LMFDB - Embedded newform 768.4.f.h.383.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("768.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$12$
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^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.7
Root			$-0.910871i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.4.f.h.383.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.98169 - 4.80343i) q^{3} -11.9846 q^{5} +22.6995i q^{7} +(-19.1458 - 19.0378i) q^{9} +O(q^{10})$	\(q+(1.98169 - 4.80343i) q^{3} -11.9846 q^{5} +22.6995i q^{7} +(-19.1458 - 19.0378i) q^{9} -61.9410i q^{11} +71.3317i q^{13} +(-23.7497 + 57.5672i) q^{15} +74.3305i q^{17} +108.076 q^{19} +(109.036 + 44.9834i) q^{21} +24.7059 q^{23} +18.6309 q^{25} +(-129.388 + 54.2388i) q^{27} +84.1331 q^{29} -130.377i q^{31} +(-297.529 - 122.748i) q^{33} -272.045i q^{35} -397.468i q^{37} +(342.637 + 141.357i) q^{39} -104.132i q^{41} +161.650 q^{43} +(229.455 + 228.160i) q^{45} +72.8162 q^{47} -172.269 q^{49} +(357.041 + 147.300i) q^{51} -136.267 q^{53} +742.339i q^{55} +(214.172 - 519.134i) q^{57} +243.465i q^{59} -358.454i q^{61} +(432.149 - 434.602i) q^{63} -854.883i q^{65} +449.508 q^{67} +(48.9593 - 118.673i) q^{69} -329.053 q^{71} +925.687 q^{73} +(36.9207 - 89.4923i) q^{75} +1406.03 q^{77} -55.9579i q^{79} +(4.12647 + 728.988i) q^{81} -928.194i q^{83} -890.823i q^{85} +(166.725 - 404.127i) q^{87} -853.930i q^{89} -1619.20 q^{91} +(-626.256 - 258.366i) q^{93} -1295.25 q^{95} -714.982 q^{97} +(-1179.22 + 1185.91i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{3} + 12 q^{5}+O(q^{10}) $ 12 * q + 2 * q^3 + 12 * q^5	$ 12 q + 2 q^{3} + 12 q^{5} - 84 q^{15} + 180 q^{19} + 156 q^{21} - 120 q^{23} + 300 q^{25} - 130 q^{27} + 588 q^{29} - 116 q^{33} + 620 q^{39} - 372 q^{43} - 740 q^{45} + 1248 q^{47} - 948 q^{49} - 360 q^{51} - 948 q^{53} + 172 q^{57} - 2744 q^{63} + 2292 q^{67} + 3280 q^{69} - 2040 q^{71} + 216 q^{73} - 2522 q^{75} + 4824 q^{77} - 1076 q^{81} + 4156 q^{87} + 3480 q^{91} - 4180 q^{93} + 5448 q^{95} - 48 q^{97} - 3048 q^{99}+O(q^{100}) $ 12 * q + 2 * q^3 + 12 * q^5 - 84 * q^15 + 180 * q^19 + 156 * q^21 - 120 * q^23 + 300 * q^25 - 130 * q^27 + 588 * q^29 - 116 * q^33 + 620 * q^39 - 372 * q^43 - 740 * q^45 + 1248 * q^47 - 948 * q^49 - 360 * q^51 - 948 * q^53 + 172 * q^57 - 2744 * q^63 + 2292 * q^67 + 3280 * q^69 - 2040 * q^71 + 216 * q^73 - 2522 * q^75 + 4824 * q^77 - 1076 * q^81 + 4156 * q^87 + 3480 * q^91 - 4180 * q^93 + 5448 * q^95 - 48 * q^97 - 3048 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.98169	−	4.80343i	0.381376	−	0.924420i
$3$	1.98169	−	4.80343i	0.381376	−	0.924420i
$4$		0			0
$5$	−11.9846			−1.07194			−0.535968	−	0.844238i	$-0.680053\pi$
$5$	−11.9846			−1.07194			−0.535968	+	0.844238i	$0.680053\pi$
$6$		0			0
$7$			22.6995i			1.22566i	0.790215	+	0.612830i	$0.209969\pi$
$7$			22.6995i			1.22566i	−0.790215	+	0.612830i	$0.790031\pi$
$8$		0			0
$9$	−19.1458	−	19.0378i	−0.709105	−	0.705103i
$10$		0			0
$11$		−	61.9410i		−	1.69781i	−0.528545	−	0.848905i	$-0.677262\pi$
$11$		−	61.9410i		−	1.69781i	0.528545	−	0.848905i	$-0.322738\pi$
$12$		0			0
$13$			71.3317i			1.52184i	0.648848	+	0.760918i	$0.275251\pi$
$13$			71.3317i			1.52184i	−0.648848	+	0.760918i	$0.724749\pi$
$14$		0			0
$15$	−23.7497	+	57.5672i	−0.408810	+	0.990920i
$16$		0			0
$17$			74.3305i			1.06046i	0.847854	+	0.530229i	$0.177894\pi$
$17$			74.3305i			1.06046i	−0.847854	+	0.530229i	$0.822106\pi$
$18$		0			0
$19$	108.076			1.30496			0.652481	−	0.757805i	$-0.273728\pi$
$19$	108.076			1.30496			0.652481	+	0.757805i	$0.273728\pi$
$20$		0			0
$21$	109.036	+	44.9834i	1.13302	+	0.467437i
$22$		0			0
$23$	24.7059			0.223980			0.111990	−	0.993709i	$-0.464278\pi$
$23$	24.7059			0.223980			0.111990	+	0.993709i	$0.464278\pi$
$24$		0			0
$25$	18.6309			0.149047
$26$		0			0
$27$	−129.388	+	54.2388i	−0.922247	+	0.386602i
$28$		0			0
$29$	84.1331			0.538728			0.269364	−	0.963038i	$-0.413186\pi$
$29$	84.1331			0.538728			0.269364	+	0.963038i	$0.413186\pi$
$30$		0			0
$31$		−	130.377i		−	0.755367i	−0.925935	−	0.377684i	$-0.876721\pi$
$31$		−	130.377i		−	0.755367i	0.925935	−	0.377684i	$-0.123279\pi$
$32$		0			0
$33$	−297.529	−	122.748i	−1.56949	−	0.647504i
$34$		0			0
$35$		−	272.045i		−	1.31383i
$36$		0			0
$37$		−	397.468i		−	1.76604i	−0.469338	−	0.883019i	$-0.655507\pi$
