0
$117$	1100.00	−	448.714i	0.869188	−	0.354561i
$118$		0			0
$119$	−208.000			−0.160230
$120$		0			0
$121$	785.000			0.589782
$122$		0			0
$123$	2496.00	−	489.506i	1.82973	−	0.358839i
$124$		0			0
$125$			1488.91i			1.06538i
$126$		0			0
$127$			2253.77i			1.57472i	0.616493	+	0.787360i	$0.288553\pi$
$127$			2253.77i			1.57472i	−0.616493	+	0.787360i	$0.711447\pi$
$128$		0			0
$129$	−1196.00	+	234.555i	−0.816294	+	0.160088i
$130$		0			0
$131$	1742.00			1.16183			0.580913	−	0.813966i	$-0.302696\pi$
$131$	1742.00			1.16183			0.580913	+	0.813966i	$0.302696\pi$
$132$		0			0
$133$	104.000			0.0678041
$134$		0			0
$135$	−1196.00	+	785.249i	−0.762484	+	0.500618i
$136$		0			0
$137$		−	2406.74i		−	1.50089i	−0.660935	−	0.750443i	$-0.729840\pi$
$137$		−	2406.74i		−	1.50089i	0.660935	−	0.750443i	$-0.270160\pi$
$138$		0			0
$139$			764.853i			0.466719i	0.972390	+	0.233360i	$0.0749719\pi$
$139$			764.853i			0.466719i	−0.972390	+	0.233360i	$0.925028\pi$
$140$		0			0
$141$	384.000	+	1958.02i	0.229352	+	1.16947i
$142$		0			0
$143$	−2024.00			−1.18360
$144$		0			0
$145$	2600.00			1.48909
$146$		0			0
$147$	−239.000	−	1218.67i	−0.134098	−	0.683768i
$148$		0			0
$149$		−	479.308i		−	0.263533i	−0.991281	−	0.131767i	$-0.957935\pi$
$149$		−	479.308i		−	0.263533i	0.991281	−	0.131767i	$-0.0420649\pi$
$150$		0			0
$151$			1499.11i			0.807920i	0.914777	+	0.403960i	$0.132367\pi$
$151$			1499.11i			0.807920i	−0.914777	+	0.403960i	$0.867633\pi$
$152$		0			0
$153$	208.000	+	509.902i	0.109907	+	0.269432i
$154$		0			0
$155$	−2184.00			−1.13176
$156$		0			0
$157$	−1948.00			−0.990238			−0.495119	−	0.868825i	$-0.664875\pi$
$157$	−1948.00			−0.990238			−0.495119	+	0.868825i	$0.664875\pi$
$158$		0			0
$159$	2340.00	−	458.912i	1.16713	−	0.228894i
$160$		0			0
$161$			897.427i			0.439299i
$162$		0			0
$163$			3395.95i			1.63185i	0.578160	+	0.815924i	$0.303771\pi$
$163$			3395.95i			1.63185i	−0.578160	+	0.815924i	$0.696229\pi$
$164$		0			0
$165$	2392.00	−	469.110i	1.12859	−	0.221334i
$166$		0			0
$167$	2008.00			0.930441			0.465221	−	0.885195i	$-0.345975\pi$
$167$	2008.00			0.930441			0.465221	+	0.885195i	$0.345975\pi$
$168$		0			0
$169$	−261.000			−0.118798
$170$		0			0
$171$	−104.000	−	254.951i	−0.0465092	−	0.114015i
$172$		0			0
$173$			1213.57i			0.533328i	0.963790	+	0.266664i	$0.0859214\pi$
$173$			1213.57i			0.533328i	−0.963790	+	0.266664i	$0.914079\pi$
$174$		0			0
$175$		−	214.159i		−	0.0925079i
$176$		0			0
$177$	630.000	+	3212.38i	0.267535	+	1.36417i
$178$		0			0
$179$	2382.00			0.994632			0.497316	−	0.867570i	$-0.334319\pi$
$179$	2382.00			0.994632			0.497316	+	0.867570i	$0.334319\pi$
$180$		0			0
$181$	−2652.00			−1.08907			−0.544535	−	0.838738i	$-0.683294\pi$
$181$	−2652.00			−1.08907			−0.544535	+	0.838738i	$0.683294\pi$
$182$		0			0
$183$	236.000	+	1203.37i	0.0953313	+	0.486096i
$184$		0			0
$185$		−	3385.75i		−	1.34554i
$186$		0			0
$187$		−	938.220i		−	0.366895i
$188$		0			0
$189$	1196.00	−	785.249i	0.460297	−	0.302214i
$190$		0			0
$191$	−4784.00			−1.81235			−0.906173	−	0.422907i	$-0.861010\pi$
$191$	−4784.00			−1.81235			−0.906173	+	0.422907i	$0.861010\pi$
$192$		0			0
$193$	−3074.00			−1.14648			−0.573242	−	0.819386i	$-0.694314\pi$
$193$	−3074.00			−1.14648			−0.573242	+	0.819386i	$0.694314\pi$
$194$		0			0
$195$	−2288.00	+	448.714i	−0.840241	+	0.164785i
$196$		0			0
$197$		−	4844.07i		−	1.75191i	−0.482396	−	0.875953i	$-0.660233\pi$
$197$		−	4844.07i		−	1.75191i	0.482396	−	0.875953i	$-0.339767\pi$
$198$		0			0
$199$		−	1927.43i		−	0.686592i	−0.939227	−	0.343296i	$-0.888457\pi$
$199$		−	1927.43i		−	0.686592i	0.939227	−	0.343296i	$-0.111543\pi$
$200$		0			0
$201$	−260.000	+	50.9902i	−0.0912387	+	0.0178934i
$202$		0			0
$203$	−2600.00			−0.898937
$204$		0			0
$205$	−4992.00			−1.70076
$206$		0			0
$207$	2200.00	−	897.427i	0.738698	−	0.301331i
$208$		0			0
$209$			469.110i			0.155258i
$210$		0			0
$211$		−	3436.74i		−	1.12130i	−0.828052	−	0.560651i	$-0.810551\pi$
$211$		−	3436.74i		−	1.12130i	0.828052	−	0.560651i	$-0.189449\pi$
$212$		0			0
$213$	−680.000	−	3467.33i	−0.218746	−	1.11539i
$214$		0			0
$215$	2392.00			0.758758
$216$		0			0
$217$	2184.00			0.683224
$218$		0			0
$219$	422.000	+	2151.79i	0.130211	+	0.663946i
$220$		0			0
$221$			897.427i			0.273156i
$222$		0			0
$223$			1233.96i			0.370548i	0.982687	+	0.185274i	$0.0593173\pi$
$223$			1233.96i			0.370548i	−0.982687	+	0.185274i	$0.940683\pi$
$224$		0			0
$225$	−525.000	+	214.159i	−0.155556	+	0.0634545i
$226$		0			0
$227$	−1270.00			−0.371334			−0.185667	−	0.982613i	$-0.559445\pi$
$227$	−1270.00			−0.371334			−0.185667	+	0.982613i	$0.559445\pi$
$228$		0			0
$229$	−4556.00			−1.31471			−0.657356	−	0.753580i	$-0.728325\pi$
$229$	−4556.00			−1.31471			−0.657356	+	0.753580i	$0.728325\pi$
$230$		0			0
