LMFDB - Embedded newform 384.4.c.c.383.5

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			383.5
Root			$2.36157i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.4.c.c.383.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.556921 - 5.16622i) q^{3} +10.6077i q^{5} +7.90379i q^{7} +(-26.3797 - 5.75435i) q^{9} +O(q^{10})$	\(q+(0.556921 - 5.16622i) q^{3} +10.6077i q^{5} +7.90379i q^{7} +(-26.3797 - 5.75435i) q^{9} +11.7652 q^{11} -30.5992 q^{13} +(54.8017 + 5.90765i) q^{15} -118.124i q^{17} -66.4680i q^{19} +(40.8327 + 4.40179i) q^{21} +166.335 q^{23} +12.4768 q^{25} +(-44.4197 + 133.079i) q^{27} -111.999i q^{29} -224.770i q^{31} +(6.55231 - 60.7818i) q^{33} -83.8410 q^{35} +70.6099 q^{37} +(-17.0414 + 158.082i) q^{39} -247.184i q^{41} +98.7603i q^{43} +(61.0404 - 279.828i) q^{45} +189.577 q^{47} +280.530 q^{49} +(-610.257 - 65.7860i) q^{51} -529.370i q^{53} +124.802i q^{55} +(-343.388 - 37.0174i) q^{57} -811.889 q^{59} -833.410 q^{61} +(45.4812 - 208.499i) q^{63} -324.587i q^{65} +2.83446i q^{67} +(92.6357 - 859.325i) q^{69} +796.627 q^{71} -875.489 q^{73} +(6.94861 - 64.4581i) q^{75} +92.9900i q^{77} +656.826i q^{79} +(662.775 + 303.596i) q^{81} +237.058 q^{83} +1253.03 q^{85} +(-578.610 - 62.3744i) q^{87} -868.956i q^{89} -241.850i q^{91} +(-1161.21 - 125.179i) q^{93} +705.072 q^{95} +755.048 q^{97} +(-310.363 - 67.7014i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.0684480\pi$
$7$			7.90379i			0.426764i	−0.976969	+	0.213382i	$0.931552\pi$
$8$		0			0
$9$	−26.3797	−	5.75435i	−0.977025	−	0.213124i
$10$		0			0
$11$	11.7652			0.322487			0.161243	−	0.986915i	$-0.448450\pi$
$11$	11.7652			0.322487			0.161243	+	0.986915i	$0.448450\pi$
$12$		0			0
$13$	−30.5992			−0.652823			−0.326412	−	0.945228i	$-0.605840\pi$
$13$	−30.5992			−0.652823			−0.326412	+	0.945228i	$0.605840\pi$
$14$		0			0
$15$	54.8017	+	5.90765i	0.943316	+	0.101690i
$16$		0			0
$17$		−	118.124i		−	1.68526i	−0.538495	−	0.842629i	$-0.681007\pi$
$17$		−	118.124i		−	1.68526i	0.538495	−	0.842629i	$-0.318993\pi$
$18$		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	$-0.774296\pi$
$31$		−	224.770i		−	1.30226i	0.758968	−	0.651128i	$-0.225704\pi$
$32$		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	$-0.449860\pi$
$37$	70.6099			0.313735			0.156868	+	0.987620i	$0.449860\pi$
$38$		0			0
$39$	−17.0414	+	158.082i	−0.0699693	+	0.649063i
$40$		0			0
$41$		−	247.184i		−	0.941551i	−0.882253	−	0.470775i	$-0.843974\pi$
$41$		−	247.184i		−	0.941551i	0.882253	−	0.470775i	$-0.156026\pi$
$42$		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	$-0.853363\pi$
$59$	−811.889			−1.79151			−0.895753	+	0.444551i	$0.853363\pi$
$60$		0			0
$61$	−833.410			−1.74930			−0.874650	−	0.484756i	$-0.838909\pi$
$61$	−833.410			−1.74930			−0.874650	+	0.484756i	$0.838909\pi$
$62$		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	$0.154922\pi$
$79$			656.826i			0.935427i	−0.883880	+	0.467713i	$0.845078\pi$
$80$		0			0
$81$	662.775	+	303.596i	0.909156	+	0.416455i
$82$		0			0
$83$	237.058			0.313500			0.156750	−	0.987638i	$-0.449898\pi$