$37$		−	397.468i		−	1.76604i	0.469338	−	0.883019i	$-0.344493\pi$
$38$		0			0
$39$	342.637	+	141.357i	1.40682	+	0.580391i
$40$		0			0
$41$		−	104.132i		−	0.396649i	−0.980136	−	0.198325i	$-0.936450\pi$
$41$		−	104.132i		−	0.396649i	0.980136	−	0.198325i	$-0.0635500\pi$
$42$		0			0
$43$	161.650			0.573289			0.286644	−	0.958037i	$-0.407460\pi$
$43$	161.650			0.573289			0.286644	+	0.958037i	$0.407460\pi$
$44$		0			0
$45$	229.455	+	228.160i	0.760116	+	0.755825i
$46$		0			0
$47$	72.8162			0.225986			0.112993	−	0.993596i	$-0.463956\pi$
$47$	72.8162			0.225986			0.112993	+	0.993596i	$0.463956\pi$
$48$		0			0
$49$	−172.269			−0.502242
$50$		0			0
$51$	357.041	+	147.300i	0.980310	+	0.404433i
$52$		0			0
$53$	−136.267			−0.353164			−0.176582	−	0.984286i	$-0.556504\pi$
$53$	−136.267			−0.353164			−0.176582	+	0.984286i	$0.556504\pi$
$54$		0			0
$55$			742.339i			1.81995i
$56$		0			0
$57$	214.172	−	519.134i	0.497681	−	1.20633i
$58$		0			0
$59$			243.465i			0.537228i	0.963248	+	0.268614i	$0.0865656\pi$
$59$			243.465i			0.537228i	−0.963248	+	0.268614i	$0.913434\pi$
$60$		0			0
$61$		−	358.454i		−	0.752383i	−0.926542	−	0.376192i	$-0.877233\pi$
$61$		−	358.454i		−	0.752383i	0.926542	−	0.376192i	$-0.122767\pi$
$62$		0			0
$63$	432.149	−	434.602i	0.864216	−	0.869122i
$64$		0			0
$65$		−	854.883i		−	1.63131i
$66$		0			0
$67$	449.508			0.819645			0.409822	−	0.912165i	$-0.365591\pi$
$67$	449.508			0.819645			0.409822	+	0.912165i	$0.365591\pi$
$68$		0			0
$69$	48.9593	−	118.673i	0.0854205	−	0.207052i
$70$		0			0
$71$	−329.053			−0.550020			−0.275010	−	0.961441i	$-0.588681\pi$
$71$	−329.053			−0.550020			−0.275010	+	0.961441i	$0.588681\pi$
$72$		0			0
$73$	925.687			1.48416			0.742079	−	0.670313i	$-0.233840\pi$
$73$	925.687			1.48416			0.742079	+	0.670313i	$0.233840\pi$
$74$		0			0
$75$	36.9207	−	89.4923i	0.0568431	−	0.137782i
$76$		0			0
$77$	1406.03			2.08094
$78$		0			0
$79$		−	55.9579i		−	0.0796931i	−0.999206	−	0.0398465i	$-0.987313\pi$
$79$		−	55.9579i		−	0.0796931i	0.999206	−	0.0398465i	$-0.0126869\pi$
$80$		0			0
$81$	4.12647	+	728.988i	0.00566045	+	0.999984i
$82$		0			0
$83$		−	928.194i		−	1.22750i	−0.789500	−	0.613750i	$-0.789660\pi$
$83$		−	928.194i		−	1.22750i	0.789500	−	0.613750i	$-0.210340\pi$
$84$		0			0
$85$		−	890.823i		−	1.13674i
$86$		0			0
$87$	166.725	−	404.127i	0.205458	−	0.498011i
$88$		0			0
$89$		−	853.930i		−	1.01704i	−0.861051	−	0.508519i	$-0.830193\pi$
$89$		−	853.930i		−	1.01704i	0.861051	−	0.508519i	$-0.169807\pi$
$90$		0			0
$91$	−1619.20			−1.86525
$92$		0			0
$93$	−626.256	−	258.366i	−0.698277	−	0.288079i
$94$		0			0
$95$	−1295.25			−1.39884
$96$		0			0
$97$	−714.982			−0.748406			−0.374203	−	0.927347i	$-0.622084\pi$
$97$	−714.982			−0.748406			−0.374203	+	0.927347i	$0.622084\pi$
$98$		0			0
$99$	−1179.22	+	1185.91i	−1.19713	+	1.20393i
$100$		0			0
$101$	1702.69			1.67746			0.838730	−	0.544547i	$-0.183299\pi$
$101$	1702.69			1.67746			0.838730	+	0.544547i	$0.183299\pi$
$102$		0			0
$103$		−	655.930i		−	0.627482i	−0.949509	−	0.313741i	$-0.898418\pi$
$103$		−	655.930i		−	0.627482i	0.949509	−	0.313741i	$-0.101582\pi$
$104$		0			0
$105$	−1306.75	−	539.108i	−1.21453	−	0.501062i
$106$		0			0
$107$			202.008i			0.182513i	0.995827	+	0.0912563i	$0.0290883\pi$
$107$			202.008i			0.182513i	−0.995827	+	0.0912563i	$0.970912\pi$
$108$		0			0
$109$			588.724i			0.517335i	0.965966	+	0.258667i	$0.0832834\pi$
$109$			588.724i			0.517335i	−0.965966	+	0.258667i	$0.916717\pi$
$110$		0			0
$111$	−1909.21	−	787.657i	−1.63256	−	0.673524i
$112$		0			0
$113$		−	2228.76i		−	1.85544i	−0.373280	−	0.927719i	$-0.621767\pi$
$113$		−	2228.76i		−	1.85544i	0.373280	−	0.927719i	$-0.378233\pi$
$114$		0			0
$115$	−296.091			−0.240092
$116$		0			0
$117$	1358.00	−	1365.71i	1.07305	−	1.07914i
$118$		0			0
$119$	−1687.27			−1.29976
$120$		0			0
$121$	−2505.69			−1.88256
$122$		0			0
$123$	−500.188	−	206.356i	−0.366670	−	0.151272i
$124$		0			0
$125$	1274.79			0.912167
$126$		0			0
$127$			1345.93i			0.940409i	0.882558	+	0.470204i	$0.155820\pi$
$127$			1345.93i			0.940409i	−0.882558	+	0.470204i	$0.844180\pi$
$128$		0			0
$129$	320.340	−	776.475i	0.218638	−	0.529960i
$130$		0			0
$131$			679.531i			0.453213i	0.973986	+	0.226607i	$0.0727632\pi$
$131$			679.531i			0.453213i	−0.973986	+	0.226607i	$0.927237\pi$
$132$		0			0
$133$			2453.27i			1.59944i
$134$		0			0