$231$	−2392.00	+	469.110i	−0.681308	+	0.133615i
$232$		0			0
$233$			877.031i			0.246593i	0.992370	+	0.123297i	$0.0393467\pi$
$233$			877.031i			0.246593i	−0.992370	+	0.123297i	$0.960653\pi$
$234$		0			0
$235$		−	3916.05i		−	1.08704i
$236$		0			0
$237$	3796.00	−	744.457i	1.04041	−	0.204041i
$238$		0			0
$239$	1920.00			0.519642			0.259821	−	0.965657i	$-0.416336\pi$
$239$	1920.00			0.519642			0.259821	+	0.965657i	$0.416336\pi$
$240$		0			0
$241$	−1618.00			−0.432467			−0.216233	−	0.976342i	$-0.569377\pi$
$241$	−1618.00			−0.432467			−0.216233	+	0.976342i	$0.569377\pi$
$242$		0			0
$243$	−3121.00	−	2146.69i	−0.823919	−	0.566708i
$244$		0			0
$245$			2437.33i			0.635573i
$246$		0			0
$247$		−	448.714i		−	0.115591i
$248$		0			0
$249$	−186.000	−	948.418i	−0.0473384	−	0.241380i
$250$		0			0
$251$	−1586.00			−0.398834			−0.199417	−	0.979915i	$-0.563905\pi$
$251$	−1586.00			−0.398834			−0.199417	+	0.979915i	$0.563905\pi$
$252$		0			0
$253$	−4048.00			−1.00591
$254$		0			0
$255$	−208.000	−	1060.60i	−0.0510803	−	0.260459i
$256$		0			0
$257$		−	3956.84i		−	0.960392i	−0.877161	−	0.480196i	$-0.840565\pi$
$257$		−	3956.84i		−	0.960392i	0.877161	−	0.480196i	$-0.159435\pi$
$258$		0			0
$259$			3385.75i			0.812279i
$260$		0			0
$261$	2600.00	+	6373.77i	0.616613	+	1.51160i
$262$		0			0
$263$	−4344.00			−1.01849			−0.509244	−	0.860622i	$-0.670075\pi$
$263$	−4344.00			−1.01849			−0.509244	+	0.860622i	$0.670075\pi$
$264$		0			0
$265$	−4680.00			−1.08487
$266$		0			0
$267$	4888.00	−	958.616i	1.12038	−	0.219724i
$268$		0			0
$269$		−	2743.27i		−	0.621785i	−0.950445	−	0.310893i	$-0.899372\pi$
$269$		−	2743.27i		−	0.621785i	0.950445	−	0.310893i	$-0.100628\pi$
$270$		0			0
$271$		−	3253.17i		−	0.729211i	−0.931162	−	0.364606i	$-0.881204\pi$
$271$		−	3253.17i		−	0.729211i	0.931162	−	0.364606i	$-0.118796\pi$
$272$		0			0
$273$	2288.00	−	448.714i	0.507238	−	0.0994776i
$274$		0			0
$275$	966.000			0.211825
$276$		0			0
$277$	−4956.00			−1.07501			−0.537504	−	0.843261i	$-0.680633\pi$
$277$	−4956.00			−1.07501			−0.537504	+	0.843261i	$0.680633\pi$
$278$		0			0
$279$	−2184.00	−	5353.97i	−0.468648	−	1.14887i
$280$		0			0
$281$			1978.42i			0.420009i	0.977700	+	0.210005i	$0.0673479\pi$
$281$			1978.42i			0.420009i	−0.977700	+	0.210005i	$0.932652\pi$
$282$		0			0
$283$			7985.06i			1.67725i	0.544706	+	0.838627i	$0.316641\pi$
$283$			7985.06i			1.67725i	−0.544706	+	0.838627i	$0.683359\pi$
$284$		0			0
$285$	104.000	+	530.298i	0.0216155	+	0.110218i
$286$		0			0
$287$	4992.00			1.02672
$288$		0			0
$289$	4497.00			0.915327
$290$		0			0
$291$	1062.00	+	5415.16i	0.213937	+	1.09087i
$292$		0			0
$293$			1845.85i			0.368039i	0.982923	+	0.184019i	$0.0589110\pi$
$293$			1845.85i			0.368039i	−0.982923	+	0.184019i	$0.941089\pi$
$294$		0			0
$295$		−	6424.76i		−	1.26801i
$296$		0			0
$297$	3542.00	+	5394.76i	0.692012	+	1.05399i
$298$		0			0
$299$	3872.00			0.748908
$300$		0			0
$301$	−2392.00			−0.458048
$302$		0			0
$303$	52.0000	−	10.1980i	0.00985915	−	0.00193354i
$304$		0			0
$305$		−	2406.74i		−	0.451834i
$306$		0			0
$307$		−	8291.01i		−	1.54134i	−0.637232	−	0.770672i	$-0.719921\pi$
$307$		−	8291.01i		−	1.54134i	0.637232	−	0.770672i	$-0.280079\pi$
$308$		0			0
$309$	10140.0	−	1988.62i	1.86681	−	0.366112i
$310$		0			0
$311$	10104.0			1.84227			0.921134	−	0.389246i	$-0.127264\pi$
$311$	10104.0			1.84227			0.921134	+	0.389246i	$0.127264\pi$
$312$		0			0
$313$	−1198.00			−0.216342			−0.108171	−	0.994132i	$-0.534499\pi$
$313$	−1198.00			−0.216342			−0.108171	+	0.994132i	$0.534499\pi$
$314$		0			0
$315$	−2600.00	+	1060.60i	−0.465058	+	0.189707i
$316$		0			0
$317$		−	5027.63i		−	0.890789i	−0.895335	−	0.445394i	$-0.853063\pi$
$317$		−	5027.63i		−	0.890789i	0.895335	−	0.445394i	$-0.146937\pi$
$318$		0			0
$319$		−	11727.7i		−	2.05839i
$320$		0			0
$321$	−234.000	−	1193.17i	−0.0406872	−	0.207465i
$322$		0			0
$323$	208.000			0.0358311
$324$		0			0
$325$	−924.000			−0.157706
$326$		0			0
$327$	−724.000	−	3691.69i	−0.122438	−	0.624315i
$328$		0			0
$329$			3916.05i			0.656227i
$330$		0			0
$331$		−	6129.02i		−	1.01777i	−0.860835	−	0.508884i	$-0.830058\pi$
$331$		−	6129.02i		−	1.01777i	0.860835	−	0.508884i	$-0.169942\pi$
$332$		0			0
$333$	8300.00	−	3385.75i	1.36588	−	0.557171i
$334$		0			0
$335$	520.000			0.0848079
$336$		0			0
$337$	390.000			0.0630405			0.0315203	−	0.999503i	$-0.489965\pi$
$337$	390.000			0.0630405			0.0315203	+	0.999503i	$0.489965\pi$
$338$		0			0
$339$	5200.00	−	1019.80i	0.833113	−	0.163387i
$340$		0			0
$341$			9851.31i			1.56445i
$342$		0			0
$343$		−	5935.26i		−	0.934326i
$344$		0			0
$345$	−4576.00	+	897.427i	−0.714097	+	0.140046i
$346$		0			0
$347$	4366.00			0.675444			0.337722	−	0.941246i	$-0.390344\pi$
$347$	4366.00			0.675444			0.337722	+	0.941246i	$0.390344\pi$
$348$		0			0