$83$	237.058			0.313500			0.156750	+	0.987638i	$0.449898\pi$
$84$		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	$-0.826875\pi$
$89$		−	868.956i		−	1.03493i	0.855703	−	0.517467i	$-0.173125\pi$
$90$		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	$-0.760919\pi$
$103$		−	1426.76i		−	1.36488i	0.730942	−	0.682440i	$-0.239081\pi$
$104$		0			0
$105$	−46.6928	+	433.141i	−0.0433976	+	0.402574i
$106$		0			0
$107$	1503.34			1.35826			0.679128	−	0.734020i	$-0.262358\pi$
$107$	1503.34			1.35826			0.679128	+	0.734020i	$0.262358\pi$
$108$		0			0
$109$	446.011			0.391928			0.195964	−	0.980611i	$-0.437216\pi$
$109$	446.011			0.391928			0.195964	+	0.980611i	$0.437216\pi$
$110$		0			0
$111$	39.3242	−	364.787i	0.0336260	−	0.311928i
$112$		0			0
$113$		−	1413.62i		−	1.17684i	−0.808557	−	0.588418i	$-0.799751\pi$
$113$		−	1413.62i		−	1.17684i	0.808557	−	0.588418i	$-0.200249\pi$
$114$		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	$0.151975\pi$
$127$			1315.32i			0.919020i	−0.888173	+	0.459510i	$0.848025\pi$
$128$		0			0
$129$	510.218	+	55.0017i	0.348234	+	0.0375398i
$130$		0			0
$131$	2183.27			1.45613			0.728067	−	0.685506i	$-0.240419\pi$
$131$	2183.27			1.45613			0.728067	+	0.685506i	$0.240419\pi$
$132$		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	$0.0752990\pi$
$137$			751.608i			0.468717i	−0.972150	+	0.234359i	$0.924701\pi$
$138$		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	$0.0839704\pi$
$151$			967.660i			0.521504i	−0.965406	+	0.260752i	$0.916030\pi$
$152$		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	$-0.875882\pi$
$157$	−3639.08			−1.84987			−0.924936	+	0.380123i	$0.875882\pi$
$158$		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	$-0.444119\pi$
$179$	836.542			0.349308			0.174654	+	0.984630i	$0.444119\pi$
$180$		0			0
$181$	1716.39			0.704853			0.352426	−	0.935840i	$-0.385357\pi$
$181$	1716.39			0.704853			0.352426	+	0.935840i	$0.385357\pi$
$182$		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	$-0.797447\pi$
$199$		−	3336.44i		−	1.18851i	0.804276	−	0.594256i	$-0.202553\pi$
$200$		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	$-0.920436\pi$
$223$		−	1647.48i		−	0.494723i	0.968923	−	0.247362i	$-0.0795635\pi$
$224$		0			0
$225$	−329.135	−	71.7962i	−0.0975215	−	0.0212729i
$226$		0			0
$227$	340.265			0.0994897			0.0497449	−	0.998762i	$-0.484159\pi$
$227$	340.265			0.0994897			0.0497449	+	0.998762i	$0.484159\pi$
$228$		0			0
$229$	−4031.94			−1.16349			−0.581743	−	0.813373i	$-0.697629\pi$
$229$	−4031.94			−1.16349			−0.581743	+	0.813373i	$0.697629\pi$
$230$		0			0
$231$	480.407	+	51.7881i	0.136833	+	0.0147507i
$232$		0			0
$233$		−	1464.29i		−	0.411710i	−0.978582	−	0.205855i	$-0.934002\pi$
$233$		−	1464.29i		−	0.411710i	0.978582	−	0.205855i	$-0.0659976\pi$
$234$		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	$-0.230364\pi$
$251$	5959.74			1.49871			0.749354	+	0.662170i	$0.230364\pi$
$252$		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	$0.0537280\pi$
$257$			1384.26i			0.335982i	−0.985788	+	0.167991i	$0.946272\pi$
$258$		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	$0.317943\pi$
