LMFDB - Embedded newform 384.4.c.a.383.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(383,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("384.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 $ x^12 + 39x^10 + 549x^8 + 3500x^6 + 10236x^4 + 11952*x^2 + 4624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.7
Root			$2.36157i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.4.c.a.383.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.556921 - 5.16622i) q^{3} +10.6077i q^{5} -7.90379i q^{7} +(-26.3797 + 5.75435i) q^{9} +O(q^{10})$	\(q+(-0.556921 - 5.16622i) q^{3} +10.6077i q^{5} -7.90379i q^{7} +(-26.3797 + 5.75435i) q^{9} -11.7652 q^{11} +30.5992 q^{13} +(54.8017 - 5.90765i) q^{15} +118.124i q^{17} -66.4680i q^{19} +(-40.8327 + 4.40179i) q^{21} +166.335 q^{23} +12.4768 q^{25} +(44.4197 + 133.079i) q^{27} -111.999i q^{29} +224.770i q^{31} +(6.55231 + 60.7818i) q^{33} +83.8410 q^{35} -70.6099 q^{37} +(-17.0414 - 158.082i) q^{39} +247.184i q^{41} +98.7603i q^{43} +(-61.0404 - 279.828i) q^{45} +189.577 q^{47} +280.530 q^{49} +(610.257 - 65.7860i) q^{51} -529.370i q^{53} -124.802i q^{55} +(-343.388 + 37.0174i) q^{57} +811.889 q^{59} +833.410 q^{61} +(45.4812 + 208.499i) q^{63} +324.587i q^{65} +2.83446i q^{67} +(-92.6357 - 859.325i) q^{69} +796.627 q^{71} -875.489 q^{73} +(-6.94861 - 64.4581i) q^{75} +92.9900i q^{77} -656.826i q^{79} +(662.775 - 303.596i) q^{81} -237.058 q^{83} -1253.03 q^{85} +(-578.610 + 62.3744i) q^{87} +868.956i q^{89} -241.850i q^{91} +(1161.21 - 125.179i) q^{93} +705.072 q^{95} +755.048 q^{97} +(310.363 - 67.7014i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{3}+O(q^{10}) $ 12 * q - 10 * q^3	$ 12 q - 10 q^{3} - 36 q^{11} - 84 q^{15} + 136 q^{21} + 120 q^{23} - 300 q^{25} + 266 q^{27} - 116 q^{33} - 432 q^{35} + 528 q^{37} - 620 q^{39} + 440 q^{45} + 1248 q^{47} - 948 q^{49} + 1072 q^{51} - 172 q^{57} - 2508 q^{59} + 624 q^{61} - 2744 q^{63} - 24 q^{69} + 2040 q^{71} - 216 q^{73} + 3894 q^{75} - 1076 q^{81} - 4572 q^{83} + 480 q^{85} - 4156 q^{87} - 112 q^{93} + 5448 q^{95} - 48 q^{97} + 6044 q^{99}+O(q^{100}) $ 12 * q - 10 * q^3 - 36 * q^11 - 84 * q^15 + 136 * q^21 + 120 * q^23 - 300 * q^25 + 266 * q^27 - 116 * q^33 - 432 * q^35 + 528 * q^37 - 620 * q^39 + 440 * q^45 + 1248 * q^47 - 948 * q^49 + 1072 * q^51 - 172 * q^57 - 2508 * q^59 + 624 * q^61 - 2744 * q^63 - 24 * q^69 + 2040 * q^71 - 216 * q^73 + 3894 * q^75 - 1076 * q^81 - 4572 * q^83 + 480 * q^85 - 4156 * q^87 - 112 * q^93 + 5448 * q^95 - 48 * q^97 + 6044 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.556921	−	5.16622i	−0.107180	−	0.994240i
$3$	−0.556921	−	5.16622i	−0.107180	−	0.994240i
$4$		0			0
$5$			10.6077i			0.948781i	0.880315	+	0.474390i	$0.157332\pi$
$5$			10.6077i			0.948781i	−0.880315	+	0.474390i	$0.842668\pi$
$6$		0			0
$7$		−	7.90379i		−	0.426764i	−0.976969	−	0.213382i	$-0.931552\pi$
$7$		−	7.90379i		−	0.426764i	0.976969	−	0.213382i	$-0.0684480\pi$
$8$		0			0
$9$	−26.3797	+	5.75435i	−0.977025	+	0.213124i
$10$		0			0
$11$	−11.7652			−0.322487			−0.161243	−	0.986915i	$-0.551550\pi$
$11$	−11.7652			−0.322487			−0.161243	+	0.986915i	$0.551550\pi$
$12$		0			0
$13$	30.5992			0.652823			0.326412	−	0.945228i	$-0.394160\pi$
$13$	30.5992			0.652823			0.326412	+	0.945228i	$0.394160\pi$
$14$		0			0
$15$	54.8017	−	5.90765i	0.943316	−	0.101690i
$16$		0			0
$17$			118.124i			1.68526i	0.538495	+	0.842629i	$0.318993\pi$
$17$			118.124i			1.68526i	−0.538495	+	0.842629i	$0.681007\pi$
$18$		0			0
$19$		−	66.4680i		−	0.802568i	−0.915954	−	0.401284i	$-0.868564\pi$
$19$		−	66.4680i		−	0.802568i	0.915954	−	0.401284i	$-0.131436\pi$
$20$		0			0
$21$	−40.8327	+	4.40179i	−0.424306	+	0.0457404i
$22$		0			0
$23$	166.335			1.50797			0.753985	−	0.656891i	$-0.228129\pi$
$23$	166.335			1.50797			0.753985	+	0.656891i	$0.228129\pi$
$24$		0			0
$25$	12.4768			0.0998147
$26$		0			0
$27$	44.4197	+	133.079i	0.316614	+	0.948555i
$28$		0			0
$29$		−	111.999i		−	0.717159i	−0.933499	−	0.358580i	$-0.883261\pi$
$29$		−	111.999i		−	0.717159i	0.933499	−	0.358580i	$-0.116739\pi$
$30$		0			0
$31$			224.770i			1.30226i	0.758968	+	0.651128i	$0.225704\pi$
$31$			224.770i			1.30226i	−0.758968	+	0.651128i	$0.774296\pi$
$32$		0			0
$33$	6.55231	+	60.7818i	0.0345640	+	0.320629i
$34$		0			0
$35$	83.8410			0.404906
$36$		0			0
$37$	−70.6099			−0.313735			−0.156868	−	0.987620i	$-0.550140\pi$
$37$	−70.6099			−0.313735			−0.156868	+	0.987620i	$0.550140\pi$
$38$		0			0
$39$	−17.0414	−	158.082i	−0.0699693	−	0.649063i
$40$		0			0
$41$			247.184i			0.941551i	0.882253	+	0.470775i	$0.156026\pi$
$41$			247.184i			0.941551i	−0.882253	+	0.470775i	$0.843974\pi$
$42$		0			0
$43$			98.7603i			0.350251i	0.984546	+	0.175126i	$0.0560332\pi$
$43$			98.7603i			0.350251i	−0.984546	+	0.175126i	$0.943967\pi$
$44$		0			0
$45$	−61.0404	−	279.828i	−0.202208	−	0.926983i
$46$		0			0
$47$	189.577			0.588355			0.294178	−	0.955751i	$-0.404954\pi$
$47$	189.577			0.588355			0.294178	+	0.955751i	$0.404954\pi$
$48$		0			0
$49$	280.530			0.817872
$50$		0			0
$51$	610.257	−	65.7860i	1.67555	−	0.180625i
$52$		0			0
$53$		−	529.370i		−	1.37197i	−0.727614	−	0.685987i	$-0.759371\pi$
$53$		−	529.370i		−	1.37197i	0.727614	−	0.685987i	$-0.240629\pi$
$54$		0			0
$55$		−	124.802i		−	0.305969i
$56$		0			0
$57$	−343.388	+	37.0174i	−0.797945	+	0.0860189i
$58$		0			0
$59$	811.889			1.79151			0.895753	−	0.444551i	$-0.146637\pi$
$59$	811.889			1.79151			0.895753	+	0.444551i	$0.146637\pi$
$60$		0			0
$61$	833.410			1.74930			0.874650	−	0.484756i	$-0.161091\pi$
$61$	833.410			1.74930			0.874650	+	0.484756i	$0.161091\pi$
$62$		0			0
$63$	45.4812	+	208.499i	0.0909538	+	0.416960i
$64$		0			0
$65$			324.587i			0.619386i
$66$		0			0
$67$			2.83446i			0.00516843i	0.999997	+	0.00258421i	$0.000822582\pi$
$67$			2.83446i			0.00516843i	−0.999997	+	0.00258421i	$0.999177\pi$
$68$		0			0