$349$	5492.00			0.842350			0.421175	−	0.906979i	$-0.361618\pi$
$349$	5492.00			0.842350			0.421175	+	0.906979i	$0.361618\pi$
$350$		0			0
$351$	−3388.00	−	5160.21i	−0.515208	−	0.784705i
$352$		0			0
$353$			81.5843i			0.0123011i	0.999981	+	0.00615056i	$0.00195780\pi$
$353$			81.5843i			0.0123011i	−0.999981	+	0.00615056i	$0.998042\pi$
$354$		0			0
$355$			6934.67i			1.03677i
$356$		0			0
$357$	208.000	+	1060.60i	0.0308362	+	0.157234i
$358$		0			0
$359$	2216.00			0.325783			0.162891	−	0.986644i	$-0.447918\pi$
$359$	2216.00			0.325783			0.162891	+	0.986644i	$0.447918\pi$
$360$		0			0
$361$	6755.00			0.984837
$362$		0			0
$363$	−785.000	−	4002.73i	−0.113504	−	0.578757i
$364$		0			0
$365$		−	4303.57i		−	0.617149i
$366$		0			0
$367$		−	7128.43i		−	1.01390i	−0.861976	−	0.506950i	$-0.830773\pi$
$367$		−	7128.43i		−	1.01390i	0.861976	−	0.506950i	$-0.169227\pi$
$368$		0			0
$369$	−4992.00	−	12237.6i	−0.704263	−	1.72647i
$370$		0			0
$371$	4680.00			0.654915
$372$		0			0
$373$	−2028.00			−0.281517			−0.140759	−	0.990044i	$-0.544954\pi$
$373$	−2028.00			−0.281517			−0.140759	+	0.990044i	$0.544954\pi$
$374$		0			0
$375$	7592.00	−	1488.91i	1.04546	−	0.205032i
$376$		0			0
$377$			11217.8i			1.53249i
$378$		0			0
$379$			1315.55i			0.178298i	0.996018	+	0.0891492i	$0.0284148\pi$
$379$			1315.55i			0.178298i	−0.996018	+	0.0891492i	$0.971585\pi$
$380$		0			0
$381$	11492.0	−	2253.77i	1.54528	−	0.303055i
$382$		0			0
$383$	−2128.00			−0.283905			−0.141953	−	0.989873i	$-0.545338\pi$
$383$	−2128.00			−0.283905			−0.141953	+	0.989873i	$0.545338\pi$
$384$		0			0
$385$	4784.00			0.633286
$386$		0			0
$387$	2392.00	+	5863.87i	0.314192	+	0.770226i
$388$		0			0
$389$			1927.43i			0.251220i	0.992080	+	0.125610i	$0.0400888\pi$
$389$			1927.43i			0.251220i	−0.992080	+	0.125610i	$0.959911\pi$
$390$		0			0
$391$			1794.85i			0.232148i
$392$		0			0
$393$	−1742.00	−	8882.49i	−0.223594	−	1.14011i
$394$		0			0
$395$	−7592.00			−0.967076
$396$		0			0
$397$	−6220.00			−0.786330			−0.393165	−	0.919468i	$-0.628620\pi$
$397$	−6220.00			−0.786330			−0.393165	+	0.919468i	$0.628620\pi$
$398$		0			0
$399$	−104.000	−	530.298i	−0.0130489	−	0.0665366i
$400$		0			0
$401$		−	1937.63i		−	0.241298i	−0.992695	−	0.120649i	$-0.961502\pi$
$401$		−	1937.63i		−	0.241298i	0.992695	−	0.120649i	$-0.0384976\pi$
$402$		0			0
$403$		−	9422.99i		−	1.16475i
$404$		0			0
$405$	5200.00	+	5313.18i	0.638000	+	0.651886i
$406$		0			0
$407$	−15272.0			−1.85996
$408$		0			0
$409$	1270.00			0.153539			0.0767695	−	0.997049i	$-0.475539\pi$
$409$	1270.00			0.153539			0.0767695	+	0.997049i	$0.475539\pi$
$410$		0			0
$411$	−12272.0	+	2406.74i	−1.47283	+	0.288846i
$412$		0			0
$413$			6424.76i			0.765477i
$414$		0			0
$415$			1896.84i			0.224366i
$416$		0			0
$417$	3900.00	−	764.853i	0.457995	−	0.0898202i
$418$		0			0
$419$	−9126.00			−1.06404			−0.532022	−	0.846731i	$-0.678568\pi$
$419$	−9126.00			−1.06404			−0.532022	+	0.846731i	$0.678568\pi$
$420$		0			0
$421$	−3436.00			−0.397768			−0.198884	−	0.980023i	$-0.563732\pi$
$421$	−3436.00			−0.397768			−0.198884	+	0.980023i	$0.563732\pi$
$422$		0			0
$423$	9600.00	−	3916.05i	1.10347	−	0.450129i
$424$		0			0
$425$		−	428.318i		−	0.0488858i
$426$		0			0
$427$			2406.74i			0.272764i
$428$		0			0
$429$	2024.00	+	10320.4i	0.227785	+	1.16148i
$430$		0			0
$431$	1920.00			0.214578			0.107289	−	0.994228i	$-0.465783\pi$
$431$	1920.00			0.214578			0.107289	+	0.994228i	$0.465783\pi$
$432$		0			0
$433$	−15766.0			−1.74981			−0.874903	−	0.484299i	$-0.839075\pi$
$433$	−15766.0			−1.74981			−0.874903	+	0.484299i	$0.839075\pi$
$434$		0			0
$435$	−2600.00	−	13257.5i	−0.286576	−	1.46126i
$436$		0			0
$437$		−	897.427i		−	0.0982375i
$438$		0			0
$439$		−	7311.99i		−	0.794949i	−0.917613	−	0.397474i	$-0.869887\pi$
$439$		−	7311.99i		−	0.794949i	0.917613	−	0.397474i	$-0.130113\pi$
$440$		0			0
$441$	−5975.00	+	2437.33i	−0.645179	+	0.263182i
$442$		0			0
$443$	8190.00			0.878372			0.439186	−	0.898396i	$-0.355267\pi$
$443$	8190.00			0.878372			0.439186	+	0.898396i	$0.355267\pi$
$444$		0			0
$445$	−9776.00			−1.04141
$446$		0			0
$447$	−2444.00	+	479.308i	−0.258607	+	0.0507170i
$448$		0			0
$449$			13767.4i			1.44704i	0.690303	+	0.723521i	$0.257477\pi$
$449$			13767.4i			1.44704i	−0.690303	+	0.723521i	$0.742523\pi$
$450$		0			0
$451$			22517.3i			2.35099i
$452$		0			0
$453$	7644.00	−	1499.11i	0.792818	−	0.155484i
$454$		0			0
$455$	−4576.00			−0.471486
$456$		0			0
$457$	−10682.0			−1.09340			−0.546699	−	0.837329i	$-0.684116\pi$
$457$	−10682.0			−1.09340			−0.546699	+	0.837329i	$0.684116\pi$
$458$		0			0
$459$	2392.00	−	1570.50i	0.243244	−	0.159705i
$460$		0			0
$461$			3090.01i			0.312182i	0.987743	+	0.156091i	$0.0498893\pi$
$461$			3090.01i			0.312182i	−0.987743	+	0.156091i	$0.950111\pi$
$462$		0			0