$135$	1550.66	−	650.031i	0.988590	−	0.414413i
$136$		0			0
$137$		−	614.487i		−	0.383205i	−0.981473	−	0.191603i	$-0.938631\pi$
$137$		−	614.487i		−	0.383205i	0.981473	−	0.191603i	$-0.0613685\pi$
$138$		0			0
$139$	3163.89			1.93063			0.965315	−	0.261088i	$-0.0840813\pi$
$139$	3163.89			1.93063			0.965315	+	0.261088i	$0.0840813\pi$
$140$		0			0
$141$	144.299	−	349.768i	0.0861855	−	0.208906i
$142$		0			0
$143$	4418.36			2.58379
$144$		0			0
$145$	−1008.30			−0.577483
$146$		0			0
$147$	−341.383	+	827.482i	−0.191543	+	0.464283i
$148$		0			0
$149$	−1068.79			−0.587640			−0.293820	−	0.955861i	$-0.594927\pi$
$149$	−1068.79			−0.587640			−0.293820	+	0.955861i	$0.594927\pi$
$150$		0			0
$151$		−	1343.30i		−	0.723949i	−0.932188	−	0.361974i	$-0.882103\pi$
$151$		−	1343.30i		−	0.723949i	0.932188	−	0.361974i	$-0.117897\pi$
$152$		0			0
$153$	1415.09	−	1423.12i	0.747732	−	0.751977i
$154$		0			0
$155$			1562.52i			0.809706i
$156$		0			0
$157$		−	2626.27i		−	1.33503i	−0.744598	−	0.667513i	$-0.767359\pi$
$157$		−	2626.27i		−	1.33503i	0.744598	−	0.667513i	$-0.232641\pi$
$158$		0			0
$159$	−270.038	+	654.548i	−0.134688	+	0.326472i
$160$		0			0
$161$			560.813i			0.274523i
$162$		0			0
$163$	967.832			0.465070			0.232535	−	0.972588i	$-0.425298\pi$
$163$	967.832			0.465070			0.232535	+	0.972588i	$0.425298\pi$
$164$		0			0
$165$	3565.77	+	1471.08i	1.68239	+	0.694083i
$166$		0			0
$167$	−66.2186			−0.0306835			−0.0153418	−	0.999882i	$-0.504884\pi$
$167$	−66.2186			−0.0306835			−0.0153418	+	0.999882i	$0.504884\pi$
$168$		0			0
$169$	−2891.22			−1.31598
$170$		0			0
$171$	−2069.20	−	2057.52i	−0.925355	−	0.920132i
$172$		0			0
$173$	3061.46			1.34543			0.672713	−	0.739904i	$-0.265129\pi$
$173$	3061.46			1.34543			0.672713	+	0.739904i	$0.265129\pi$
$174$		0			0
$175$			422.913i			0.182681i
$176$		0			0
$177$	1169.47	+	482.471i	0.496625	+	0.204886i
$178$		0			0
$179$			949.274i			0.396380i	0.980164	+	0.198190i	$0.0635064\pi$
$179$			949.274i			0.396380i	−0.980164	+	0.198190i	$0.936494\pi$
$180$		0			0
$181$			874.923i			0.359296i	0.983731	+	0.179648i	$0.0574958\pi$
$181$			874.923i			0.359296i	−0.983731	+	0.179648i	$0.942504\pi$
$182$		0			0
$183$	−1721.81	−	710.344i	−0.695518	−	0.286941i
$184$		0			0
$185$			4763.50i			1.89308i
$186$		0			0
$187$	4604.11			1.80046
$188$		0			0
$189$	−1231.20	−	2937.04i	−0.473843	−	1.13036i
$190$		0			0
$191$	683.182			0.258813			0.129407	−	0.991592i	$-0.458693\pi$
$191$	683.182			0.258813			0.129407	+	0.991592i	$0.458693\pi$
$192$		0			0
$193$	3029.86			1.13002			0.565011	−	0.825083i	$-0.308872\pi$
$193$	3029.86			1.13002			0.565011	+	0.825083i	$0.308872\pi$
$194$		0			0
$195$	−4106.37	−	1694.11i	−1.50802	−	0.622142i
$196$		0			0
$197$	−2661.33			−0.962496			−0.481248	−	0.876584i	$-0.659816\pi$
$197$	−2661.33			−0.962496			−0.481248	+	0.876584i	$0.659816\pi$
$198$		0			0
$199$			2975.37i			1.05989i	0.848031	+	0.529946i	$0.177788\pi$
$199$			2975.37i			1.05989i	−0.848031	+	0.529946i	$0.822212\pi$
$200$		0			0
$201$	890.784	−	2159.18i	0.312592	−	0.757696i
$202$		0			0
$203$			1909.78i			0.660298i
$204$		0			0
$205$			1247.98i			0.425183i
$206$		0			0
$207$	−473.015	−	470.345i	−0.158825	−	0.157929i
$208$		0			0
$209$		−	6694.32i		−	2.21558i
$210$		0			0
$211$	−2601.38			−0.848751			−0.424376	−	0.905486i	$-0.639506\pi$
$211$	−2601.38			−0.848751			−0.424376	+	0.905486i	$0.639506\pi$
$212$		0			0
$213$	−652.080	+	1580.58i	−0.209764	+	0.508450i
$214$		0			0
$215$	−1937.31			−0.614529
$216$		0			0
$217$	2959.50			0.925824
$218$		0			0
$219$	1834.42	−	4446.47i	0.566022	−	1.37199i
$220$		0			0
$221$	−5302.13			−1.61384
$222$		0			0
$223$			4031.63i			1.21066i	0.795973	+	0.605332i	$0.206960\pi$
$223$			4031.63i			1.21066i	−0.795973	+	0.605332i	$0.793040\pi$
$224$		0			0
$225$	−356.705	−	354.691i	−0.105690	−	0.105094i
$226$		0			0
$227$		−	4990.76i		−	1.45924i	−0.683851	−	0.729622i	$-0.739696\pi$
$227$		−	4990.76i		−	1.45924i	0.683851	−	0.729622i	$-0.260304\pi$
$228$		0			0
$229$		−	5091.07i		−	1.46912i	−0.678546	−	0.734558i	$-0.737390\pi$
$229$		−	5091.07i		−	1.46912i	0.678546	−	0.734558i	$-0.262610\pi$
$230$		0			0
$231$	2786.31	−	6753.78i	0.793619	−	1.92366i
$232$		0			0
$233$			229.872i			0.0646326i	0.999478	+	0.0323163i	$0.0102884\pi$
$233$			229.872i			0.0646326i	−0.999478	+	0.0323163i	$0.989712\pi$
$234$		0			0
$235$	−872.674			−0.242243