$69$	−92.6357	−	859.325i	−0.161624	−	1.49928i
$70$		0			0
$71$	796.627			1.33158			0.665790	−	0.746139i	$-0.268095\pi$
$71$	796.627			1.33158			0.665790	+	0.746139i	$0.268095\pi$
$72$		0			0
$73$	−875.489			−1.40367			−0.701837	−	0.712338i	$-0.747637\pi$
$73$	−875.489			−1.40367			−0.701837	+	0.712338i	$0.747637\pi$
$74$		0			0
$75$	−6.94861	−	64.4581i	−0.0106981	−	0.0992397i
$76$		0			0
$77$			92.9900i			0.137626i
$78$		0			0
$79$		−	656.826i		−	0.935427i	−0.883880	−	0.467713i	$-0.845078\pi$
$79$		−	656.826i		−	0.935427i	0.883880	−	0.467713i	$-0.154922\pi$
$80$		0			0
$81$	662.775	−	303.596i	0.909156	−	0.416455i
$82$		0			0
$83$	−237.058			−0.313500			−0.156750	−	0.987638i	$-0.550102\pi$
$83$	−237.058			−0.313500			−0.156750	+	0.987638i	$0.550102\pi$
$84$		0			0
$85$	−1253.03			−1.59894
$86$		0			0
$87$	−578.610	+	62.3744i	−0.713028	+	0.0768648i
$88$		0			0
$89$			868.956i			1.03493i	0.855703	+	0.517467i	$0.173125\pi$
$89$			868.956i			1.03493i	−0.855703	+	0.517467i	$0.826875\pi$
$90$		0			0
$91$		−	241.850i		−	0.278602i
$92$		0			0
$93$	1161.21	−	125.179i	1.29476	−	0.139575i
$94$		0			0
$95$	705.072			0.761461
$96$		0			0
$97$	755.048			0.790345			0.395173	−	0.918607i	$-0.370685\pi$
$97$	755.048			0.790345			0.395173	+	0.918607i	$0.370685\pi$
$98$		0			0
$99$	310.363	−	67.7014i	0.315078	−	0.0687297i
$100$		0			0
$101$			1159.48i			1.14230i	0.820846	+	0.571149i	$0.193502\pi$
$101$			1159.48i			1.14230i	−0.820846	+	0.571149i	$0.806498\pi$
$102$		0			0
$103$			1426.76i			1.36488i	0.730942	+	0.682440i	$0.239081\pi$
$103$			1426.76i			1.36488i	−0.730942	+	0.682440i	$0.760919\pi$
$104$		0			0
$105$	−46.6928	−	433.141i	−0.0433976	−	0.402574i
$106$		0			0
$107$	−1503.34			−1.35826			−0.679128	−	0.734020i	$-0.737642\pi$
$107$	−1503.34			−1.35826			−0.679128	+	0.734020i	$0.737642\pi$
$108$		0			0
$109$	−446.011			−0.391928			−0.195964	−	0.980611i	$-0.562784\pi$
$109$	−446.011			−0.391928			−0.195964	+	0.980611i	$0.562784\pi$
$110$		0			0
$111$	39.3242	+	364.787i	0.0336260	+	0.311928i
$112$		0			0
$113$			1413.62i			1.17684i	0.808557	+	0.588418i	$0.200249\pi$
$113$			1413.62i			1.17684i	−0.808557	+	0.588418i	$0.799751\pi$
$114$		0			0
$115$			1764.44i			1.43073i
$116$		0			0
$117$	−807.198	+	176.079i	−0.637825	+	0.139132i
$118$		0			0
$119$	933.630			0.719208
$120$		0			0
$121$	−1192.58			−0.896002
$122$		0			0
$123$	1277.00	−	137.662i	0.936127	−	0.100915i
$124$		0			0
$125$			1458.31i			1.04348i
$126$		0			0
$127$		−	1315.32i		−	0.919020i	−0.888173	−	0.459510i	$-0.848025\pi$
$127$		−	1315.32i		−	0.919020i	0.888173	−	0.459510i	$-0.151975\pi$
$128$		0			0
$129$	510.218	−	55.0017i	0.348234	−	0.0375398i
$130$		0			0
$131$	−2183.27			−1.45613			−0.728067	−	0.685506i	$-0.759581\pi$
$131$	−2183.27			−1.45613			−0.728067	+	0.685506i	$0.759581\pi$
$132$		0			0
$133$	−525.349			−0.342508
$134$		0			0
$135$	−1411.66	+	471.190i	−0.899971	+	0.300397i
$136$		0			0
$137$		−	751.608i		−	0.468717i	−0.972150	−	0.234359i	$-0.924701\pi$
$137$		−	751.608i		−	0.468717i	0.972150	−	0.234359i	$-0.0752990\pi$
$138$		0			0
$139$		−	1745.14i		−	1.06490i	−0.846463	−	0.532448i	$-0.821272\pi$
$139$		−	1745.14i		−	1.06490i	0.846463	−	0.532448i	$-0.178728\pi$
$140$		0			0
$141$	−105.580	−	979.399i	−0.0630596	−	0.584966i
$142$		0			0
$143$	−360.008			−0.210527
$144$		0			0
$145$	1188.05			0.680427
$146$		0			0
$147$	−156.233	−	1449.28i	−0.0876591	−	0.813161i
$148$		0			0
$149$			1859.32i			1.02229i	0.859494	+	0.511145i	$0.170779\pi$
$149$			1859.32i			1.02229i	−0.859494	+	0.511145i	$0.829221\pi$
$150$		0			0
$151$		−	967.660i		−	0.521504i	−0.965406	−	0.260752i	$-0.916030\pi$
$151$		−	967.660i		−	0.521504i	0.965406	−	0.260752i	$-0.0839704\pi$
$152$		0			0
$153$	−679.730	−	3116.08i	−0.359169	−	1.64654i
$154$		0			0
$155$	−2384.30			−1.23556
$156$		0			0
$157$	3639.08			1.84987			0.924936	−	0.380123i	$-0.124118\pi$
$157$	3639.08			1.84987			0.924936	+	0.380123i	$0.124118\pi$
$158$		0			0
$159$	−2734.84	+	294.817i	−1.36407	+	0.147047i
$160$		0			0
$161$		−	1314.68i		−	0.643548i
$162$		0			0
$163$			337.359i			0.162110i	0.996710	+	0.0810551i	$0.0258290\pi$
$163$			337.359i			0.162110i	−0.996710	+	0.0810551i	$0.974171\pi$
$164$		0			0
$165$	−644.755	+	69.5049i	−0.304207	+	0.0327936i
$166$		0			0
$167$	−2686.84			−1.24499			−0.622497	−	0.782622i	$-0.713882\pi$
$167$	−2686.84			−1.24499			−0.622497	+	0.782622i	$0.713882\pi$
$168$		0			0
$169$	−1260.69			−0.573822
$170$		0			0
$171$	382.480	+	1753.40i	0.171047	+	0.784129i
$172$		0			0
$173$			1970.03i			0.865773i	0.901449	+	0.432886i	$0.142505\pi$
$173$			1970.03i			0.865773i	−0.901449	+	0.432886i	$0.857495\pi$
$174$		0			0
$175$		−	98.6143i		−	0.0425974i
$176$		0			0
$177$	−452.158	−	4194.40i	−0.192013	−	1.78119i
$178$		0			0
$179$	−836.542			−0.349308			−0.174654	−	0.984630i	$-0.555881\pi$
$179$	−836.542			−0.349308			−0.174654	+	0.984630i	$0.555881\pi$
$180$		0			0
$181$	−1716.39			−0.704853			−0.352426	−	0.935840i	$-0.614643\pi$
$181$	−1716.39			−0.704853			−0.352426	+	0.935840i	$0.614643\pi$
$182$		0			0
$183$	−464.144	−	4305.58i	−0.187489	−	1.73922i
$184$		0			0
$185$		−	749.009i		−	0.297666i
$186$		0			0
$187$		−	1389.76i		−	0.543473i
$188$		0			0
$189$	1051.82	−	351.084i	0.404809	−	0.135119i
$190$		0			0
$191$	358.884			0.135958			0.0679788	−	0.997687i	$-0.478345\pi$
$191$	358.884			0.135958			0.0679788	+	0.997687i	$0.478345\pi$
$192$		0			0
$193$	2670.56			0.996015			0.498007	−	0.867173i	$-0.334065\pi$
$193$	2670.56			0.996015			0.498007	+	0.867173i	$0.334065\pi$
$194$		0			0
$195$	1676.89	−	180.770i	0.615819	−	0.0663855i
$196$		0			0
$197$			597.410i			0.216059i	0.994148	+	0.108030i	$0.0344542\pi$
$197$			597.410i			0.216059i	−0.994148	+	0.108030i	$0.965546\pi$
$198$		0			0
$199$			3336.44i			1.18851i	0.804276	+	0.594256i	$0.202553\pi$
$199$			3336.44i			1.18851i	−0.804276	+	0.594256i	$0.797447\pi$