$463$			2661.69i			0.267169i	0.991037	+	0.133584i	$0.0426487\pi$
$463$			2661.69i			0.267169i	−0.991037	+	0.133584i	$0.957351\pi$
$464$		0			0
$465$	2184.00	+	11136.3i	0.217808	+	1.11061i
$466$		0			0
$467$	−12246.0			−1.21344			−0.606721	−	0.794915i	$-0.707515\pi$
$467$	−12246.0			−1.21344			−0.606721	+	0.794915i	$0.707515\pi$
$468$		0			0
$469$	−520.000			−0.0511969
$470$		0			0
$471$	1948.00	+	9932.89i	0.190571	+	0.971727i
$472$		0			0
$473$		−	10789.5i		−	1.04884i
$474$		0			0
$475$			214.159i			0.0206869i
$476$		0			0
$477$	−4680.00	−	11472.8i	−0.449230	−	1.10126i
$478$		0			0
$479$	15008.0			1.43159			0.715796	−	0.698309i	$-0.246064\pi$
$479$	15008.0			1.43159			0.715796	+	0.698309i	$0.246064\pi$
$480$		0			0
$481$	14608.0			1.38476
$482$		0			0
$483$	4576.00	−	897.427i	0.431087	−	0.0845432i
$484$		0			0
$485$		−	10830.3i		−	1.01398i
$486$		0			0
$487$		−	5721.10i		−	0.532336i	−0.963927	−	0.266168i	$-0.914242\pi$
$487$		−	5721.10i		−	0.532336i	0.963927	−	0.266168i	$-0.0857576\pi$
$488$		0			0
$489$	17316.0	−	3395.95i	1.60134	−	0.314049i
$490$		0			0
$491$	5018.00			0.461220			0.230610	−	0.973046i	$-0.425928\pi$
$491$	5018.00			0.461220			0.230610	+	0.973046i	$0.425928\pi$
$492$		0			0
$493$	−5200.00			−0.475043
$494$		0			0
$495$	−4784.00	−	11727.7i	−0.434394	−	1.06489i
$496$		0			0
$497$		−	6934.67i		−	0.625880i
$498$		0			0
$499$		−	16082.3i		−	1.44277i	−0.692533	−	0.721386i	$-0.743506\pi$
$499$		−	16082.3i		−	1.44277i	0.692533	−	0.721386i	$-0.256494\pi$
$500$		0			0
$501$	−2008.00	−	10238.8i	−0.179064	−	0.913048i
$502$		0			0
$503$	−10632.0			−0.942460			−0.471230	−	0.882010i	$-0.656190\pi$
$503$	−10632.0			−0.942460			−0.471230	+	0.882010i	$0.656190\pi$
$504$		0			0
$505$	−104.000			−0.00916424
$506$		0			0
$507$	261.000	+	1330.84i	0.0228628	+	0.116578i
$508$		0			0
$509$			3579.51i			0.311707i	0.987780	+	0.155854i	$0.0498128\pi$
$509$			3579.51i			0.311707i	−0.987780	+	0.155854i	$0.950187\pi$
$510$		0			0
$511$			4303.57i			0.372561i
$512$		0			0
$513$	−1196.00	+	785.249i	−0.102933	+	0.0675820i
$514$		0			0
$515$	−20280.0			−1.73523
$516$		0			0
$517$	−17664.0			−1.50263
$518$		0			0
$519$	6188.00	−	1213.57i	0.523358	−	0.102639i
$520$		0			0
$521$			1590.89i			0.133778i	0.997760	+	0.0668890i	$0.0213073\pi$
$521$			1590.89i			0.133778i	−0.997760	+	0.0668890i	$0.978693\pi$
$522$		0			0
$523$		−	4109.81i		−	0.343613i	−0.985131	−	0.171806i	$-0.945040\pi$
$523$		−	4109.81i		−	0.343613i	0.985131	−	0.171806i	$-0.0549603\pi$
$524$		0			0
$525$	−1092.00	+	214.159i	−0.0907786	+	0.0178032i
$526$		0			0
$527$	4368.00			0.361049
$528$		0			0
$529$	−4423.00			−0.363524
$530$		0			0
$531$	15750.0	−	6424.76i	1.28718	−	0.525068i
$532$		0			0
$533$		−	21538.3i		−	1.75033i
$534$		0			0
$535$			2386.34i			0.192842i
$536$		0			0
$537$	−2382.00	−	12145.9i	−0.191417	−	0.976039i
$538$		0			0
$539$	10994.0			0.878562
$540$		0			0
$541$	−9484.00			−0.753695			−0.376848	−	0.926275i	$-0.622992\pi$
$541$	−9484.00			−0.753695			−0.376848	+	0.926275i	$0.622992\pi$
$542$		0			0
$543$	2652.00	+	13522.6i	0.209592	+	1.06871i
$544$		0			0
$545$			7383.38i			0.580311i
$546$		0			0
$547$			5476.35i			0.428065i	0.976827	+	0.214033i	$0.0686599\pi$
$547$			5476.35i			0.428065i	−0.976827	+	0.214033i	$0.931340\pi$
$548$		0			0
$549$	5900.00	−	2406.74i	0.458663	−	0.187098i
$550$		0			0
$551$	2600.00			0.201023
$552$		0			0
$553$	7592.00			0.583806
$554$		0			0
$555$	−17264.0	+	3385.75i	−1.32039	+	0.258950i
$556$		0			0
$557$			1091.19i			0.0830076i	0.999138	+	0.0415038i	$0.0132149\pi$
$557$			1091.19i			0.0830076i	−0.999138	+	0.0415038i	$0.986785\pi$
$558$		0			0
$559$			10320.4i			0.780871i
$560$		0			0
$561$	−4784.00	+	938.220i	−0.360037	+	0.0706090i
$562$		0			0
$563$	17898.0			1.33981			0.669903	−	0.742449i	$-0.266336\pi$
$563$	17898.0			1.33981			0.669903	+	0.742449i	$0.266336\pi$
$564$		0			0
$565$	−10400.0			−0.774392
$566$		0			0
$567$	−5200.00	−	5313.18i	−0.385149	−	0.393532i
$568$		0			0
$569$		−	21089.5i		−	1.55381i	−0.629616	−	0.776907i	$-0.716788\pi$
$569$		−	21089.5i		−	1.55381i	0.629616	−	0.776907i	$-0.283212\pi$
$570$		0			0
$571$			3334.76i			0.244405i	0.992505	+	0.122203i	$0.0389958\pi$
$571$			3334.76i			0.244405i	−0.992505	+	0.122203i	$0.961004\pi$
$572$		0			0
$573$	4784.00	+	24393.7i	0.348786	+	1.77847i
$574$		0			0
$575$	−1848.00			−0.134029
$576$		0			0
$577$	−5066.00			−0.365512			−0.182756	−	0.983158i	$-0.558502\pi$
$577$	−5066.00			−0.365512			−0.182756	+	0.983158i	$0.558502\pi$
$578$		0			0
$579$	3074.00	+	15674.4i	0.220641	+	1.12505i
$580$		0			0
$581$		−	1896.84i		−	0.135446i
$582$		0			0
$583$			21109.9i			1.49963i
$584$		0			0
$585$	4576.00	+	11217.8i	0.323409	+	0.792822i
$586$		0			0
$587$	20042.0			1.40924			0.704618	−	0.709587i	$-0.251118\pi$