$236$		0			0
$237$	−268.790	−	110.891i	−0.0736699	−	0.0303930i
$238$		0			0
$239$	1523.39			0.412300			0.206150	−	0.978520i	$-0.433907\pi$
$239$	1523.39			0.412300			0.206150	+	0.978520i	$0.433907\pi$
$240$		0			0
$241$	1736.76			0.464210			0.232105	−	0.972691i	$-0.425439\pi$
$241$	1736.76			0.464210			0.232105	+	0.972691i	$0.425439\pi$
$242$		0			0
$243$	3509.82	+	1424.80i	0.926564	+	0.376137i
$244$		0			0
$245$	2064.58			0.538371
$246$		0			0
$247$			7709.23i			1.98594i
$248$		0			0
$249$	−4458.51	−	1839.39i	−1.13473	−	0.468139i
$250$		0			0
$251$		−	1246.85i		−	0.313547i	−0.987635	−	0.156774i	$-0.949891\pi$
$251$		−	1246.85i		−	0.313547i	0.987635	−	0.156774i	$-0.0501093\pi$
$252$		0			0
$253$		−	1530.31i		−	0.380275i
$254$		0			0
$255$	−4279.00	−	1765.33i	−1.05083	−	0.433527i
$256$		0			0
$257$			6505.35i			1.57896i	0.613777	+	0.789479i	$0.289649\pi$
$257$			6505.35i			1.57896i	−0.613777	+	0.789479i	$0.710351\pi$
$258$		0			0
$259$	9022.35			2.16456
$260$		0			0
$261$	−1610.80	−	1601.71i	−0.382015	−	0.379859i
$262$		0			0
$263$	7974.40			1.86967			0.934834	−	0.355086i	$-0.115549\pi$
$263$	7974.40			1.86967			0.934834	+	0.355086i	$0.115549\pi$
$264$		0			0
$265$	1633.11			0.378569
$266$		0			0
$267$	−4101.79	−	1692.22i	−0.940171	−	0.387874i
$268$		0			0
$269$	569.640			0.129114			0.0645568	−	0.997914i	$-0.479437\pi$
$269$	569.640			0.129114			0.0645568	+	0.997914i	$0.479437\pi$
$270$		0			0
$271$			6729.09i			1.50835i	0.656673	+	0.754175i	$0.271963\pi$
$271$			6729.09i			1.50835i	−0.656673	+	0.754175i	$0.728037\pi$
$272$		0			0
$273$	−3208.74	+	7777.70i	−0.711362	+	1.72428i
$274$		0			0
$275$		−	1154.02i		−	0.253054i
$276$		0			0
$277$			3054.27i			0.662502i	0.943543	+	0.331251i	$0.107471\pi$
$277$			3054.27i			0.662502i	−0.943543	+	0.331251i	$0.892529\pi$
$278$		0			0
$279$	−2482.09	+	2496.18i	−0.532612	+	0.535635i
$280$		0			0
$281$		−	673.673i		−	0.143018i	−0.997440	−	0.0715088i	$-0.977219\pi$
$281$		−	673.673i		−	0.143018i	0.997440	−	0.0715088i	$-0.0227814\pi$
$282$		0			0
$283$	1972.26			0.414270			0.207135	−	0.978312i	$-0.433586\pi$
$283$	1972.26			0.414270			0.207135	+	0.978312i	$0.433586\pi$
$284$		0			0
$285$	−2566.77	+	6221.62i	−0.533482	+	1.29311i
$286$		0			0
$287$	2363.74			0.486157
$288$		0			0
$289$	−612.028			−0.124573
$290$		0			0
$291$	−1416.87	+	3434.36i	−0.285424	+	0.691842i
$292$		0			0
$293$	−3641.01			−0.725974			−0.362987	−	0.931794i	$-0.618243\pi$
$293$	−3641.01			−0.725974			−0.362987	+	0.931794i	$0.618243\pi$
$294$		0			0
$295$		−	2917.84i		−	0.575874i
$296$		0			0
$297$	3359.61	+	8014.40i	0.656377	+	1.56580i
$298$		0			0
$299$			1762.32i			0.340861i
$300$		0			0
$301$			3669.38i			0.702657i
$302$		0			0
$303$	3374.19	−	8178.72i	0.639743	−	1.55068i
$304$		0			0
$305$			4295.94i			0.806507i
$306$		0			0
$307$	2847.42			0.529351			0.264676	−	0.964337i	$-0.414735\pi$
$307$	2847.42			0.529351			0.264676	+	0.964337i	$0.414735\pi$
$308$		0			0
$309$	−3150.71	−	1299.85i	−0.580057	−	0.239307i
$310$		0			0
$311$	−8784.69			−1.60172			−0.800859	−	0.598853i	$-0.795623\pi$
$311$	−8784.69			−1.60172			−0.800859	+	0.598853i	$0.795623\pi$
$312$		0			0
$313$	4790.21			0.865043			0.432522	−	0.901624i	$-0.357624\pi$
$313$	4790.21			0.865043			0.432522	+	0.901624i	$0.357624\pi$
$314$		0			0
$315$	−5179.13	+	5208.53i	−0.926385	+	0.931643i
$316$		0			0
$317$	6390.34			1.13223			0.566116	−	0.824326i	$-0.308446\pi$
$317$	6390.34			1.13223			0.566116	+	0.824326i	$0.308446\pi$
$318$		0			0
$319$		−	5211.29i		−	0.914659i
$320$		0			0
$321$	970.331	+	400.316i	0.168718	+	0.0696059i
$322$		0			0
$323$			8033.33i			1.38386i
$324$		0			0
$325$			1328.98i			0.226826i
$326$		0			0
$327$	2827.89	+	1166.67i	0.478235	+	0.197299i
$328$		0			0
$329$			1652.89i			0.276982i
$330$		0			0
$331$	−3472.47			−0.576629			−0.288315	−	0.957536i	$-0.593095\pi$
$331$	−3472.47			−0.576629			−0.288315	+	0.957536i	$0.593095\pi$
$332$		0			0
$333$	−7566.91	+	7609.87i	−1.24524	+	1.25231i
$334$		0			0
$335$	−5387.18			−0.878607
$336$		0			0
$337$	672.232			0.108661			0.0543305	−	0.998523i	$-0.482698\pi$
$337$	672.232			0.108661			0.0543305	+	0.998523i	$0.482698\pi$
$338$		0			0
$339$	−10705.7	−	4416.71i	−1.71520	−	0.707619i
$340$		0			0
$341$	−8075.68			−1.28247
$342$		0			0
$343$			3875.51i			0.610082i
$344$		0			0
$345$	−586.759	+	1422.25i	−0.0915653	+	0.221946i