$200$		0			0
$201$	14.6435	−	1.57857i	0.00513865	−	0.000553949i
$202$		0			0
$203$	−885.213			−0.306058
$204$		0			0
$205$	−2622.05			−0.893326
$206$		0			0
$207$	−4387.87	+	957.153i	−1.47333	+	0.321385i
$208$		0			0
$209$			782.012i			0.258818i
$210$		0			0
$211$		−	1234.24i		−	0.402695i	−0.979520	−	0.201348i	$-0.935468\pi$
$211$		−	1234.24i		−	0.402695i	0.979520	−	0.201348i	$-0.0645321\pi$
$212$		0			0
$213$	−443.658	−	4115.55i	−0.142718	−	1.32391i
$214$		0			0
$215$	−1047.62			−0.332312
$216$		0			0
$217$	1776.54			0.555757
$218$		0			0
$219$	487.578	+	4522.97i	0.150445	+	1.39559i
$220$		0			0
$221$			3614.52i			1.10018i
$222$		0			0
$223$			1647.48i			0.494723i	0.968923	+	0.247362i	$0.0795635\pi$
$223$			1647.48i			0.494723i	−0.968923	+	0.247362i	$0.920436\pi$
$224$		0			0
$225$	−329.135	+	71.7962i	−0.0975215	+	0.0212729i
$226$		0			0
$227$	−340.265			−0.0994897			−0.0497449	−	0.998762i	$-0.515841\pi$
$227$	−340.265			−0.0994897			−0.0497449	+	0.998762i	$0.515841\pi$
$228$		0			0
$229$	4031.94			1.16349			0.581743	−	0.813373i	$-0.302371\pi$
$229$	4031.94			1.16349			0.581743	+	0.813373i	$0.302371\pi$
$230$		0			0
$231$	480.407	−	51.7881i	0.136833	−	0.0147507i
$232$		0			0
$233$			1464.29i			0.411710i	0.978582	+	0.205855i	$0.0659976\pi$
$233$			1464.29i			0.411710i	−0.978582	+	0.205855i	$0.934002\pi$
$234$		0			0
$235$			2010.98i			0.558220i
$236$		0			0
$237$	−3393.31	+	365.800i	−0.930038	+	0.100259i
$238$		0			0
$239$	3357.75			0.908765			0.454382	−	0.890807i	$-0.349860\pi$
$239$	3357.75			0.908765			0.454382	+	0.890807i	$0.349860\pi$
$240$		0			0
$241$	−2736.93			−0.731541			−0.365771	−	0.930705i	$-0.619195\pi$
$241$	−2736.93			−0.731541			−0.365771	+	0.930705i	$0.619195\pi$
$242$		0			0
$243$	−1937.56	−	3254.96i	−0.511499	−	0.859284i
$244$		0			0
$245$			2975.78i			0.775981i
$246$		0			0
$247$		−	2033.87i		−	0.523935i
$248$		0			0
$249$	132.023	+	1224.70i	0.0336008	+	0.311694i
$250$		0			0
$251$	−5959.74			−1.49871			−0.749354	−	0.662170i	$-0.769636\pi$
$251$	−5959.74			−1.49871			−0.749354	+	0.662170i	$0.769636\pi$
$252$		0			0
$253$	−1956.98			−0.486301
$254$		0			0
$255$	697.837	+	6473.42i	0.171374	+	1.58973i
$256$		0			0
$257$		−	1384.26i		−	0.335982i	−0.985788	−	0.167991i	$-0.946272\pi$
$257$		−	1384.26i		−	0.335982i	0.985788	−	0.167991i	$-0.0537280\pi$
$258$		0			0
$259$			558.086i			0.133891i
$260$		0			0
$261$	644.480	+	2954.49i	0.152844	+	0.700683i
$262$		0			0
$263$	6426.06			1.50664			0.753322	−	0.657651i	$-0.228450\pi$
$263$	6426.06			1.50664			0.753322	+	0.657651i	$0.228450\pi$
$264$		0			0
$265$	5615.40			1.30170
$266$		0			0
$267$	4489.22	−	483.940i	1.02897	−	0.110924i
$268$		0			0
$269$		−	1594.21i		−	0.361341i	−0.983544	−	0.180671i	$-0.942173\pi$
$269$		−	1594.21i		−	0.361341i	0.983544	−	0.180671i	$-0.0578268\pi$
$270$		0			0
$271$		−	7502.42i		−	1.68170i	−0.541271	−	0.840848i	$-0.682057\pi$
$271$		−	7502.42i		−	1.68170i	0.541271	−	0.840848i	$-0.317943\pi$
$272$		0			0
$273$	−1249.45	+	134.691i	−0.276997	+	0.0298604i
$274$		0			0
$275$	−146.793			−0.0321889
$276$		0			0
$277$	1099.05			0.238395			0.119198	−	0.992871i	$-0.461968\pi$
$277$	1099.05			0.238395			0.119198	+	0.992871i	$0.461968\pi$
$278$		0			0
$279$	−1293.41	−	5929.37i	−0.277542	−	1.27234i
$280$		0			0
$281$			6848.46i			1.45390i	0.686692	+	0.726949i	$0.259062\pi$
$281$			6848.46i			1.45390i	−0.686692	+	0.726949i	$0.740938\pi$
$282$		0			0
$283$		−	4354.29i		−	0.914614i	−0.889309	−	0.457307i	$-0.848814\pi$
$283$		−	4354.29i		−	0.914614i	0.889309	−	0.457307i	$-0.151186\pi$
$284$		0			0
$285$	−392.669	−	3642.56i	−0.0816131	−	0.757075i
$286$		0			0
$287$	1953.69			0.401820
$288$		0			0
$289$	−9040.37			−1.84009
$290$		0			0
$291$	−420.502	−	3900.74i	−0.0847088	−	0.785793i
$292$		0			0
$293$		−	2845.10i		−	0.567278i	−0.958931	−	0.283639i	$-0.908458\pi$
$293$		−	2845.10i		−	0.567278i	0.958931	−	0.283639i	$-0.0915417\pi$
$294$		0			0
$295$			8612.27i			1.69975i
$296$		0			0
$297$	−522.608	−	1565.70i	−0.102104	−	0.305896i
$298$		0			0
$299$	5089.74			0.984439
$300$		0			0
$301$	780.581			0.149475
$302$		0			0
$303$	5990.11	−	645.737i	1.13572	−	0.122431i
$304$		0			0
$305$			8840.56i			1.65970i
$306$		0			0
$307$			7798.95i			1.44987i	0.688819	+	0.724934i	$0.258130\pi$
$307$			7798.95i			1.44987i	−0.688819	+	0.724934i	$0.741870\pi$
$308$		0			0
$309$	7370.95	−	794.592i	1.35702	−	0.146287i
$310$		0			0
$311$	4799.43			0.875084			0.437542	−	0.899198i	$-0.355849\pi$
$311$	4799.43			0.875084			0.437542	+	0.899198i	$0.355849\pi$
$312$		0			0
$313$	−2400.65			−0.433524			−0.216762	−	0.976224i	$-0.569550\pi$
$313$	−2400.65			−0.433524			−0.216762	+	0.976224i	$0.569550\pi$
$314$		0			0
$315$	−2211.70	+	482.451i	−0.395603	+	0.0862953i
$316$		0			0
$317$		−	5599.88i		−	0.992179i	−0.868271	−	0.496090i	$-0.834769\pi$
$317$		−	5599.88i		−	0.992179i	0.868271	−	0.496090i	$-0.165231\pi$
$318$		0			0
$319$			1317.69i			0.231274i
$320$		0			0
$321$	837.242	+	7766.59i	0.145577	+	1.35043i
$322$		0			0
$323$	7851.49			1.35253
$324$		0			0
$325$	381.782			0.0651614
$326$		0			0
$327$	248.393	+	2304.19i	0.0420067	+	0.389670i
$328$		0			0
$329$		−	1498.38i		−	0.251089i
$330$		0			0
$331$		−	5568.28i		−	0.924654i	−0.886709	−	0.462327i	$-0.847015\pi$
$331$		−	5568.28i		−	0.924654i	0.886709	−	0.462327i	$-0.152985\pi$
$332$		0			0
$333$	1862.67	−	406.315i	0.306527	−	0.0668646i
$334$		0			0
$335$	−30.0671			−0.00490370
$336$		0			0
$337$	−1366.22			−0.220839			−0.110420	−	0.993885i	$-0.535220\pi$
$337$	−1366.22			−0.220839			−0.110420	+	0.993885i	$0.535220\pi$
$338$		0			0
$339$	7303.09	−	787.276i	1.17006	−	0.126133i
$340$		0			0
$341$		−	2644.48i		−	0.419961i
$342$		0			0
$343$		−	4928.25i		−	0.775803i
$344$		0			0
$345$	9115.46	−	982.651i	1.42249	−	0.153345i
$346$		0			0
$347$	9688.05			1.49879			0.749397	−	0.662121i	$-0.230343\pi$