$271$			7502.42i			1.68170i	−0.541271	+	0.840848i	$0.682057\pi$
$272$		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	$-0.538032\pi$
$277$	−1099.05			−0.238395			−0.119198	+	0.992871i	$0.538032\pi$
$278$		0			0
$279$	−1293.41	+	5929.37i	−0.277542	+	1.27234i
$280$		0			0
$281$		−	6848.46i		−	1.45390i	−0.686692	−	0.726949i	$-0.740938\pi$
$281$		−	6848.46i		−	1.45390i	0.686692	−	0.726949i	$-0.259062\pi$
$282$		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	$-0.769657\pi$
$347$	−9688.05			−1.49879			−0.749397	+	0.662121i	$0.769657\pi$
$348$		0			0
$349$	11427.5			1.75272			0.876361	−	0.481654i	$-0.159964\pi$
$349$	11427.5			1.75272			0.876361	+	0.481654i	$0.159964\pi$
$350$		0			0
$351$	1359.21	−	4072.10i	0.206693	−	0.619239i
$352$		0			0
$353$			630.357i			0.0950440i	0.998870	+	0.0475220i	$0.0151324\pi$
$353$			630.357i			0.0950440i	−0.998870	+	0.0475220i	$0.984868\pi$
$354$		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	$0.312796\pi$
$367$			11698.9i			1.66397i	−0.554797	+	0.831986i	$0.687204\pi$
$368$		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	$-0.617473\pi$
$373$	−5197.32			−0.721466			−0.360733	+	0.932669i	$0.617473\pi$
$374$		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	$-0.554808\pi$
$397$	−2710.55			−0.342667			−0.171333	+	0.985213i	$0.554808\pi$
$398$		0			0
$399$	292.578	−	2714.07i	0.0367098	−	0.340535i
$400$		0			0
$401$			4661.17i			0.580468i	0.956956	+	0.290234i	$0.0937331\pi$
$401$			4661.17i			0.580468i	−0.956956	+	0.290234i	$0.906267\pi$
$402$		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	$-0.509800\pi$
$419$	−528.053			−0.0615682			−0.0307841	+	0.999526i	$0.509800\pi$
$420$		0			0
$421$	−8788.97			−1.01745			−0.508727	−	0.860928i	$-0.669884\pi$
$421$	−8788.97			−1.01745			−0.508727	+	0.860928i	$0.669884\pi$
$422$		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	$0.244667\pi$
$439$			12788.3i			1.39032i	−0.718854	+	0.695161i	$0.755333\pi$
$440$		0			0
$441$	−7400.29	−	1614.27i	−0.799082	−	0.174308i
$442$		0			0
$443$	−5382.41			−0.577259			−0.288630	−	0.957441i	$-0.593200\pi$
$443$	−5382.41			−0.577259			−0.288630	+	0.957441i	$0.593200\pi$
$444$		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	$0.165726\pi$
$449$			9465.38i			0.994875i	−0.867500	+	0.497437i	$0.834274\pi$
$450$		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	$-0.996445\pi$
$463$		−	222.541i		−	0.0223377i	0.999938	−	0.0111689i	$-0.00355523\pi$
$464$		0			0
$465$	1327.86	−	12317.8i	0.132426	−	1.22844i
$466$		0			0
$467$	−5784.61			−0.573190			−0.286595	−	0.958052i	$-0.592523\pi$
$467$	−5784.61			−0.573190			−0.286595	+	0.958052i	$0.592523\pi$
$468$		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	$-0.688030\pi$
$487$		−	17851.9i		−	1.66109i	0.556955	−	0.830543i	$-0.311970\pi$
$488$		0			0
$489$	1742.87	+	187.882i	0.161176	+	0.0173749i
$490$		0			0
$491$	12979.1			1.19295			0.596473	−	0.802633i	$-0.296568\pi$
$491$	12979.1			1.19295			0.596473	+	0.802633i	$0.296568\pi$
$492$		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	$0.194778\pi$