$346$		0			0
$347$			606.106i			0.0937679i	0.998900	+	0.0468839i	$0.0149291\pi$
$347$			606.106i			0.0937679i	−0.998900	+	0.0468839i	$0.985071\pi$
$348$		0			0
$349$			7881.87i			1.20890i	0.796643	+	0.604451i	$0.206607\pi$
$349$			7881.87i			1.20890i	−0.796643	+	0.604451i	$0.793393\pi$
$350$		0			0
$351$	−3868.95	−	9229.44i	−0.588345	−	1.40351i
$352$		0			0
$353$			528.013i			0.0796128i	0.999207	+	0.0398064i	$0.0126741\pi$
$353$			528.013i			0.0796128i	−0.999207	+	0.0398064i	$0.987326\pi$
$354$		0			0
$355$	3943.58			0.589587
$356$		0			0
$357$	−3343.64	+	8104.67i	−0.495698	+	1.20153i
$358$		0			0
$359$	9239.79			1.35838			0.679188	−	0.733964i	$-0.262332\pi$
$359$	9239.79			1.35838			0.679188	+	0.733964i	$0.262332\pi$
$360$		0			0
$361$	4821.37			0.702926
$362$		0			0
$363$	−4965.49	+	12035.9i	−0.717963	+	1.74028i
$364$		0			0
$365$	−11094.0			−1.59092
$366$		0			0
$367$		−	8258.02i		−	1.17456i	−0.809382	−	0.587282i	$-0.800198\pi$
$367$		−	8258.02i		−	1.17456i	0.809382	−	0.587282i	$-0.199802\pi$
$368$		0			0
$369$	−1982.43	+	1993.69i	−0.279678	+	0.281266i
$370$		0			0
$371$		−	3093.19i		−	0.432859i
$372$		0			0
$373$		−	2343.62i		−	0.325330i	−0.986681	−	0.162665i	$-0.947991\pi$
$373$		−	2343.62i		−	0.325330i	0.986681	−	0.162665i	$-0.0520090\pi$
$374$		0			0
$375$	2526.24	−	6123.37i	0.347878	−	0.843226i
$376$		0			0
$377$			6001.36i			0.819856i
$378$		0			0
$379$	778.226			0.105474			0.0527372	−	0.998608i	$-0.483205\pi$
$379$	778.226			0.105474			0.0527372	+	0.998608i	$0.483205\pi$
$380$		0			0
$381$	6465.07	+	2667.21i	0.869333	+	0.358649i
$382$		0			0
$383$	−9755.66			−1.30154			−0.650772	−	0.759274i	$-0.725554\pi$
$383$	−9755.66			−1.30154			−0.650772	+	0.759274i	$0.725554\pi$
$384$		0			0
$385$	−16850.8			−2.23063
$386$		0			0
$387$	−3094.93	−	3077.46i	−0.406522	−	0.404227i
$388$		0			0
$389$	−3566.20			−0.464816			−0.232408	−	0.972618i	$-0.574660\pi$
$389$	−3566.20			−0.464816			−0.232408	+	0.972618i	$0.574660\pi$
$390$		0			0
$391$			1836.40i			0.237521i
$392$		0			0
$393$	3264.08	+	1346.62i	0.418960	+	0.172845i
$394$		0			0
$395$			670.634i			0.0854259i
$396$		0			0
$397$		−	8958.31i		−	1.13251i	−0.824232	−	0.566253i	$-0.808393\pi$
$397$		−	8958.31i		−	1.13251i	0.824232	−	0.566253i	$-0.191607\pi$
$398$		0			0
$399$	11784.1	+	4861.61i	1.47855	+	0.609987i
$400$		0			0
$401$			12946.7i			1.61229i	0.591716	+	0.806146i	$0.298451\pi$
$401$			12946.7i			1.61229i	−0.591716	+	0.806146i	$0.701549\pi$
$402$		0			0
$403$	9300.01			1.14955
$404$		0			0
$405$	−49.4541	−	8736.64i	−0.00606764	−	1.07192i
$406$		0			0
$407$	−24619.6			−2.99840
$408$		0			0
$409$	7979.27			0.964669			0.482335	−	0.875987i	$-0.339789\pi$
$409$	7979.27			0.964669			0.482335	+	0.875987i	$0.339789\pi$
$410$		0			0
$411$	−2951.64	−	1217.72i	−0.354243	−	0.146145i
$412$		0			0
$413$	−5526.55			−0.658459
$414$		0			0
$415$			11124.0i			1.31580i
$416$		0			0
$417$	6269.83	−	15197.5i	0.736295	−	1.78471i
$418$		0			0
$419$		−	8671.29i		−	1.01103i	−0.862819	−	0.505514i	$-0.831303\pi$
$419$		−	8671.29i		−	1.01103i	0.862819	−	0.505514i	$-0.168697\pi$
$420$		0			0
$421$		−	2908.89i		−	0.336747i	−0.985723	−	0.168373i	$-0.946149\pi$
$421$		−	2908.89i		−	0.336747i	0.985723	−	0.168373i	$-0.0538514\pi$
$422$		0			0
$423$	−1394.13	−	1386.26i	−0.160248	−	0.159343i
$424$		0			0
$425$			1384.85i			0.158059i
$426$		0			0
$427$	8136.75			0.922166
$428$		0			0
$429$	8755.80	−	21223.3i	0.985394	−	2.38851i
$430$		0			0
$431$	−4968.52			−0.555279			−0.277639	−	0.960685i	$-0.589552\pi$
$431$	−4968.52			−0.555279			−0.277639	+	0.960685i	$0.589552\pi$
$432$		0			0
$433$	9208.60			1.02203			0.511013	−	0.859573i	$-0.329271\pi$
$433$	9208.60			1.02203			0.511013	+	0.859573i	$0.329271\pi$
$434$		0			0
$435$	−1998.14	+	4843.31i	−0.220238	+	0.533837i
$436$		0			0
$437$	2670.11			0.292285
$438$		0			0
$439$			16534.4i			1.79760i	0.438361	+	0.898799i	$0.355559\pi$
$439$			16534.4i			1.79760i	−0.438361	+	0.898799i	$0.644441\pi$
$440$		0			0
$441$	3298.23	+	3279.62i	0.356142	+	0.354132i
$442$		0			0
$443$		−	9024.44i		−	0.967865i	−0.875105	−	0.483933i	$-0.839208\pi$
$443$		−	9024.44i		−	0.967865i	0.875105	−	0.483933i	$-0.160792\pi$
$444$		0			0
$445$			10234.0i			1.09020i
$446$		0			0
$447$	−2118.00	+	5133.83i	−0.224111	+	0.543226i
$448$		0			0
$449$			8971.07i			0.942920i	0.881888	+	0.471460i	$0.156273\pi$