$347$	9688.05			1.49879			0.749397	+	0.662121i	$0.230343\pi$
$348$		0			0
$349$	−11427.5			−1.75272			−0.876361	−	0.481654i	$-0.840036\pi$
$349$	−11427.5			−1.75272			−0.876361	+	0.481654i	$0.840036\pi$
$350$		0			0
$351$	1359.21	+	4072.10i	0.206693	+	0.619239i
$352$		0			0
$353$		−	630.357i		−	0.0950440i	−0.998870	−	0.0475220i	$-0.984868\pi$
$353$		−	630.357i		−	0.0950440i	0.998870	−	0.0475220i	$-0.0151324\pi$
$354$		0			0
$355$			8450.37i			1.26338i
$356$		0			0
$357$	−519.958	−	4823.34i	−0.0770843	−	0.715065i
$358$		0			0
$359$	−7429.40			−1.09222			−0.546112	−	0.837712i	$-0.683893\pi$
$359$	−7429.40			−1.09222			−0.546112	+	0.837712i	$0.683893\pi$
$360$		0			0
$361$	2441.01			0.355884
$362$		0			0
$363$	664.172	+	6161.13i	0.0960331	+	0.890841i
$364$		0			0
$365$		−	9286.92i		−	1.33178i
$366$		0			0
$367$		−	11698.9i		−	1.66397i	−0.554797	−	0.831986i	$-0.687204\pi$
$367$		−	11698.9i		−	1.66397i	0.554797	−	0.831986i	$-0.312796\pi$
$368$		0			0
$369$	−1422.38	−	6520.62i	−0.200667	−	0.919919i
$370$		0			0
$371$	−4184.03			−0.585510
$372$		0			0
$373$	5197.32			0.721466			0.360733	−	0.932669i	$-0.382527\pi$
$373$	5197.32			0.721466			0.360733	+	0.932669i	$0.382527\pi$
$374$		0			0
$375$	7533.96	−	812.165i	1.03747	−	0.111840i
$376$		0			0
$377$		−	3427.07i		−	0.468178i
$378$		0			0
$379$		−	2510.74i		−	0.340284i	−0.985420	−	0.170142i	$-0.945577\pi$
$379$		−	2510.74i		−	0.340284i	0.985420	−	0.170142i	$-0.0544227\pi$
$380$		0			0
$381$	−6795.22	+	732.528i	−0.913726	+	0.0985001i
$382$		0			0
$383$	−4920.99			−0.656530			−0.328265	−	0.944586i	$-0.606464\pi$
$383$	−4920.99			−0.656530			−0.328265	+	0.944586i	$0.606464\pi$
$384$		0			0
$385$	−986.409			−0.130577
$386$		0			0
$387$	−568.302	−	2605.27i	−0.0746471	−	0.342204i
$388$		0			0
$389$		−	3477.90i		−	0.453308i	−0.973975	−	0.226654i	$-0.927221\pi$
$389$		−	3477.90i		−	0.453308i	0.973975	−	0.226654i	$-0.0727786\pi$
$390$		0			0
$391$			19648.3i			2.54132i
$392$		0			0
$393$	1215.91	+	11279.3i	0.156068	+	1.44775i
$394$		0			0
$395$	6967.41			0.887515
$396$		0			0
$397$	2710.55			0.342667			0.171333	−	0.985213i	$-0.445192\pi$
$397$	2710.55			0.342667			0.171333	+	0.985213i	$0.445192\pi$
$398$		0			0
$399$	292.578	+	2714.07i	0.0367098	+	0.340535i
$400$		0			0
$401$		−	4661.17i		−	0.580468i	−0.956956	−	0.290234i	$-0.906267\pi$
$401$		−	4661.17i		−	0.580468i	0.956956	−	0.290234i	$-0.0937331\pi$
$402$		0			0
$403$			6877.80i			0.850144i
$404$		0			0
$405$	3220.45	+	7030.51i	0.395125	+	0.862590i
$406$		0			0
$407$	830.743			0.101175
$408$		0			0
$409$	12349.9			1.49307			0.746533	−	0.665348i	$-0.231717\pi$
$409$	12349.9			1.49307			0.746533	+	0.665348i	$0.231717\pi$
$410$		0			0
$411$	−3882.97	+	418.586i	−0.466017	+	0.0502369i
$412$		0			0
$413$		−	6417.00i		−	0.764551i
$414$		0			0
$415$		−	2514.64i		−	0.297443i
$416$		0			0
$417$	−9015.76	+	971.903i	−1.05876	+	0.114135i
$418$		0			0
$419$	528.053			0.0615682			0.0307841	−	0.999526i	$-0.490200\pi$
$419$	528.053			0.0615682			0.0307841	+	0.999526i	$0.490200\pi$
$420$		0			0
$421$	8788.97			1.01745			0.508727	−	0.860928i	$-0.330116\pi$
$421$	8788.97			1.01745			0.508727	+	0.860928i	$0.330116\pi$
$422$		0			0
$423$	−5000.99	+	1090.90i	−0.574838	+	0.125393i
$424$		0			0
$425$			1473.82i			0.168213i
$426$		0			0
$427$		−	6587.10i		−	0.746539i
$428$		0			0
$429$	200.496	+	1859.88i	0.0225642	+	0.209314i
$430$		0			0
$431$	3660.00			0.409040			0.204520	−	0.978862i	$-0.434437\pi$
$431$	3660.00			0.409040			0.204520	+	0.978862i	$0.434437\pi$
$432$		0			0
$433$	−4322.55			−0.479742			−0.239871	−	0.970805i	$-0.577105\pi$
$433$	−4322.55			−0.479742			−0.239871	+	0.970805i	$0.577105\pi$
$434$		0			0
$435$	−661.648	−	6137.71i	−0.0729278	−	0.676508i
$436$		0			0
$437$		−	11056.0i		−	1.21025i
$438$		0			0
$439$		−	12788.3i		−	1.39032i	−0.718854	−	0.695161i	$-0.755333\pi$
$439$		−	12788.3i		−	1.39032i	0.718854	−	0.695161i	$-0.244667\pi$
$440$		0			0
$441$	−7400.29	+	1614.27i	−0.799082	+	0.174308i
$442$		0			0
$443$	5382.41			0.577259			0.288630	−	0.957441i	$-0.406800\pi$
$443$	5382.41			0.577259			0.288630	+	0.957441i	$0.406800\pi$
$444$		0			0
$445$	−9217.62			−0.981926
$446$		0			0
$447$	9605.65	−	1035.49i	1.01640	−	0.109569i
$448$		0			0
$449$		−	9465.38i		−	0.994875i	−0.867500	−	0.497437i	$-0.834274\pi$
$449$		−	9465.38i		−	0.994875i	0.867500	−	0.497437i	$-0.165726\pi$
$450$		0			0
$451$		−	2908.17i		−	0.303638i
$452$		0			0
$453$	−4999.15	+	538.910i	−0.518500	+	0.0558945i
$454$		0			0
$455$	2565.47			0.264332
$456$		0			0
$457$	−716.322			−0.0733219			−0.0366610	−	0.999328i	$-0.511672\pi$
$457$	−716.322			−0.0733219			−0.0366610	+	0.999328i	$0.511672\pi$
$458$		0			0
$459$	−15719.8	+	5247.05i	−1.59856	+	0.533575i
$460$		0			0
$461$			19191.7i			1.93893i	0.245226	+	0.969466i	$0.421138\pi$
$461$			19191.7i			1.93893i	−0.245226	+	0.969466i	$0.578862\pi$
$462$		0			0
$463$			222.541i			0.0223377i	0.999938	+	0.0111689i	$0.00355523\pi$
$463$			222.541i			0.0223377i	−0.999938	+	0.0111689i	$0.996445\pi$
$464$		0			0
$465$	1327.86	+	12317.8i	0.132426	+	1.22844i
$466$		0			0
$467$	5784.61			0.573190			0.286595	−	0.958052i	$-0.407477\pi$
$467$	5784.61			0.573190			0.286595	+	0.958052i	$0.407477\pi$
$468$		0			0
$469$	22.4030			0.00220570
$470$		0			0
$471$	−2026.68	−	18800.3i	−0.198268	−	1.83922i
$472$		0			0
$473$		−	1161.94i		−	0.112951i
$474$		0			0
$475$		−	829.310i		−	0.0801081i
$476$		0			0
$477$	3046.18	+	13964.6i	0.292401	+	1.34045i
$478$		0			0
$479$	−9218.02			−0.879294			−0.439647	−	0.898171i	$-0.644897\pi$
$479$	−9218.02			−0.879294			−0.439647	+	0.898171i	$0.644897\pi$
$480$		0			0
$481$	−2160.61			−0.204814
$482$		0			0
$483$	−6791.93	+	732.173i	−0.639841	+	0.0689752i
$484$		0			0
$485$			8009.32i			0.749864i
$486$		0			0
$487$			17851.9i			1.66109i	0.556955	+	0.830543i	$0.311970\pi$