$587$	20042.0			1.40924			0.704618	+	0.709587i	$0.251118\pi$
$588$		0			0
$589$	−2184.00			−0.152785
$590$		0			0
$591$	−24700.0	+	4844.07i	−1.71916	+	0.337155i
$592$		0			0
$593$			26351.7i			1.82485i	0.409244	+	0.912425i	$0.365792\pi$
$593$			26351.7i			1.82485i	−0.409244	+	0.912425i	$0.634208\pi$
$594$		0			0
$595$		−	2121.19i		−	0.146152i
$596$		0			0
$597$	−9828.00	+	1927.43i	−0.673758	+	0.132135i
$598$		0			0
$599$	−4344.00			−0.296312			−0.148156	−	0.988964i	$-0.547334\pi$
$599$	−4344.00			−0.296312			−0.148156	+	0.988964i	$0.547334\pi$
$600$		0			0
$601$	18250.0			1.23866			0.619329	−	0.785132i	$-0.287405\pi$
$601$	18250.0			1.23866			0.619329	+	0.785132i	$0.287405\pi$
$602$		0			0
$603$	520.000	+	1274.75i	0.0351178	+	0.0860896i
$604$		0			0
$605$			8005.46i			0.537964i
$606$		0			0
$607$			5272.39i			0.352553i	0.984341	+	0.176276i	$0.0564053\pi$
$607$			5272.39i			0.352553i	−0.984341	+	0.176276i	$0.943595\pi$
$608$		0			0
$609$	2600.00	+	13257.5i	0.173001	+	0.882133i
$610$		0			0
$611$	16896.0			1.11872
$612$		0			0
$613$	11748.0			0.774058			0.387029	−	0.922068i	$-0.373501\pi$
$613$	11748.0			0.774058			0.387029	+	0.922068i	$0.373501\pi$
$614$		0			0
$615$	4992.00	+	25454.3i	0.327312	+	1.66897i
$616$		0			0
$617$		−	3895.65i		−	0.254186i	−0.991891	−	0.127093i	$-0.959435\pi$
$617$		−	3895.65i		−	0.254186i	0.991891	−	0.127093i	$-0.0405647\pi$
$618$		0			0
$619$		−	7189.62i		−	0.466842i	−0.972376	−	0.233421i	$-0.925008\pi$
$619$		−	7189.62i		−	0.466842i	0.972376	−	0.233421i	$-0.0749920\pi$
$620$		0			0
$621$	−6776.00	−	10320.4i	−0.437861	−	0.666899i
$622$		0			0
$623$	9776.00			0.628679
$624$		0			0
$625$	−12559.0			−0.803776
$626$		0			0
$627$	2392.00	−	469.110i	0.152356	−	0.0298795i
$628$		0			0
$629$			6771.50i			0.429248i
$630$		0			0
$631$			26831.0i			1.69275i	0.532585	+	0.846376i	$0.321221\pi$
$631$			26831.0i			1.69275i	−0.532585	+	0.846376i	$0.678779\pi$
$632$		0			0
$633$	−17524.0	+	3436.74i	−1.10034	+	0.215795i
$634$		0			0
$635$	−22984.0			−1.43637
$636$		0			0
$637$	−10516.0			−0.654096
$638$		0			0
$639$	−17000.0	+	6934.67i	−1.05244	+	0.429313i
$640$		0			0
$641$		−	23639.1i		−	1.45661i	−0.685254	−	0.728305i	$-0.740309\pi$
$641$		−	23639.1i		−	1.45661i	0.685254	−	0.728305i	$-0.259691\pi$
$642$		0			0
$643$			16245.5i			0.996359i	0.867074	+	0.498180i	$0.165998\pi$
$643$			16245.5i			0.996359i	−0.867074	+	0.498180i	$0.834002\pi$
$644$		0			0
$645$	−2392.00	−	12196.9i	−0.146023	−	0.744575i
$646$		0			0
$647$	7384.00			0.448679			0.224339	−	0.974511i	$-0.427978\pi$
$647$	7384.00			0.448679			0.224339	+	0.974511i	$0.427978\pi$
$648$		0			0
$649$	−28980.0			−1.75280
$650$		0			0
$651$	−2184.00	−	11136.3i	−0.131486	−	0.670452i
$652$		0			0
$653$			15082.9i			0.903889i	0.892046	+	0.451944i	$0.149269\pi$
$653$			15082.9i			0.903889i	−0.892046	+	0.451944i	$0.850731\pi$
$654$		0			0
$655$			17765.0i			1.05975i
$656$		0			0
$657$	10550.0	−	4303.57i	0.626476	−	0.255553i
$658$		0			0
$659$	−20434.0			−1.20788			−0.603942	−	0.797028i	$-0.706404\pi$
$659$	−20434.0			−1.20788			−0.603942	+	0.797028i	$0.706404\pi$
$660$		0			0
$661$	12148.0			0.714830			0.357415	−	0.933946i	$-0.383658\pi$
$661$	12148.0			0.714830			0.357415	+	0.933946i	$0.383658\pi$
$662$		0			0
$663$	4576.00	−	897.427i	0.268050	−	0.0525689i
$664$		0			0
$665$			1060.60i			0.0618468i
$666$		0			0
$667$			22435.7i			1.30242i
$668$		0			0
$669$	6292.00	−	1233.96i	0.363621	−	0.0713120i
$670$		0			0
$671$	−10856.0			−0.624577
$672$		0			0
$673$	12730.0			0.729131			0.364566	−	0.931178i	$-0.381217\pi$
$673$	12730.0			0.729131			0.364566	+	0.931178i	$0.381217\pi$
$674$		0			0
$675$	1617.00	+	2462.83i	0.0922050	+	0.140436i
$676$		0			0
$677$			19672.0i			1.11678i	0.829580	+	0.558388i	$0.188580\pi$
$677$			19672.0i			1.11678i	−0.829580	+	0.558388i	$0.811420\pi$
$678$		0			0
$679$			10830.3i			0.612120i
$680$		0			0
$681$	1270.00	+	6475.75i	0.0714633	+	0.364393i
$682$		0			0
$683$	6558.00			0.367401			0.183701	−	0.982982i	$-0.441192\pi$
$683$	6558.00			0.367401			0.183701	+	0.982982i	$0.441192\pi$
$684$		0			0
$685$	24544.0			1.36902
$686$		0			0
$687$	4556.00	+	23231.1i	0.253016	+	1.29014i
$688$		0			0
$689$		−	20192.1i		−	1.11649i
$690$		0			0
$691$		−	30461.5i		−	1.67701i	−0.544896	−	0.838503i	$-0.683431\pi$
$691$		−	30461.5i		−	1.67701i	0.544896	−	0.838503i	$-0.316569\pi$
$692$		0			0
$693$	4784.00	+	11727.7i	0.262235	+	0.642857i
$694$		0			0
$695$	−7800.00			−0.425713
$696$		0			0
$697$	9984.00			0.542570
$698$		0			0
$699$	4472.00	−	877.031i	0.241984	−	0.0474569i
$700$		0			0
$701$		−	31461.0i		−	1.69510i	−0.530717	−	0.847549i	$-0.678077\pi$
$701$		−	31461.0i		−	1.69510i	0.530717	−	0.847549i	$-0.321923\pi$
$702$		0			0
$703$		−	3385.75i		−	0.181644i
$704$		0			0
$705$	−19968.0	+	3916.05i	−1.06672	+	0.209201i