$449$			8971.07i			0.942920i	−0.881888	+	0.471460i	$0.843727\pi$
$450$		0			0
$451$	−6450.01			−0.673435
$452$		0			0
$453$	−6452.45	−	2662.00i	−0.669233	−	0.276096i
$454$		0			0
$455$	19405.5			1.99943
$456$		0			0
$457$	−11467.8			−1.17383			−0.586915	−	0.809648i	$-0.699658\pi$
$457$	−11467.8			−1.17383			−0.586915	+	0.809648i	$0.699658\pi$
$458$		0			0
$459$	−4031.60	−	9617.45i	−0.409976	−	0.978005i
$460$		0			0
$461$	−14270.2			−1.44171			−0.720855	−	0.693086i	$-0.756251\pi$
$461$	−14270.2			−1.44171			−0.720855	+	0.693086i	$0.756251\pi$
$462$		0			0
$463$		−	557.295i		−	0.0559389i	−0.999609	−	0.0279694i	$-0.991096\pi$
$463$		−	557.295i		−	0.0559389i	0.999609	−	0.0279694i	$-0.00890411\pi$
$464$		0			0
$465$	7505.44	+	3096.42i	0.748508	+	0.308802i
$466$		0			0
$467$			9659.10i			0.957109i	0.878058	+	0.478554i	$0.158839\pi$
$467$			9659.10i			0.957109i	−0.878058	+	0.478554i	$0.841161\pi$
$468$		0			0
$469$			10203.6i			1.00461i
$470$		0			0
$471$	−12615.1	−	5204.44i	−1.23412	−	0.509146i
$472$		0			0
$473$		−	10012.8i		−	0.973336i
$474$		0			0
$475$	2013.55			0.194501
$476$		0			0
$477$	2608.94	+	2594.22i	0.250430	+	0.249017i
$478$		0			0
$479$	−4353.23			−0.415249			−0.207625	−	0.978209i	$-0.566573\pi$
$479$	−4353.23			−0.415249			−0.207625	+	0.978209i	$0.566573\pi$
$480$		0			0
$481$	28352.1			2.68762
$482$		0			0
$483$	2693.82	+	1111.35i	0.253775	+	0.104696i
$484$		0			0
$485$	8568.78			0.802244
$486$		0			0
$487$		−	12368.5i		−	1.15087i	−0.817849	−	0.575433i	$-0.804833\pi$
$487$		−	12368.5i		−	1.15087i	0.817849	−	0.575433i	$-0.195167\pi$
$488$		0			0
$489$	1917.94	−	4648.91i	0.177367	−	0.429920i
$490$		0			0
$491$		−	19510.2i		−	1.79324i	−0.442798	−	0.896621i	$-0.646014\pi$
$491$		−	19510.2i		−	1.79324i	0.442798	−	0.896621i	$-0.353986\pi$
$492$		0			0
$493$			6253.66i			0.571299i
$494$		0			0
$495$	14132.5	−	14212.7i	1.28325	−	1.29053i
$496$		0			0
$497$		−	7469.36i		−	0.674138i
$498$		0			0
$499$	1269.62			0.113899			0.0569497	−	0.998377i	$-0.481863\pi$
$499$	1269.62			0.113899			0.0569497	+	0.998377i	$0.481863\pi$
$500$		0			0
$501$	−131.224	+	318.076i	−0.0117020	+	0.0283645i
$502$		0			0
$503$	14901.0			1.32088			0.660442	−	0.750877i	$-0.270369\pi$
$503$	14901.0			1.32088			0.660442	+	0.750877i	$0.270369\pi$
$504$		0			0
$505$	−20406.0			−1.79813
$506$		0			0
$507$	−5729.48	+	13887.8i	−0.501884	+	1.21652i
$508$		0			0
$509$	−1853.05			−0.161365			−0.0806826	−	0.996740i	$-0.525710\pi$
$509$	−1853.05			−0.161365			−0.0806826	+	0.996740i	$0.525710\pi$
$510$		0			0
$511$			21012.7i			1.81907i
$512$		0			0
$513$	−13983.7	+	5861.90i	−1.20350	+	0.504501i
$514$		0			0
$515$			7861.07i			0.672621i
$516$		0			0
$517$		−	4510.31i		−	0.383681i
$518$		0			0
$519$	6066.85	−	14705.5i	0.513112	−	1.24374i
$520$		0			0
$521$		−	1482.78i		−	0.124686i	−0.998055	−	0.0623432i	$-0.980143\pi$
$521$		−	1482.78i		−	0.124686i	0.998055	−	0.0623432i	$-0.0198573\pi$
$522$		0			0
$523$	16235.8			1.35744			0.678721	−	0.734396i	$-0.262535\pi$
$523$	16235.8			1.35744			0.678721	+	0.734396i	$0.262535\pi$
$524$		0			0
$525$	2031.43	+	838.082i	0.168874	+	0.0696703i
$526$		0			0
$527$	9690.99			0.801036
$528$		0			0
$529$	−11556.6			−0.949833
$530$		0			0
$531$	4635.03	−	4661.35i	0.378801	−	0.380951i
$532$		0			0
$533$	7427.88			0.603635
$534$		0			0
$535$		−	2420.99i		−	0.195642i
$536$		0			0
$537$	4559.77	+	1881.16i	0.366422	+	0.151170i
$538$		0			0
$539$			10670.5i			0.852712i
$540$		0			0
$541$		−	13039.6i		−	1.03626i	−0.855301	−	0.518131i	$-0.826628\pi$
$541$		−	13039.6i		−	1.03626i	0.855301	−	0.518131i	$-0.173372\pi$
$542$		0			0
$543$	4202.63	+	1733.82i	0.332140	+	0.137027i
$544$		0			0
$545$		−	7055.63i		−	0.554550i
$546$		0			0
$547$	−18647.7			−1.45762			−0.728811	−	0.684715i	$-0.759926\pi$
$547$	−18647.7			−1.45762			−0.728811	+	0.684715i	$0.759926\pi$
$548$		0			0
$549$	−6824.17	+	6862.91i	−0.530507	+	0.533519i
$550$		0			0
$551$	9092.75			0.703020
$552$		0			0
$553$	1270.22			0.0976766
$554$		0			0
$555$	22881.1	+	9439.77i	1.75000	+	0.721975i
$556$		0			0
$557$	−6780.36			−0.515786			−0.257893	−	0.966173i	$-0.583028\pi$
$557$	−6780.36			−0.515786			−0.257893	+	0.966173i	$0.583028\pi$
$558$		0			0
$559$			11530.8i			0.872451i
$560$		0			0
$561$	9123.90	−	22115.5i	0.686651	−	1.66438i
$562$		0			0