$487$			17851.9i			1.66109i	−0.556955	+	0.830543i	$0.688030\pi$
$488$		0			0
$489$	1742.87	−	187.882i	0.161176	−	0.0173749i
$490$		0			0
$491$	−12979.1			−1.19295			−0.596473	−	0.802633i	$-0.703432\pi$
$491$	−12979.1			−1.19295			−0.596473	+	0.802633i	$0.703432\pi$
$492$		0			0
$493$	13229.8			1.20860
$494$		0			0
$495$	718.155	+	3292.24i	0.0652095	+	0.298940i
$496$		0			0
$497$		−	6296.37i		−	0.568271i
$498$		0			0
$499$		−	17513.6i		−	1.57117i	−0.618752	−	0.785586i	$-0.712361\pi$
$499$		−	17513.6i		−	1.57117i	0.618752	−	0.785586i	$-0.287639\pi$
$500$		0			0
$501$	1496.36	+	13880.8i	0.133438	+	1.23782i
$502$		0			0
$503$	−7405.59			−0.656459			−0.328230	−	0.944598i	$-0.606452\pi$
$503$	−7405.59			−0.656459			−0.328230	+	0.944598i	$0.606452\pi$
$504$		0			0
$505$	−12299.4			−1.08379
$506$		0			0
$507$	702.103	+	6512.98i	0.0615019	+	0.570516i
$508$		0			0
$509$			2162.88i			0.188346i	0.995556	+	0.0941728i	$0.0300206\pi$
$509$			2162.88i			0.188346i	−0.995556	+	0.0941728i	$0.969979\pi$
$510$		0			0
$511$			6919.68i			0.599038i
$512$		0			0
$513$	8845.46	−	2952.48i	0.761280	−	0.254104i
$514$		0			0
$515$	−15134.6			−1.29497
$516$		0			0
$517$	−2230.42			−0.189737
$518$		0			0
$519$	10177.6	−	1097.15i	0.860786	−	0.0927931i
$520$		0			0
$521$		−	13662.4i		−	1.14887i	−0.818550	−	0.574435i	$-0.805222\pi$
$521$		−	13662.4i		−	1.14887i	0.818550	−	0.574435i	$-0.194778\pi$
$522$		0			0
$523$		−	3574.54i		−	0.298860i	−0.988772	−	0.149430i	$-0.952256\pi$
$523$		−	3574.54i		−	0.298860i	0.988772	−	0.149430i	$-0.0477438\pi$
$524$		0			0
$525$	−509.463	+	54.9204i	−0.0423520	+	0.00456556i
$526$		0			0
$527$	−26550.9			−2.19464
$528$		0			0
$529$	15500.5			1.27398
$530$		0			0
$531$	−21417.4	+	4671.89i	−1.75035	+	0.381813i
$532$		0			0
$533$			7563.63i			0.614666i
$534$		0			0
$535$		−	15947.0i		−	1.28869i
$536$		0			0
$537$	465.888	+	4321.76i	0.0374386	+	0.347295i
$538$		0			0
$539$	−3300.51			−0.263753
$540$		0			0
$541$	−5965.93			−0.474114			−0.237057	−	0.971496i	$-0.576183\pi$
$541$	−5965.93			−0.474114			−0.237057	+	0.971496i	$0.576183\pi$
$542$		0			0
$543$	955.895	+	8867.26i	0.0755458	+	0.700793i
$544$		0			0
$545$		−	4731.15i		−	0.371854i
$546$		0			0
$547$			6960.57i			0.544081i	0.962286	+	0.272040i	$0.0876985\pi$
$547$			6960.57i			0.544081i	−0.962286	+	0.272040i	$0.912302\pi$
$548$		0			0
$549$	−21985.1	+	4795.74i	−1.70911	+	0.372818i
$550$		0			0
$551$	−7444.32			−0.575569
$552$		0			0
$553$	−5191.41			−0.399207
$554$		0			0
$555$	−3869.54	+	417.139i	−0.295951	+	0.0319037i
$556$		0			0
$557$		−	6929.11i		−	0.527102i	−0.964645	−	0.263551i	$-0.915106\pi$
$557$		−	6929.11i		−	0.527102i	0.964645	−	0.263551i	$-0.0848937\pi$
$558$		0			0
$559$			3021.99i			0.228652i
$560$		0			0
$561$	−7179.82	+	773.988i	−0.540343	+	0.0582492i
$562$		0			0
$563$	20207.5			1.51269			0.756344	−	0.654175i	$-0.226984\pi$
$563$	20207.5			1.51269			0.756344	+	0.654175i	$0.226984\pi$
$564$		0			0
$565$	−14995.3			−1.11656
$566$		0			0
$567$	−2399.56	−	5238.43i	−0.177728	−	0.387995i
$568$		0			0
$569$		−	14954.6i		−	1.10181i	−0.834567	−	0.550906i	$-0.814282\pi$
$569$		−	14954.6i		−	1.10181i	0.834567	−	0.550906i	$-0.185718\pi$
$570$		0			0
$571$			11065.5i			0.810996i	0.914096	+	0.405498i	$0.132902\pi$
$571$			11065.5i			0.810996i	−0.914096	+	0.405498i	$0.867098\pi$
$572$		0			0
$573$	−199.870	−	1854.07i	−0.0145719	−	0.135174i
$574$		0			0
$575$	2075.34			0.150518
$576$		0			0
$577$	20741.1			1.49647			0.748235	−	0.663433i	$-0.230901\pi$
$577$	20741.1			1.49647			0.748235	+	0.663433i	$0.230901\pi$
$578$		0			0
$579$	−1487.29	−	13796.7i	−0.106752	−	0.990277i
$580$		0			0
$581$			1873.66i			0.133791i
$582$		0			0
$583$			6228.17i			0.442443i
$584$		0			0
$585$	−1867.79	−	8562.51i	−0.132006	−	0.605156i
$586$		0			0
$587$	−8044.96			−0.565675			−0.282837	−	0.959168i	$-0.591276\pi$
$587$	−8044.96			−0.565675			−0.282837	+	0.959168i	$0.591276\pi$
$588$		0			0
$589$	14940.0			1.04515
$590$		0			0
$591$	3086.35	−	332.710i	0.214815	−	0.0231571i
$592$		0			0
$593$		−	11007.3i		−	0.762255i	−0.924522	−	0.381128i	$-0.875536\pi$
$593$		−	11007.3i		−	0.762255i	0.924522	−	0.381128i	$-0.124464\pi$
$594$		0			0
$595$			9903.66i			0.682371i
$596$		0			0
$597$	17236.8	−	1858.13i	1.18167	−	0.127384i
$598$		0			0
$599$	−15693.5			−1.07048			−0.535242	−	0.844699i	$-0.679780\pi$
$599$	−15693.5			−1.07048			−0.535242	+	0.844699i	$0.679780\pi$
$600$		0			0
$601$	6984.08			0.474021			0.237010	−	0.971507i	$-0.423832\pi$
$601$	6984.08			0.474021			0.237010	+	0.971507i	$0.423832\pi$
$602$		0			0
$603$	−16.3105	−	74.7722i	−0.00110152	−	0.00504968i
$604$		0			0
$605$		−	12650.5i		−	0.850110i
$606$		0			0
$607$			2251.13i			0.150528i	0.997164	+	0.0752641i	$0.0239800\pi$
$607$			2251.13i			0.150528i	−0.997164	+	0.0752641i	$0.976020\pi$
$608$		0			0
$609$	492.994	+	4573.21i	0.0328032	+	0.304295i
$610$		0			0
$611$	5800.93			0.384092
$612$		0			0
$613$	−4638.50			−0.305624			−0.152812	−	0.988255i	$-0.548833\pi$
$613$	−4638.50			−0.305624			−0.152812	+	0.988255i	$0.548833\pi$
$614$		0			0
$615$	1460.27	+	13546.1i	0.0957462	+	0.888180i
$616$		0			0
$617$		−	16218.9i		−	1.05826i	−0.848540	−	0.529132i	$-0.822518\pi$
$617$		−	16218.9i		−	1.05826i	0.848540	−	0.529132i	$-0.177482\pi$
$618$		0			0
$619$			11838.8i			0.768727i	0.923182	+	0.384363i	$0.125579\pi$
$619$			11838.8i			0.768727i	−0.923182	+	0.384363i	$0.874421\pi$
$620$		0			0
$621$	7388.56	+	22135.7i	0.477444	+	1.43039i
$622$		0			0
$623$	6868.05			0.441673
$624$		0			0
$625$	−13909.7			−0.890222
$626$		0			0
$627$	4040.04	−	435.519i	0.257327	−	0.0277399i
$628$		0			0
$629$		−	8340.76i		−	0.528724i
$630$		0			0
$631$		−	14214.6i		−	0.896791i	−0.893835	−	0.448395i	$-0.851996\pi$
$631$		−	14214.6i		−	0.896791i	0.893835	−	0.448395i	$-0.148004\pi$
$632$		0			0
$633$	−6376.36	+	687.375i	−0.400375	+	0.0431607i