$706$		0			0
$707$	104.000			0.00553228
$708$		0			0
$709$	25396.0			1.34523			0.672614	−	0.739993i	$-0.265171\pi$
$709$	25396.0			1.34523			0.672614	+	0.739993i	$0.265171\pi$
$710$		0			0
$711$	−7592.00	−	18611.4i	−0.400453	−	0.981692i
$712$		0			0
$713$		−	18846.0i		−	0.989884i
$714$		0			0
$715$		−	20640.8i		−	1.07961i
$716$		0			0
$717$	−1920.00	−	9790.12i	−0.100005	−	0.509928i
$718$		0			0
$719$	−7888.00			−0.409142			−0.204571	−	0.978852i	$-0.565580\pi$
$719$	−7888.00			−0.409142			−0.204571	+	0.978852i	$0.565580\pi$
$720$		0			0
$721$	20280.0			1.04753
$722$		0			0
$723$	1618.00	+	8250.21i	0.0832283	+	0.424383i
$724$		0			0
$725$		−	5353.97i		−	0.274264i
$726$		0			0
$727$			6475.75i			0.330361i	0.986263	+	0.165181i	$0.0528207\pi$
$727$			6475.75i			0.330361i	−0.986263	+	0.165181i	$0.947179\pi$
$728$		0			0
$729$	−7825.00	+	18060.7i	−0.397551	+	0.917580i
$730$		0			0
$731$	−4784.00			−0.242056
$732$		0			0
$733$	14772.0			0.744361			0.372180	−	0.928160i	$-0.378610\pi$
$733$	14772.0			0.744361			0.372180	+	0.928160i	$0.378610\pi$
$734$		0			0
$735$	12428.0	−	2437.33i	0.623692	−	0.122316i
$736$		0			0
$737$		−	2345.55i		−	0.117231i
$738$		0			0
$739$		−	22364.3i		−	1.11324i	−0.830767	−	0.556620i	$-0.812098\pi$
$739$		−	22364.3i		−	1.11324i	0.830767	−	0.556620i	$-0.187902\pi$
$740$		0			0
$741$	−2288.00	+	448.714i	−0.113430	+	0.0222455i
$742$		0			0
$743$	−19080.0			−0.942096			−0.471048	−	0.882108i	$-0.656124\pi$
$743$	−19080.0			−0.942096			−0.471048	+	0.882108i	$0.656124\pi$
$744$		0			0
$745$	4888.00			0.240379
$746$		0			0
$747$	−4650.00	+	1896.84i	−0.227757	+	0.0929071i
$748$		0			0
$749$		−	2386.34i		−	0.116415i
$750$		0			0
$751$			20324.7i			0.987561i	0.869586	+	0.493781i	$0.164385\pi$
$751$			20324.7i			0.987561i	−0.869586	+	0.493781i	$0.835615\pi$
$752$		0			0
$753$	1586.00	+	8087.04i	0.0767557	+	0.391379i
$754$		0			0
$755$	−15288.0			−0.736937
$756$		0			0
$757$	−4188.00			−0.201077			−0.100539	−	0.994933i	$-0.532057\pi$
$757$	−4188.00			−0.201077			−0.100539	+	0.994933i	$0.532057\pi$
$758$		0			0
$759$	4048.00	+	20640.8i	0.193588	+	0.987108i
$760$		0			0
$761$			30757.3i			1.46511i	0.680706	+	0.732556i	$0.261673\pi$
$761$			30757.3i			1.46511i	−0.680706	+	0.732556i	$0.738327\pi$
$762$		0			0
$763$		−	7383.38i		−	0.350323i
$764$		0			0
$765$	−5200.00	+	2121.19i	−0.245760	+	0.100251i
$766$		0			0
$767$	27720.0			1.30497
$768$		0			0
$769$	−10370.0			−0.486283			−0.243142	−	0.969991i	$-0.578178\pi$
$769$	−10370.0			−0.486283			−0.243142	+	0.969991i	$0.578178\pi$
$770$		0			0
$771$	−20176.0	+	3956.84i	−0.942440	+	0.184828i
$772$		0			0
$773$			10779.3i			0.501559i	0.968044	+	0.250780i	$0.0806870\pi$
$773$			10779.3i			0.501559i	−0.968044	+	0.250780i	$0.919313\pi$
$774$		0			0
$775$			4497.34i			0.208450i
$776$		0			0
$777$	17264.0	−	3385.75i	0.797095	−	0.156323i
$778$		0			0
$779$	−4992.00			−0.229598
$780$		0			0
$781$	31280.0			1.43315
$782$		0			0
$783$	29900.0	−	19631.2i	1.36467	−	0.895993i
$784$		0			0
$785$		−	19865.8i		−	0.903236i
$786$		0			0
$787$			23384.1i			1.05915i	0.848262	+	0.529576i	$0.177649\pi$
$787$			23384.1i			1.05915i	−0.848262	+	0.529576i	$0.822351\pi$
$788$		0			0
$789$	4344.00	+	22150.1i	0.196008	+	0.999450i
$790$		0			0
$791$	10400.0			0.467486
$792$		0			0
$793$	10384.0			0.465002
$794$		0			0
$795$	4680.00	+	23863.4i	0.208783	+	1.06459i
$796$		0			0
$797$		−	3232.78i		−	0.143677i	−0.997416	−	0.0718387i	$-0.977113\pi$
$797$		−	3232.78i		−	0.143677i	0.997416	−	0.0718387i	$-0.0228867\pi$
$798$		0			0
$799$			7832.09i			0.346783i
$800$		0			0
$801$	−9776.00	−	23965.4i	−0.431233	−	1.05715i
$802$		0			0
$803$	−19412.0			−0.853094
$804$		0			0
$805$	−9152.00			−0.400703
$806$		0			0
$807$	−13988.0	+	2743.27i	−0.610162	+	0.119663i
$808$		0			0
$809$			33857.5i			1.47140i	0.677305	+	0.735702i	$0.263148\pi$
$809$			33857.5i			1.47140i	−0.677305	+	0.735702i	$0.736852\pi$
$810$		0			0
$811$			10493.8i			0.454361i	0.973853	+	0.227180i	$0.0729507\pi$
$811$			10493.8i			0.454361i	−0.973853	+	0.227180i	$0.927049\pi$
$812$		0			0
$813$	−16588.0	+	3253.17i	−0.715580	+	0.140337i
$814$		0			0
$815$	−34632.0			−1.48847
$816$		0			0
$817$	2392.00			0.102430
$818$		0			0
$819$	−4576.00	−	11217.8i	−0.195236	−	0.478612i
$820$		0			0
$821$			15103.3i			0.642032i	0.947074	+	0.321016i	$0.104024\pi$
$821$			15103.3i			0.642032i	−0.947074	+	0.321016i	$0.895976\pi$
$822$		0			0
$823$		−	17958.7i		−	0.760635i	−0.924856	−	0.380317i	$-0.875815\pi$
$823$		−	17958.7i		−	0.760635i	0.924856	−	0.380317i	$-0.124185\pi$
$824$		0			0
$825$	−966.000	−	4925.65i	−0.0407658	−	0.207866i
$826$		0			0
$827$	−6598.00			−0.277430			−0.138715	−	0.990332i	$-0.544297\pi$
$827$	−6598.00			−0.277430			−0.138715	+	0.990332i	$0.544297\pi$
$828$		0			0