$563$			5028.45i			0.376419i	0.982129	+	0.188209i	$0.0602683\pi$
$563$			5028.45i			0.376419i	−0.982129	+	0.188209i	$0.939732\pi$
$564$		0			0
$565$			26710.9i			1.98891i
$566$		0			0
$567$	−16547.7	+	93.6689i	−1.22564	+	0.00693779i
$568$		0			0
$569$			3264.38i			0.240510i	0.992743	+	0.120255i	$0.0383712\pi$
$569$			3264.38i			0.240510i	−0.992743	+	0.120255i	$0.961629\pi$
$570$		0			0
$571$	3622.18			0.265470			0.132735	−	0.991152i	$-0.457624\pi$
$571$	3622.18			0.265470			0.132735	+	0.991152i	$0.457624\pi$
$572$		0			0
$573$	1353.85	−	3281.62i	0.0987050	−	0.239252i
$574$		0			0
$575$	460.294			0.0333836
$576$		0			0
$577$	−9604.14			−0.692939			−0.346469	−	0.938061i	$-0.612620\pi$
$577$	−9604.14			−0.692939			−0.346469	+	0.938061i	$0.612620\pi$
$578$		0			0
$579$	6004.24	−	14553.7i	0.430963	−	1.04462i
$580$		0			0
$581$	21069.6			1.50450
$582$		0			0
$583$			8440.51i			0.599606i
$584$		0			0
$585$	−16275.1	+	16367.5i	−1.15024	+	1.15677i
$586$		0			0
$587$		−	8942.98i		−	0.628818i	−0.949288	−	0.314409i	$-0.898194\pi$
$587$		−	8942.98i		−	0.628818i	0.949288	−	0.314409i	$-0.101806\pi$
$588$		0			0
$589$		−	14090.6i		−	0.985726i
$590$		0			0
$591$	−5273.92	+	12783.5i	−0.367073	+	0.889751i
$592$		0			0
$593$		−	9534.33i		−	0.660250i	−0.943937	−	0.330125i	$-0.892909\pi$
$593$		−	9534.33i		−	0.660250i	0.943937	−	0.330125i	$-0.107091\pi$
$594$		0			0
$595$	20221.3			1.39326
$596$		0			0
$597$	14292.0	+	5896.25i	0.979786	+	0.404217i
$598$		0			0
$599$	−5691.25			−0.388210			−0.194105	−	0.980981i	$-0.562180\pi$
$599$	−5691.25			−0.388210			−0.194105	+	0.980981i	$0.562180\pi$
$600$		0			0
$601$	−13566.3			−0.920769			−0.460385	−	0.887720i	$-0.652288\pi$
$601$	−13566.3			−0.920769			−0.460385	+	0.887720i	$0.652288\pi$
$602$		0			0
$603$	−8606.21	−	8557.64i	−0.581214	−	0.577934i
$604$		0			0
$605$	30029.7			2.01799
$606$		0			0
$607$		−	813.549i		−	0.0544002i	−0.999630	−	0.0272001i	$-0.991341\pi$
$607$		−	813.549i		−	0.0544002i	0.999630	−	0.0272001i	$-0.00865913\pi$
$608$		0			0
$609$	9173.50	+	3784.59i	0.610393	+	0.251821i
$610$		0			0
$611$			5194.11i			0.343913i
$612$		0			0
$613$		−	25165.4i		−	1.65811i	−0.559167	−	0.829055i	$-0.688879\pi$
$613$		−	25165.4i		−	1.65811i	0.559167	−	0.829055i	$-0.311121\pi$
$614$		0			0
$615$	5994.56	+	2473.10i	0.393047	+	0.162154i
$616$		0			0
$617$		−	24993.6i		−	1.63080i	−0.578898	−	0.815400i	$-0.696517\pi$
$617$		−	24993.6i		−	1.63080i	0.578898	−	0.815400i	$-0.303483\pi$
$618$		0			0
$619$	7970.28			0.517532			0.258766	−	0.965940i	$-0.416684\pi$
$619$	7970.28			0.517532			0.258766	+	0.965940i	$0.416684\pi$
$620$		0			0
$621$	−3196.64	+	1340.02i	−0.206565	+	0.0865911i
$622$		0			0
$623$	19383.8			1.24654
$624$		0			0
$625$	−17606.8			−1.12683
$626$		0			0
$627$	−32155.7	−	13266.0i	−2.04813	−	0.844968i
$628$		0			0
$629$	29544.0			1.87281
$630$		0			0
$631$			4608.28i			0.290734i	0.989378	+	0.145367i	$0.0464362\pi$
$631$			4608.28i			0.290734i	−0.989378	+	0.145367i	$0.953564\pi$
$632$		0			0
$633$	−5155.12	+	12495.5i	−0.323693	+	0.784603i
$634$		0			0
$635$		−	16130.4i		−	1.00806i
$636$		0			0
$637$		−	12288.2i		−	0.764330i
$638$		0			0
$639$	6300.00	+	6264.44i	0.390022	+	0.387821i
$640$		0			0
$641$			4200.53i			0.258832i	0.991590	+	0.129416i	$0.0413102\pi$
$641$			4200.53i			0.258832i	−0.991590	+	0.129416i	$0.958690\pi$
$642$		0			0
$643$	−3546.92			−0.217538			−0.108769	−	0.994067i	$-0.534691\pi$
$643$	−3546.92			−0.217538			−0.108769	+	0.994067i	$0.534691\pi$
$644$		0			0
$645$	−3839.15	+	9305.75i	−0.234366	+	0.568083i
$646$		0			0
$647$	2851.58			0.173272			0.0866362	−	0.996240i	$-0.472388\pi$
$647$	2851.58			0.173272			0.0866362	+	0.996240i	$0.472388\pi$
$648$		0			0
$649$	15080.5			0.912112
$650$		0			0
$651$	5864.79	−	14215.7i	0.353087	−	0.855850i
$652$		0			0
$653$	33080.9			1.98247			0.991237	−	0.132098i	$-0.0421715\pi$
$653$	33080.9			1.98247			0.991237	+	0.132098i	$0.0421715\pi$
$654$		0			0
$655$		−	8143.92i		−	0.485816i
$656$		0			0
$657$	−17723.1	−	17623.0i	−1.05242	−	1.04648i
$658$		0			0
$659$		−	11761.6i		−	0.695244i	−0.937635	−	0.347622i	$-0.886989\pi$
$659$		−	11761.6i		−	0.695244i	0.937635	−	0.347622i	$-0.113011\pi$
$660$		0			0
$661$			2478.98i			0.145871i	0.997337	+	0.0729357i	$0.0232368\pi$
$661$			2478.98i			0.145871i	−0.997337	+	0.0729357i	$0.976763\pi$
$662$		0			0
$663$	−10507.1	+	25468.4i	−0.615481	+	1.49187i