$634$		0			0
$635$	13952.5			0.871949
$636$		0			0
$637$	8584.01			0.533926
$638$		0			0
$639$	−21014.8	+	4584.07i	−1.30099	+	0.283792i
$640$		0			0
$641$			13075.6i			0.805704i	0.915265	+	0.402852i	$0.131981\pi$
$641$			13075.6i			0.805704i	−0.915265	+	0.402852i	$0.868019\pi$
$642$		0			0
$643$		−	5272.61i		−	0.323377i	−0.986842	−	0.161689i	$-0.948306\pi$
$643$		−	5272.61i		−	0.323377i	0.986842	−	0.161689i	$-0.0516940\pi$
$644$		0			0
$645$	583.441	+	5412.23i	0.0356170	+	0.330398i
$646$		0			0
$647$	−32739.3			−1.98936			−0.994679	−	0.103027i	$-0.967147\pi$
$647$	−32739.3			−1.98936			−0.994679	+	0.103027i	$0.967147\pi$
$648$		0			0
$649$	−9552.07			−0.577737
$650$		0			0
$651$	−989.391	−	9177.99i	−0.0595657	−	0.552555i
$652$		0			0
$653$		−	14895.3i		−	0.892649i	−0.894871	−	0.446324i	$-0.852733\pi$
$653$		−	14895.3i		−	0.892649i	0.894871	−	0.446324i	$-0.147267\pi$
$654$		0			0
$655$		−	23159.5i		−	1.38155i
$656$		0			0
$657$	23095.1	−	5037.87i	1.37142	−	0.299157i
$658$		0			0
$659$	−13070.1			−0.772595			−0.386297	−	0.922374i	$-0.626246\pi$
$659$	−13070.1			−0.772595			−0.386297	+	0.922374i	$0.626246\pi$
$660$		0			0
$661$	−24322.7			−1.43123			−0.715616	−	0.698494i	$-0.753854\pi$
$661$	−24322.7			−1.43123			−0.715616	+	0.698494i	$0.753854\pi$
$662$		0			0
$663$	18673.4	−	2013.00i	1.09384	−	0.117916i
$664$		0			0
$665$		−	5572.74i		−	0.324965i
$666$		0			0
$667$		−	18629.3i		−	1.08146i
$668$		0			0
$669$	8511.23	−	917.515i	0.491873	−	0.0530242i
$670$		0			0
$671$	−9805.27			−0.564126
$672$		0			0
$673$	−1854.52			−0.106221			−0.0531104	−	0.998589i	$-0.516914\pi$
$673$	−1854.52			−0.106221			−0.0531104	+	0.998589i	$0.516914\pi$
$674$		0			0
$675$	554.217	+	1660.40i	0.0316027	+	0.0946797i
$676$		0			0
$677$		−	6481.24i		−	0.367938i	−0.982932	−	0.183969i	$-0.941105\pi$
$677$		−	6481.24i		−	0.367938i	0.982932	−	0.183969i	$-0.0588946\pi$
$678$		0			0
$679$		−	5967.74i		−	0.337291i
$680$		0			0
$681$	189.501	+	1757.88i	0.0106633	+	0.0989166i
$682$		0			0
$683$	7618.89			0.426836			0.213418	−	0.976961i	$-0.431540\pi$
$683$	7618.89			0.426836			0.213418	+	0.976961i	$0.431540\pi$
$684$		0			0
$685$	7972.83			0.444710
$686$		0			0
$687$	−2245.47	−	20829.9i	−0.124702	−	1.15678i
$688$		0			0
$689$		−	16198.3i		−	0.895657i
$690$		0			0
$691$		−	8461.55i		−	0.465836i	−0.972496	−	0.232918i	$-0.925173\pi$
$691$		−	8461.55i		−	0.465836i	0.972496	−	0.232918i	$-0.0748273\pi$
$692$		0			0
$693$	−535.097	−	2453.05i	−0.0293314	−	0.134464i
$694$		0			0
$695$	18511.9			1.01035
$696$		0			0
$697$	−29198.4			−1.58676
$698$		0			0
$699$	7564.82	−	815.492i	0.409339	−	0.0441269i
$700$		0			0
$701$			19113.8i			1.02984i	0.857238	+	0.514921i	$0.172179\pi$
$701$			19113.8i			1.02984i	−0.857238	+	0.514921i	$0.827821\pi$
$702$		0			0
$703$			4693.30i			0.251794i
$704$		0			0
$705$	10389.2	−	1119.96i	0.555005	−	0.0598298i
$706$		0			0
$707$	9164.25			0.487493
$708$		0			0
$709$	7854.59			0.416058			0.208029	−	0.978123i	$-0.433295\pi$
$709$	7854.59			0.416058			0.208029	+	0.978123i	$0.433295\pi$
$710$		0			0
$711$	3779.61	+	17326.9i	0.199362	+	0.913935i
$712$		0			0
$713$			37387.3i			1.96377i
$714$		0			0
$715$		−	3818.85i		−	0.199744i
$716$		0			0
$717$	−1870.00	−	17346.9i	−0.0974010	−	0.903530i
$718$		0			0
$719$	27011.1			1.40103			0.700517	−	0.713635i	$-0.252953\pi$
$719$	27011.1			1.40103			0.700517	+	0.713635i	$0.252953\pi$
$720$		0			0
$721$	11276.8			0.582482
$722$		0			0
$723$	1524.26	+	14139.6i	0.0784062	+	0.727327i
$724$		0			0
$725$		−	1397.39i		−	0.0715830i
$726$		0			0
$727$			22178.9i			1.13146i	0.824591	+	0.565729i	$0.191405\pi$
$727$			22178.9i			1.13146i	−0.824591	+	0.565729i	$0.808595\pi$
$728$		0			0
$729$	−15736.8	+	11822.6i	−0.799512	+	0.600651i
$730$		0			0
$731$	−11666.0			−0.590264
$732$		0			0
$733$	9103.17			0.458708			0.229354	−	0.973343i	$-0.426339\pi$
$733$	9103.17			0.458708			0.229354	+	0.973343i	$0.426339\pi$
$734$		0			0
$735$	15373.5	−	1657.27i	0.771512	−	0.0831693i
$736$		0			0
$737$		−	33.3481i		−	0.00166675i
$738$		0			0
$739$			19226.3i			0.957036i	0.878078	+	0.478518i	$0.158826\pi$
$739$			19226.3i			0.957036i	−0.878078	+	0.478518i	$0.841174\pi$
$740$		0			0
$741$	−10507.4	+	1132.70i	−0.520917	+	0.0561551i
$742$		0			0
$743$	22501.9			1.11106			0.555528	−	0.831498i	$-0.312516\pi$
$743$	22501.9			1.11106			0.555528	+	0.831498i	$0.312516\pi$
$744$		0			0
$745$	−19723.1			−0.969930
$746$		0			0
$747$	6253.52	−	1364.12i	0.306298	−	0.0668145i
$748$		0			0
$749$			11882.1i			0.579655i
$750$		0			0
$751$			2080.69i			0.101099i	0.998722	+	0.0505495i	$0.0160973\pi$
$751$			2080.69i			0.101099i	−0.998722	+	0.0505495i	$0.983903\pi$
$752$		0			0
$753$	3319.10	+	30789.3i	0.160631	+	1.49007i
$754$		0			0
$755$	10264.6			0.494793
$756$		0			0
$757$	−26222.3			−1.25900			−0.629501	−	0.777000i	$-0.716741\pi$
$757$	−26222.3			−1.25900			−0.629501	+	0.777000i	$0.716741\pi$
$758$		0			0
$759$	1089.88	+	10110.2i	0.0521215	+	0.483499i
$760$		0			0
$761$			11443.7i			0.545118i	0.962139	+	0.272559i	$0.0878700\pi$
$761$			11443.7i			0.545118i	−0.962139	+	0.272559i	$0.912130\pi$
$762$		0			0
$763$			3525.18i			0.167261i
$764$		0			0
$765$	33054.5	−	7210.36i	1.56220	−	0.340773i
$766$		0			0
$767$	24843.2			1.16954
$768$		0			0
$769$	18991.7			0.890582			0.445291	−	0.895386i	$-0.353100\pi$
$769$	18991.7			0.890582			0.445291	+	0.895386i	$0.353100\pi$
$770$		0			0
$771$	−7151.37	+	770.921i	−0.334047	+	0.0360104i
$772$		0			0
$773$		−	36956.6i		−	1.71958i	−0.510645	−	0.859791i	$-0.670593\pi$
$773$		−	36956.6i		−	1.71958i	0.510645	−	0.859791i	$-0.329407\pi$
$774$		0			0
$775$			2804.42i			0.129984i
$776$		0			0
$777$	2883.20	−	310.810i	0.133120	−	0.0143504i
$778$		0			0
$779$	16429.8			0.755659
$780$		0			0
$781$	−9372.51			−0.429417
$782$		0			0
$783$	14904.6	−	4974.94i	0.680265	−	0.227062i
$784$		0			0