$521$			13662.4i			1.14887i	−0.818550	+	0.574435i	$0.805222\pi$
$522$		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	$-0.423817\pi$
$541$	5965.93			0.474114			0.237057	+	0.971496i	$0.423817\pi$
$542$		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	$-0.773016\pi$
$563$	−20207.5			−1.51269			−0.756344	+	0.654175i	$0.773016\pi$
$564$		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	$0.185718\pi$
$569$			14954.6i			1.10181i	−0.834567	+	0.550906i	$0.814282\pi$
$570$		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	$-0.408724\pi$
$587$	8044.96			0.565675			0.282837	+	0.959168i	$0.408724\pi$
$588$		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	$0.124464\pi$
$593$			11007.3i			0.762255i	−0.924522	+	0.381128i	$0.875536\pi$
$594$		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	$-0.976020\pi$
$607$		−	2251.13i		−	0.150528i	0.997164	−	0.0752641i	$-0.0239800\pi$
$608$		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	$-0.451167\pi$
$613$	4638.50			0.305624			0.152812	+	0.988255i	$0.451167\pi$
$614$		0			0
$615$	1460.27	−	13546.1i	0.0957462	−	0.888180i
$616$		0			0
$617$			16218.9i			1.05826i	0.848540	+	0.529132i	$0.177482\pi$
$617$			16218.9i			1.05826i	−0.848540	+	0.529132i	$0.822518\pi$
$618$		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	$0.148004\pi$
$631$			14214.6i			0.896791i	−0.893835	+	0.448395i	$0.851996\pi$
$632$		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	$-0.868019\pi$
$641$		−	13075.6i		−	0.805704i	0.915265	−	0.402852i	$-0.131981\pi$
$642$		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	$-0.373754\pi$
$659$	13070.1			0.772595			0.386297	+	0.922374i	$0.373754\pi$
$660$		0			0
$661$	24322.7			1.43123			0.715616	−	0.698494i	$-0.246146\pi$
$661$	24322.7			1.43123			0.715616	+	0.698494i	$0.246146\pi$
$662$		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	$-0.568460\pi$
$683$	−7618.89			−0.426836			−0.213418	+	0.976961i	$0.568460\pi$
$684$		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	$-0.566705\pi$
$709$	−7854.59			−0.416058			−0.208029	+	0.978123i	$0.566705\pi$
$710$		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	$-0.808595\pi$
$727$		−	22178.9i		−	1.13146i	0.824591	−	0.565729i	$-0.191405\pi$
$728$		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	$-0.573661\pi$
$733$	−9103.17			−0.458708			−0.229354	+	0.973343i	$0.573661\pi$
$734$		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	$-0.983903\pi$
$751$		−	2080.69i		−	0.101099i	0.998722	−	0.0505495i	$-0.0160973\pi$
$752$		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	$-0.283259\pi$
$757$	26222.3			1.25900			0.629501	+	0.777000i	$0.283259\pi$
$758$		0			0
$759$	1089.88	−	10110.2i	0.0521215	−	0.483499i
$760$		0			0
$761$		−	11443.7i		−	0.545118i	−0.962139	−	0.272559i	$-0.912130\pi$
$761$		−	11443.7i		−	0.545118i	0.962139	−	0.272559i	$-0.0878700\pi$
$762$		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	$-0.865171\pi$
$809$		−	18915.6i		−	0.822050i	0.911624	−	0.411025i	$-0.134829\pi$
$810$		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	$0.124241\pi$