$829$	41028.0			1.71889			0.859446	−	0.511227i	$-0.170809\pi$
$829$	41028.0			1.71889			0.859446	+	0.511227i	$0.170809\pi$
$830$		0			0
$831$	4956.00	+	25270.7i	0.206885	+	1.05491i
$832$		0			0
$833$		−	4874.66i		−	0.202758i
$834$		0			0
$835$			20477.7i			0.848693i
$836$		0			0
$837$	−25116.0	+	16490.2i	−1.03720	+	0.680987i
$838$		0			0
$839$	−17816.0			−0.733107			−0.366553	−	0.930397i	$-0.619462\pi$
$839$	−17816.0			−0.733107			−0.366553	+	0.930397i	$0.619462\pi$
$840$		0			0
$841$	−40611.0			−1.66514
$842$		0			0
$843$	10088.0	−	1978.42i	0.412158	−	0.0808308i
$844$		0			0
$845$		−	2661.69i		−	0.108361i
$846$		0			0
$847$		−	8005.46i		−	0.324759i
$848$		0			0
$849$	40716.0	−	7985.06i	1.64590	−	0.322788i
$850$		0			0
$851$	29216.0			1.17686
$852$		0			0
$853$	28308.0			1.13628			0.568140	−	0.822932i	$-0.307663\pi$
$853$	28308.0			1.13628			0.568140	+	0.822932i	$0.307663\pi$
$854$		0			0
$855$	2600.00	−	1060.60i	0.103998	−	0.0424229i
$856$		0			0
$857$			33735.1i			1.34466i	0.740254	+	0.672328i	$0.234705\pi$
$857$			33735.1i			1.34466i	−0.740254	+	0.672328i	$0.765295\pi$
$858$		0			0
$859$			336.535i			0.0133672i	0.999978	+	0.00668361i	$0.00212747\pi$
$859$			336.535i			0.0133672i	−0.999978	+	0.00668361i	$0.997873\pi$
$860$		0			0
$861$	−4992.00	−	25454.3i	−0.197592	−	1.00753i
$862$		0			0
$863$	−48208.0			−1.90153			−0.950764	−	0.309915i	$-0.899700\pi$
$863$	−48208.0			−1.90153			−0.950764	+	0.309915i	$0.899700\pi$
$864$		0			0
$865$	−12376.0			−0.486470
$866$		0			0
$867$	−4497.00	−	22930.3i	−0.176155	−	0.898216i
$868$		0			0
$869$			34245.0i			1.33680i
$870$		0			0
$871$			2243.57i			0.0872795i
$872$		0			0
$873$	26550.0	−	10830.3i	1.02930	−	0.419875i
$874$		0			0
$875$	15184.0			0.586643
$876$		0			0
$877$	−42844.0			−1.64965			−0.824823	−	0.565391i	$-0.808725\pi$
$877$	−42844.0			−1.64965			−0.824823	+	0.565391i	$0.808725\pi$
$878$		0			0
$879$	9412.00	−	1845.85i	0.361159	−	0.0708291i
$880$		0			0
$881$			6975.46i			0.266753i	0.991065	+	0.133376i	$0.0425819\pi$
$881$			6975.46i			0.266753i	−0.991065	+	0.133376i	$0.957418\pi$
$882$		0			0
$883$		−	4803.28i		−	0.183061i	−0.995802	−	0.0915306i	$-0.970824\pi$
$883$		−	4803.28i		−	0.183061i	0.995802	−	0.0915306i	$-0.0291759\pi$
$884$		0			0
$885$	−32760.0	+	6424.76i	−1.24431	+	0.244029i
$886$		0			0
$887$	37336.0			1.41333			0.706663	−	0.707550i	$-0.250200\pi$
$887$	37336.0			1.41333			0.706663	+	0.707550i	$0.250200\pi$
$888$		0			0
$889$	22984.0			0.867108
$890$		0			0
$891$	23966.0	−	23455.5i	0.901112	−	0.881917i
$892$		0			0
$893$		−	3916.05i		−	0.146747i
$894$		0			0
$895$			24291.7i			0.907244i
$896$		0			0
$897$	−3872.00	−	19743.4i	−0.144127	−	0.734909i
$898$		0			0
$899$	54600.0			2.02560
$900$		0			0
$901$	9360.00			0.346090
$902$		0			0
$903$	2392.00	+	12196.9i	0.0881515	+	0.449486i
$904$		0			0
$905$		−	27045.2i		−	0.993384i
$906$		0			0
$907$			16490.2i			0.603692i	0.953357	+	0.301846i	$0.0976029\pi$
$907$			16490.2i			0.603692i	−0.953357	+	0.301846i	$0.902397\pi$
$908$		0			0
$909$	−104.000	−	254.951i	−0.00379479	−	0.00930274i
$910$		0			0
$911$	35360.0			1.28598			0.642991	−	0.765874i	$-0.277693\pi$
$911$	35360.0			1.28598			0.642991	+	0.765874i	$0.277693\pi$
$912$		0			0
$913$	8556.00			0.310145
$914$		0			0
$915$	−12272.0	+	2406.74i	−0.443388	+	0.0869555i
$916$		0			0
$917$		−	17765.0i		−	0.639751i
$918$		0			0
$919$			4395.35i			0.157769i	0.996884	+	0.0788843i	$0.0251358\pi$
$919$			4395.35i			0.157769i	−0.996884	+	0.0788843i	$0.974864\pi$
$920$		0			0
$921$	−42276.0	+	8291.01i	−1.51253	+	0.296632i
$922$		0			0
$923$	−29920.0			−1.06699
$924$		0			0
$925$	−6972.00			−0.247825
$926$		0			0
$927$	−20280.0	−	49715.4i	−0.718536	−	1.76146i
$928$		0			0
$929$			1815.25i			0.0641081i	0.999486	+	0.0320541i	$0.0102049\pi$
$929$			1815.25i			0.0641081i	−0.999486	+	0.0320541i	$0.989795\pi$
$930$		0			0
$931$			2437.33i			0.0858005i
$932$		0			0
$933$	−10104.0	−	51520.5i	−0.354545	−	1.80783i
$934$		0			0
$935$	9568.00			0.334660
$936$		0			0
$937$	−34034.0			−1.18660			−0.593299	−	0.804982i	$-0.702175\pi$
$937$	−34034.0			−1.18660			−0.593299	+	0.804982i	$0.702175\pi$
$938$		0			0
$939$	1198.00	+	6108.63i	0.0416350	+	0.212298i
$940$		0			0
$941$		−	36967.9i		−	1.28068i	−0.768092	−	0.640339i	$-0.778794\pi$
$941$		−	36967.9i		−	1.28068i	0.768092	−	0.640339i	$-0.221206\pi$
$942$		0			0
$943$		−	43076.5i		−	1.48756i
$944$		0			0
$945$	8008.00	+	12196.9i	0.275662	+	0.419856i
$946$		0			0
$947$	24862.0			0.853122			0.426561	−	0.904459i	$-0.359725\pi$
$947$	24862.0			0.853122			0.426561	+	0.904459i	$0.359725\pi$
$948$		0			0
$949$	18568.0			0.635135
$950$		0			0
$951$	−25636.0	+	5027.63i	−0.874137	+	0.171432i
$952$		0			0
$953$		−	22802.8i		−	0.775085i	−0.921852	−	0.387542i	$-0.873324\pi$