$785$			38602.2i			1.75512i
$786$		0			0
$787$			19962.2i			0.904162i	0.891977	+	0.452081i	$0.149318\pi$
$787$			19962.2i			0.904162i	−0.891977	+	0.452081i	$0.850682\pi$
$788$		0			0
$789$	−3578.81	−	33198.4i	−0.161481	−	1.49797i
$790$		0			0
$791$	11173.0			0.502232
$792$		0			0
$793$	25501.7			1.14198
$794$		0			0
$795$	−3127.33	−	29010.4i	−0.139516	−	1.29420i
$796$		0			0
$797$			6131.27i			0.272498i	0.990675	+	0.136249i	$0.0435047\pi$
$797$			6131.27i			0.272498i	−0.990675	+	0.136249i	$0.956495\pi$
$798$		0			0
$799$			22393.7i			0.991530i
$800$		0			0
$801$	−5000.28	−	22922.8i	−0.220570	−	1.01116i
$802$		0			0
$803$	10300.3			0.452666
$804$		0			0
$805$	13945.7			0.610586
$806$		0			0
$807$	−8236.05	+	887.850i	−0.359260	+	0.0387284i
$808$		0			0
$809$			18915.6i			0.822050i	0.911624	+	0.411025i	$0.134829\pi$
$809$			18915.6i			0.822050i	−0.911624	+	0.411025i	$0.865171\pi$
$810$		0			0
$811$			8699.98i			0.376692i	0.982103	+	0.188346i	$0.0603127\pi$
$811$			8699.98i			0.376692i	−0.982103	+	0.188346i	$0.939687\pi$
$812$		0			0
$813$	−38759.2	+	4178.26i	−1.67201	+	0.180243i
$814$		0			0
$815$	−3578.60			−0.153807
$816$		0			0
$817$	6564.40			0.281101
$818$		0			0
$819$	1391.69	+	6379.92i	0.0593768	+	0.272201i
$820$		0			0
$821$			21629.4i			0.919453i	0.888061	+	0.459726i	$0.152052\pi$
$821$			21629.4i			0.919453i	−0.888061	+	0.459726i	$0.847948\pi$
$822$		0			0
$823$		−	17966.4i		−	0.760960i	−0.924789	−	0.380480i	$-0.875759\pi$
$823$		−	17966.4i		−	0.760960i	0.924789	−	0.380480i	$-0.124241\pi$
$824$		0			0
$825$	81.7521	+	758.365i	0.00344999	+	0.0320035i
$826$		0			0
$827$	−19001.1			−0.798951			−0.399475	−	0.916744i	$-0.630808\pi$
$827$	−19001.1			−0.798951			−0.399475	+	0.916744i	$0.630808\pi$
$828$		0			0
$829$	25624.2			1.07354			0.536770	−	0.843728i	$-0.319644\pi$
$829$	25624.2			1.07354			0.536770	+	0.843728i	$0.319644\pi$
$830$		0			0
$831$	−612.083	−	5677.93i	−0.0255511	−	0.237022i
$832$		0			0
$833$			33137.4i			1.37832i
$834$		0			0
$835$		−	28501.2i		−	1.18123i
$836$		0			0
$837$	−29912.1	+	9984.23i	−1.23526	+	0.412312i
$838$		0			0
$839$	39037.6			1.60635			0.803176	−	0.595742i	$-0.203142\pi$
$839$	39037.6			1.60635			0.803176	+	0.595742i	$0.203142\pi$
$840$		0			0
$841$	11845.3			0.485683
$842$		0			0
$843$	35380.7	−	3814.05i	1.44552	−	0.155828i
$844$		0			0
$845$		−	13373.0i		−	0.544431i
$846$		0			0
$847$			9425.89i			0.382382i
$848$		0			0
$849$	−22495.2	+	2425.00i	−0.909346	+	0.0980279i
$850$		0			0
$851$	−11744.9			−0.473104
$852$		0			0
$853$	−10697.0			−0.429377			−0.214688	−	0.976683i	$-0.568874\pi$
$853$	−10697.0			−0.429377			−0.214688	+	0.976683i	$0.568874\pi$
$854$		0			0
$855$	−18599.6	+	4057.23i	−0.743967	+	0.162286i
$856$		0			0
$857$			3546.91i			0.141377i	0.997498	+	0.0706886i	$0.0225197\pi$
$857$			3546.91i			0.141377i	−0.997498	+	0.0706886i	$0.977480\pi$
$858$		0			0
$859$			37273.1i			1.48049i	0.672336	+	0.740246i	$0.265291\pi$
$859$			37273.1i			1.48049i	−0.672336	+	0.740246i	$0.734709\pi$
$860$		0			0
$861$	−1088.05	−	10093.2i	−0.0430669	−	0.399506i
$862$		0			0
$863$	12710.9			0.501372			0.250686	−	0.968068i	$-0.419344\pi$
$863$	12710.9			0.501372			0.250686	+	0.968068i	$0.419344\pi$
$864$		0			0
$865$	−20897.5			−0.821429
$866$		0			0
$867$	5034.77	+	46704.5i	0.197220	+	1.82949i
$868$		0			0
$869$			7727.72i			0.301663i
$870$		0			0
$871$			86.7324i			0.00337407i
$872$		0			0
$873$	−19917.9	+	4344.81i	−0.772187	+	0.168442i
$874$		0			0
$875$	11526.2			0.445322
$876$		0			0
$877$	−13273.5			−0.511078			−0.255539	−	0.966799i	$-0.582253\pi$
$877$	−13273.5			−0.511078			−0.255539	+	0.966799i	$0.582253\pi$
$878$		0			0
$879$	−14698.4	+	1584.49i	−0.564010	+	0.0608005i
$880$		0			0
$881$		−	24760.3i		−	0.946875i	−0.880828	−	0.473437i	$-0.843013\pi$
$881$		−	24760.3i		−	0.946875i	0.880828	−	0.473437i	$-0.156987\pi$
$882$		0			0
$883$		−	46347.6i		−	1.76639i	−0.469009	−	0.883193i	$-0.655389\pi$
$883$		−	46347.6i		−	1.76639i	0.469009	−	0.883193i	$-0.344611\pi$
$884$		0			0
$885$	44492.9	−	4796.35i	1.68996	−	0.182178i
$886$		0			0
$887$	−12454.7			−0.471461			−0.235731	−	0.971818i	$-0.575748\pi$
$887$	−12454.7			−0.471461			−0.235731	+	0.971818i	$0.575748\pi$
$888$		0			0
$889$	−10396.0			−0.392205
$890$		0			0
$891$	−7797.71	+	3571.88i	−0.293191	+	0.134301i
$892$		0			0
$893$		−	12600.8i		−	0.472195i
$894$		0			0
$895$		−	8873.78i		−	0.331416i
$896$		0			0
$897$	−2834.58	−	26294.7i	−0.105512	−	0.978768i
$898$		0			0
$899$	25174.0			0.933925
$900$		0			0
$901$	62531.5			2.31213
$902$		0			0
$903$	−434.722	−	4032.65i	−0.0160206	−	0.148614i
$904$		0			0
$905$		−	18207.0i		−	0.668751i
$906$		0			0
$907$		−	31110.3i		−	1.13892i	−0.822020	−	0.569459i	$-0.807153\pi$
$907$		−	31110.3i		−	1.13892i	0.822020	−	0.569459i	$-0.192847\pi$
$908$		0			0
$909$	−6672.04	−	30586.6i	−0.243452	−	1.11605i
$910$		0			0
$911$	−2882.81			−0.104843			−0.0524214	−	0.998625i	$-0.516694\pi$
$911$	−2882.81			−0.104843			−0.0524214	+	0.998625i	$0.516694\pi$
$912$		0			0
$913$	2789.05			0.101100
$914$		0			0
$915$	45672.3	−	4923.49i	1.65014	−	0.177886i
$916$		0			0
$917$			17256.1i			0.621426i
$918$		0			0
$919$		−	6249.30i		−	0.224315i	−0.993690	−	0.112157i	$-0.964224\pi$
$919$		−	6249.30i		−	0.224315i	0.993690	−	0.112157i	$-0.0357761\pi$
$920$		0			0
$921$	40291.1	−	4343.40i	1.44152	−	0.155396i
$922$		0			0
$923$	24376.2			0.869287
$924$		0			0
$925$	−880.989			−0.0313154
$926$		0			0
$927$	−8210.07	−	37637.4i	−0.290889	−	1.33352i
$928$		0			0
$929$			32605.2i			1.15150i	0.817627	+	0.575749i	$0.195289\pi$
$929$			32605.2i			1.15150i	−0.817627	+	0.575749i	$0.804711\pi$
$930$		0			0
$931$		−	18646.3i		−	0.656398i
$932$		0			0
$933$	−2672.91	−	24794.9i	−0.0937910	−	0.870043i
$934$		0			0
$935$	14742.2			0.515637
$936$		0			0
$937$	15176.1			0.529116			0.264558	−	0.964370i	$-0.414774\pi$