$823$			17966.4i			0.760960i	−0.924789	+	0.380480i	$0.875759\pi$
$824$		0			0
$825$	81.7521	−	758.365i	0.00344999	−	0.0320035i
$826$		0			0
$827$	19001.1			0.798951			0.399475	−	0.916744i	$-0.369192\pi$
$827$	19001.1			0.798951			0.399475	+	0.916744i	$0.369192\pi$
$828$		0			0
$829$	−25624.2			−1.07354			−0.536770	−	0.843728i	$-0.680356\pi$
$829$	−25624.2			−1.07354			−0.536770	+	0.843728i	$0.680356\pi$
$830$		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	$-0.431126\pi$
$853$	10697.0			0.429377			0.214688	+	0.976683i	$0.431126\pi$
$854$		0			0
$855$	−18599.6	−	4057.23i	−0.743967	−	0.162286i
$856$		0			0
$857$		−	3546.91i		−	0.141377i	−0.997498	−	0.0706886i	$-0.977480\pi$
$857$		−	3546.91i		−	0.141377i	0.997498	−	0.0706886i	$-0.0225197\pi$
$858$		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	$-0.417747\pi$
$877$	13273.5			0.511078			0.255539	+	0.966799i	$0.417747\pi$
$878$		0			0
$879$	−14698.4	−	1584.49i	−0.564010	−	0.0608005i
$880$		0			0
$881$			24760.3i			0.946875i	0.880828	+	0.473437i	$0.156987\pi$
$881$			24760.3i			0.946875i	−0.880828	+	0.473437i	$0.843013\pi$
$882$		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	$0.0357761\pi$
$919$			6249.30i			0.224315i	−0.993690	+	0.112157i	$0.964224\pi$
$920$		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	$-0.804711\pi$
$929$		−	32605.2i		−	1.15150i	0.817627	−	0.575749i	$-0.195289\pi$
$930$		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	$-0.111054\pi$
$947$	54773.3			1.87951			0.939754	+	0.341853i	$0.111054\pi$
$948$		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	$-0.972060\pi$
$953$		−	5158.12i		−	0.175328i	0.996150	−	0.0876642i	$-0.0279403\pi$
$954$		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	$0.278089\pi$
$967$			46108.3i			1.53334i	−0.642039	+	0.766672i	$0.721911\pi$
$968$		0			0
$969$	−4372.66	+	40562.5i	−0.144964	+	1.34474i
$970$		0			0
$971$	5437.59			0.179712			0.0898561	−	0.995955i	$-0.471359\pi$
$971$	5437.59			0.179712			0.0898561	+	0.995955i	$0.471359\pi$
$972$		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	$0.115058\pi$
$977$			21599.2i			0.707288i	−0.935380	+	0.353644i	$0.884942\pi$
$978$		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	$-0.958434\pi$
$991$		−	8124.54i		−	0.260428i	0.991486	−	0.130214i	$-0.0415665\pi$
$992$		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	$-0.670019\pi$
$997$	−32053.1			−1.01819			−0.509093	+	0.860711i	$0.670019\pi$
$998$		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.c.383.5	yes	12
3.2	odd	2		384.4.c.b.383.7	yes	12
4.3	odd	2		384.4.c.b.383.8	yes	12
8.3	odd	2		384.4.c.d.383.5	yes	12
8.5	even	2		384.4.c.a.383.8	yes	12
12.11	even	2	inner	384.4.c.c.383.6	yes	12
16.3	odd	4		768.4.f.g.383.1		12
16.5	even	4		768.4.f.h.383.1		12
16.11	odd	4		768.4.f.f.383.12		12
16.13	even	4		768.4.f.e.383.12		12
24.5	odd	2		384.4.c.d.383.6	yes	12
24.11	even	2		384.4.c.a.383.7	✓	12
48.5	odd	4		768.4.f.g.383.2		12
48.11	even	4		768.4.f.e.383.11		12
48.29	odd	4		768.4.f.f.383.11		12
48.35	even	4		768.4.f.h.383.2		12

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