$953$		−	22802.8i		−	0.775085i	0.921852	−	0.387542i	$-0.126676\pi$
$954$		0			0
$955$		−	48787.4i		−	1.65311i
$956$		0			0
$957$	−59800.0	+	11727.7i	−2.01992	+	0.396138i
$958$		0			0
$959$	−24544.0			−0.826452
$960$		0			0
$961$	−16073.0			−0.539525
$962$		0			0
$963$	−5850.00	+	2386.34i	−0.195757	+	0.0798533i
$964$		0			0
$965$		−	31348.8i		−	1.04575i
$966$		0			0
$967$		−	40068.1i		−	1.33247i	−0.745740	−	0.666237i	$-0.767904\pi$
$967$		−	40068.1i		−	1.33247i	0.745740	−	0.666237i	$-0.232096\pi$
$968$		0			0
$969$	−208.000	−	1060.60i	−0.00689569	−	0.0351613i
$970$		0			0
$971$	−43522.0			−1.43840			−0.719201	−	0.694803i	$-0.755492\pi$
$971$	−43522.0			−1.43840			−0.719201	+	0.694803i	$0.755492\pi$
$972$		0			0
$973$	7800.00			0.256995
$974$		0			0
$975$	924.000	+	4711.49i	0.0303504	+	0.154758i
$976$		0			0
$977$			17683.4i			0.579060i	0.957169	+	0.289530i	$0.0934991\pi$
$977$			17683.4i			0.579060i	−0.957169	+	0.289530i	$0.906501\pi$
$978$		0			0
$979$			44096.3i			1.43956i
$980$		0			0
$981$	−18100.0	+	7383.38i	−0.589081	+	0.240299i
$982$		0			0
$983$	42360.0			1.37444			0.687220	−	0.726450i	$-0.258831\pi$
$983$	42360.0			1.37444			0.687220	+	0.726450i	$0.258831\pi$
$984$		0			0
$985$	49400.0			1.59798
$986$		0			0
$987$	19968.0	−	3916.05i	0.643960	−	0.126291i
$988$		0			0
$989$			20640.8i			0.663640i
$990$		0			0
$991$		−	25199.4i		−	0.807754i	−0.914813	−	0.403877i	$-0.867662\pi$
$991$		−	25199.4i		−	0.807754i	0.914813	−	0.403877i	$-0.132338\pi$
$992$		0			0
$993$	−31252.0	+	6129.02i	−0.998743	+	0.195870i
$994$		0			0
$995$	19656.0			0.626268
$996$		0			0
$997$	−5356.00			−0.170137			−0.0850683	−	0.996375i	$-0.527111\pi$
$997$	−5356.00			−0.170137			−0.0850683	+	0.996375i	$0.527111\pi$
$998$		0			0
$999$	−25564.0	−	38936.1i	−0.809619	−	1.23312i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.c.d.767.1		2
3.2	odd	2		768.4.c.f.767.1		2
4.3	odd	2		768.4.c.f.767.2		2
8.3	odd	2		768.4.c.e.767.1		2
8.5	even	2		768.4.c.g.767.2		2
12.11	even	2	inner	768.4.c.d.767.2		2
16.3	odd	4		384.4.f.e.191.2	yes	4
16.5	even	4		384.4.f.f.191.2	yes	4
16.11	odd	4		384.4.f.e.191.3	yes	4
16.13	even	4		384.4.f.f.191.3	yes	4
24.5	odd	2		768.4.c.e.767.2		2
24.11	even	2		768.4.c.g.767.1		2
48.5	odd	4		384.4.f.e.191.1	✓	4
48.11	even	4		384.4.f.f.191.4	yes	4
48.29	odd	4		384.4.f.e.191.4	yes	4
48.35	even	4		384.4.f.f.191.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.f.e.191.1	✓	4	48.5	odd	4
384.4.f.e.191.2	yes	4	16.3	odd	4
384.4.f.e.191.3	yes	4	16.11	odd	4
384.4.f.e.191.4	yes	4	48.29	odd	4
384.4.f.f.191.1	yes	4	48.35	even	4
384.4.f.f.191.2	yes	4	16.5	even	4
384.4.f.f.191.3	yes	4	16.13	even	4
384.4.f.f.191.4	yes	4	48.11	even	4
768.4.c.d.767.1		2	1.1	even	1	trivial
768.4.c.d.767.2		2	12.11	even	2	inner
768.4.c.e.767.1		2	8.3	odd	2
768.4.c.e.767.2		2	24.5	odd	2
768.4.c.f.767.1		2	3.2	odd	2
768.4.c.f.767.2		2	4.3	odd	2
768.4.c.g.767.1		2	24.11	even	2
768.4.c.g.767.2		2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-1.00000 - 5.09902i) q^{3} +10.1980i q^{5} -10.1980i q^{7} +(-25.0000 + 10.1980i) q^{9} +46.0000 q^{11} -44.0000 q^{13} +(52.0000 - 10.1980i) q^{15} -20.3961i q^{17} +10.1980i q^{19} +(-52.0000 + 10.1980i) q^{21} -88.0000 q^{23} +21.0000 q^{25} +(77.0000 + 117.277i) q^{27} -254.951i q^{29} +214.159i q^{31} +(-46.0000 - 234.555i) q^{33} +104.000 q^{35} -332.000 q^{37} +(44.0000 + 224.357i) q^{39} +489.506i q^{41} -234.555i q^{43} +(-104.000 - 254.951i) q^{45} -384.000 q^{47} +239.000 q^{49} +(-104.000 + 20.3961i) q^{51} +458.912i q^{53} +469.110i q^{55} +(52.0000 - 10.1980i) q^{57} -630.000 q^{59} -236.000 q^{61} +(104.000 + 254.951i) q^{63} -448.714i q^{65} -50.9902i q^{67} +(88.0000 + 448.714i) q^{69} +680.000 q^{71} -422.000 q^{73} +(-21.0000 - 107.079i) q^{75} -469.110i q^{77} +744.457i q^{79} +(521.000 - 509.902i) q^{81} +186.000 q^{83} +208.000 q^{85} +(-1300.00 + 254.951i) q^{87} +958.616i q^{89} +448.714i q^{91} +(1092.00 - 214.159i) q^{93} -104.000 q^{95} -1062.00 q^{97} +(-1150.00 + 469.110i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 50 q^{9} + 92 q^{11} - 88 q^{13} + 104 q^{15} - 104 q^{21} - 176 q^{23} + 42 q^{25} + 154 q^{27} - 92 q^{33} + 208 q^{35} - 664 q^{37} + 88 q^{39} - 208 q^{45} - 768 q^{47} + 478 q^{49}+ \cdots - 2300 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 50 * q^9 + 92 * q^11 - 88 * q^13 + 104 * q^15 - 104 * q^21 - 176 * q^23 + 42 * q^25 + 154 * q^27 - 92 * q^33 + 208 * q^35 - 664 * q^37 + 88 * q^39 - 208 * q^45 - 768 * q^47 + 478 * q^49 - 208 * q^51 + 104 * q^57 - 1260 * q^59 - 472 * q^61 + 208 * q^63 + 176 * q^69 + 1360 * q^71 - 844 * q^73 - 42 * q^75 + 1042 * q^81 + 372 * q^83 + 416 * q^85 - 2600 * q^87 + 2184 * q^93 - 208 * q^95 - 2124 * q^97 - 2300 * q^99

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