$937$	15176.1			0.529116			0.264558	+	0.964370i	$0.414774\pi$
$938$		0			0
$939$	1336.97	+	12402.3i	0.0464649	+	0.431027i
$940$		0			0
$941$		−	37892.6i		−	1.31271i	−0.754450	−	0.656357i	$-0.772096\pi$
$941$		−	37892.6i		−	1.31271i	0.754450	−	0.656357i	$-0.227904\pi$
$942$		0			0
$943$			41115.4i			1.41983i
$944$		0			0
$945$	3724.19	+	11157.4i	0.128199	+	0.384075i
$946$		0			0
$947$	−54773.3			−1.87951			−0.939754	−	0.341853i	$-0.888946\pi$
$947$	−54773.3			−1.87951			−0.939754	+	0.341853i	$0.888946\pi$
$948$		0			0
$949$	−26789.3			−0.916351
$950$		0			0
$951$	−28930.2	+	3118.69i	−0.986464	+	0.106341i
$952$		0			0
$953$			5158.12i			0.175328i	0.996150	+	0.0876642i	$0.0279403\pi$
$953$			5158.12i			0.175328i	−0.996150	+	0.0876642i	$0.972060\pi$
$954$		0			0
$955$			3806.93i			0.128994i
$956$		0			0
$957$	6807.48	−	733.850i	0.229942	−	0.0247879i
$958$		0			0
$959$	−5940.55			−0.200032
$960$		0			0
$961$	−20730.7			−0.695872
$962$		0			0
$963$	39657.6	−	8650.76i	1.32705	−	0.289477i
$964$		0			0
$965$			28328.4i			0.945000i
$966$		0			0
$967$		−	46108.3i		−	1.53334i	−0.642039	−	0.766672i	$-0.721911\pi$
$967$		−	46108.3i		−	1.53334i	0.642039	−	0.766672i	$-0.278089\pi$
$968$		0			0
$969$	−4372.66	−	40562.5i	−0.144964	−	1.34474i
$970$		0			0
$971$	−5437.59			−0.179712			−0.0898561	−	0.995955i	$-0.528641\pi$
$971$	−5437.59			−0.179712			−0.0898561	+	0.995955i	$0.528641\pi$
$972$		0			0
$973$	−13793.2			−0.454460
$974$		0			0
$975$	−212.622	−	1972.37i	−0.00698396	−	0.0647860i
$976$		0			0
$977$		−	21599.2i		−	0.707288i	−0.935380	−	0.353644i	$-0.884942\pi$
$977$		−	21599.2i		−	0.707288i	0.935380	−	0.353644i	$-0.115058\pi$
$978$		0			0
$979$		−	10223.5i		−	0.333753i
$980$		0			0
$981$	11765.6	−	2566.51i	0.382924	−	0.0835294i
$982$		0			0
$983$	37178.0			1.20630			0.603151	−	0.797627i	$-0.293912\pi$
$983$	37178.0			1.20630			0.603151	+	0.797627i	$0.293912\pi$
$984$		0			0
$985$	−6337.14			−0.204993
$986$		0			0
$987$	−7740.96	+	834.479i	−0.249643	+	0.0269116i
$988$		0			0
$989$			16427.3i			0.528169i
$990$		0			0
$991$			8124.54i			0.260428i	0.991486	+	0.130214i	$0.0415665\pi$
$991$			8124.54i			0.260428i	−0.991486	+	0.130214i	$0.958434\pi$
$992$		0			0
$993$	−28767.0	+	3101.09i	−0.919328	+	0.0991040i
$994$		0			0
$995$	−35391.9			−1.12764
$996$		0			0
$997$	32053.1			1.01819			0.509093	−	0.860711i	$-0.329981\pi$
$997$	32053.1			1.01819			0.509093	+	0.860711i	$0.329981\pi$
$998$		0			0
$999$	−3136.47	−	9396.67i	−0.0993328	−	0.297595i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.c.a.383.7	✓	12
3.2	odd	2		384.4.c.d.383.5	yes	12
4.3	odd	2		384.4.c.d.383.6	yes	12
8.3	odd	2		384.4.c.b.383.7	yes	12
8.5	even	2		384.4.c.c.383.6	yes	12
12.11	even	2	inner	384.4.c.a.383.8	yes	12
16.3	odd	4		768.4.f.g.383.2		12
16.5	even	4		768.4.f.h.383.2		12
16.11	odd	4		768.4.f.f.383.11		12
16.13	even	4		768.4.f.e.383.11		12
24.5	odd	2		384.4.c.b.383.8	yes	12
24.11	even	2		384.4.c.c.383.5	yes	12
48.5	odd	4		768.4.f.g.383.1		12
48.11	even	4		768.4.f.e.383.12		12
48.29	odd	4		768.4.f.f.383.12		12
48.35	even	4		768.4.f.h.383.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.c.a.383.7	✓	12	1.1	even	1	trivial
384.4.c.a.383.8	yes	12	12.11	even	2	inner
384.4.c.b.383.7	yes	12	8.3	odd	2
384.4.c.b.383.8	yes	12	24.5	odd	2
384.4.c.c.383.5	yes	12	24.11	even	2
384.4.c.c.383.6	yes	12	8.5	even	2
384.4.c.d.383.5	yes	12	3.2	odd	2
384.4.c.d.383.6	yes	12	4.3	odd	2
768.4.f.e.383.11		12	16.13	even	4
768.4.f.e.383.12		12	48.11	even	4
768.4.f.f.383.11		12	16.11	odd	4
768.4.f.f.383.12		12	48.29	odd	4
768.4.f.g.383.1		12	48.5	odd	4
768.4.f.g.383.2		12	16.3	odd	4
768.4.f.h.383.1		12	48.35	even	4
768.4.f.h.383.2		12	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.556921 - 5.16622i) q^{3} +10.6077i q^{5} -7.90379i q^{7} +(-26.3797 + 5.75435i) q^{9} +O(q^{10})\)	\(q+(-0.556921 - 5.16622i) q^{3} +10.6077i q^{5} -7.90379i q^{7} +(-26.3797 + 5.75435i) q^{9} -11.7652 q^{11} +30.5992 q^{13} +(54.8017 - 5.90765i) q^{15} +118.124i q^{17} -66.4680i q^{19} +(-40.8327 + 4.40179i) q^{21} +166.335 q^{23} +12.4768 q^{25} +(44.4197 + 133.079i) q^{27} -111.999i q^{29} +224.770i q^{31} +(6.55231 + 60.7818i) q^{33} +83.8410 q^{35} -70.6099 q^{37} +(-17.0414 - 158.082i) q^{39} +247.184i q^{41} +98.7603i q^{43} +(-61.0404 - 279.828i) q^{45} +189.577 q^{47} +280.530 q^{49} +(610.257 - 65.7860i) q^{51} -529.370i q^{53} -124.802i q^{55} +(-343.388 + 37.0174i) q^{57} +811.889 q^{59} +833.410 q^{61} +(45.4812 + 208.499i) q^{63} +324.587i q^{65} +2.83446i q^{67} +(-92.6357 - 859.325i) q^{69} +796.627 q^{71} -875.489 q^{73} +(-6.94861 - 64.4581i) q^{75} +92.9900i q^{77} -656.826i q^{79} +(662.775 - 303.596i) q^{81} -237.058 q^{83} -1253.03 q^{85} +(-578.610 + 62.3744i) q^{87} +868.956i q^{89} -241.850i q^{91} +(1161.21 - 125.179i) q^{93} +705.072 q^{95} +755.048 q^{97} +(310.363 - 67.7014i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{3}+O(q^{10}) \) 12 * q - 10 * q^3	\( 12 q - 10 q^{3} - 36 q^{11} - 84 q^{15} + 136 q^{21} + 120 q^{23} - 300 q^{25} + 266 q^{27} - 116 q^{33} - 432 q^{35} + 528 q^{37} - 620 q^{39} + 440 q^{45} + 1248 q^{47} - 948 q^{49} + 1072 q^{51} - 172 q^{57} - 2508 q^{59} + 624 q^{61} - 2744 q^{63} - 24 q^{69} + 2040 q^{71} - 216 q^{73} + 3894 q^{75} - 1076 q^{81} - 4572 q^{83} + 480 q^{85} - 4156 q^{87} - 112 q^{93} + 5448 q^{95} - 48 q^{97} + 6044 q^{99}+O(q^{100}) \) 12 * q - 10 * q^3 - 36 * q^11 - 84 * q^15 + 136 * q^21 + 120 * q^23 - 300 * q^25 + 266 * q^27 - 116 * q^33 - 432 * q^35 + 528 * q^37 - 620 * q^39 + 440 * q^45 + 1248 * q^47 - 948 * q^49 + 1072 * q^51 - 172 * q^57 - 2508 * q^59 + 624 * q^61 - 2744 * q^63 - 24 * q^69 + 2040 * q^71 - 216 * q^73 + 3894 * q^75 - 1076 * q^81 - 4572 * q^83 + 480 * q^85 - 4156 * q^87 - 112 * q^93 + 5448 * q^95 - 48 * q^97 + 6044 * q^99

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