$664$		0			0
$665$		−	29401.5i		−	1.71450i
$666$		0			0
$667$	2078.58			0.120664
$668$		0			0
$669$	19365.6	+	7989.42i	1.11916	+	0.461718i
$670$		0			0
$671$	−22203.0			−1.27740
$672$		0			0
$673$	5752.56			0.329487			0.164744	−	0.986336i	$-0.447320\pi$
$673$	5752.56			0.329487			0.164744	+	0.986336i	$0.447320\pi$
$674$		0			0
$675$	−2410.61	+	1010.52i	−0.137458	+	0.0576221i
$676$		0			0
$677$	−25177.1			−1.42930			−0.714649	−	0.699483i	$-0.753414\pi$
$677$	−25177.1			−1.42930			−0.714649	+	0.699483i	$0.753414\pi$
$678$		0			0
$679$		−	16229.8i		−	0.917292i
$680$		0			0
$681$	−23972.8	−	9890.12i	−1.34895	−	0.556520i
$682$		0			0
$683$			33891.4i			1.89871i	0.314203	+	0.949356i	$0.398263\pi$
$683$			33891.4i			1.89871i	−0.314203	+	0.949356i	$0.601737\pi$
$684$		0			0
$685$			7364.38i			0.410772i
$686$		0			0
$687$	−24454.6	−	10088.9i	−1.35808	−	0.560285i
$688$		0			0
$689$		−	9720.15i		−	0.537457i
$690$		0			0
$691$	−1194.54			−0.0657634			−0.0328817	−	0.999459i	$-0.510468\pi$
$691$	−1194.54			−0.0657634			−0.0328817	+	0.999459i	$0.510468\pi$
$692$		0			0
$693$	−26919.7	−	26767.7i	−1.47560	−	1.46728i
$694$		0			0
$695$	−37918.0			−2.06951
$696$		0			0
$697$	7740.15			0.420630
$698$		0			0
$699$	1104.17	+	455.534i	0.0597477	+	0.0246493i
$700$		0			0
$701$	−23289.7			−1.25484			−0.627419	−	0.778682i	$-0.715889\pi$
$701$	−23289.7			−1.25484			−0.627419	+	0.778682i	$0.715889\pi$
$702$		0			0
$703$		−	42956.7i		−	2.30461i
$704$		0			0
$705$	−1729.37	+	4191.83i	−0.0923854	+	0.223934i
$706$		0			0
$707$			38650.2i			2.05600i
$708$		0			0
$709$		−	11916.8i		−	0.631236i	−0.948886	−	0.315618i	$-0.897788\pi$
$709$		−	11916.8i		−	0.631236i	0.948886	−	0.315618i	$-0.102212\pi$
$710$		0			0
$711$	−1065.31	+	1071.36i	−0.0561918	+	0.0565108i
$712$		0			0
$713$		−	3221.08i		−	0.169187i
$714$		0			0
$715$	−52952.3			−2.76966
$716$		0			0
$717$	3018.87	−	7317.48i	0.157241	−	0.381138i
$718$		0			0
$719$	5520.22			0.286327			0.143164	−	0.989699i	$-0.454272\pi$
$719$	5520.22			0.286327			0.143164	+	0.989699i	$0.454272\pi$
$720$		0			0
$721$	14889.3			0.769080
$722$		0			0
$723$	3441.72	−	8342.41i	0.177038	−	0.429125i
$724$		0			0
$725$	1567.48			0.0802961
$726$		0			0
$727$		−	28589.0i		−	1.45847i	−0.684264	−	0.729234i	$-0.739876\pi$
$727$		−	28589.0i		−	1.45847i	0.684264	−	0.729234i	$-0.260124\pi$
$728$		0			0
$729$	13799.3	−	14035.7i	0.701078	−	0.713085i
$730$		0			0
$731$			12015.5i			0.607949i
$732$		0			0
$733$		−	23340.5i		−	1.17613i	−0.808814	−	0.588064i	$-0.799890\pi$
$733$		−	23340.5i		−	1.17613i	0.808814	−	0.588064i	$-0.200110\pi$
$734$		0			0
$735$	4091.34	−	9917.05i	0.205322	−	0.497681i
$736$		0			0
$737$		−	27843.0i		−	1.39160i
$738$		0			0
$739$	8818.16			0.438946			0.219473	−	0.975619i	$-0.429566\pi$
$739$	8818.16			0.438946			0.219473	+	0.975619i	$0.429566\pi$
$740$		0			0
$741$	37030.7	+	15277.3i	1.83584	+	0.757388i
$742$		0			0
$743$	25836.1			1.27569			0.637843	−	0.770166i	$-0.279827\pi$
$743$	25836.1			1.27569			0.637843	+	0.770166i	$0.279827\pi$
$744$		0			0
$745$	12809.0			0.629912
$746$		0			0
$747$	−17670.8	+	17771.1i	−0.865514	+	0.870427i
$748$		0			0
$749$	−4585.49			−0.223698
$750$		0			0
$751$			19166.2i			0.931272i	0.884976	+	0.465636i	$0.154174\pi$
$751$			19166.2i			0.931272i	−0.884976	+	0.465636i	$0.845826\pi$
$752$		0			0
$753$	−5989.14	−	2470.86i	−0.289849	−	0.119579i
$754$		0			0
$755$			16098.9i			0.776027i
$756$		0			0
$757$		−	18644.0i		−	0.895150i	−0.894247	−	0.447575i	$-0.852288\pi$
$757$		−	18644.0i		−	0.895150i	0.894247	−	0.447575i	$-0.147712\pi$
$758$		0			0
$759$	−7350.73	−	3032.59i	−0.351534	−	0.145028i
$760$		0			0
$761$			5444.32i			0.259338i	0.991557	+	0.129669i	$0.0413915\pi$
$761$			5444.32i			0.259338i	−0.991557	+	0.129669i	$0.958608\pi$
$762$		0			0
$763$	−13363.8			−0.634077
$764$		0			0
$765$	−16959.3	+	17055.5i	−0.801522	+	0.806071i
$766$		0			0
$767$	−17366.8			−0.817573
$768$		0			0
$769$	−5426.95			−0.254487			−0.127244	−	0.991871i	$-0.540613\pi$
$769$	−5426.95			−0.254487			−0.127244	+	0.991871i	$0.540613\pi$
$770$		0			0
$771$	31248.0	+	12891.6i	1.45962	+	0.602176i
$772$		0			0
$773$	27580.1			1.28330			0.641648	−	0.766999i	$-0.278251\pi$
$773$	27580.1			1.28330			0.641648	+	0.766999i	$0.278251\pi$
$774$		0			0
$775$		−	2429.04i		−	0.112586i
$776$		0			0
$777$	17879.5	−	43338.2i	0.825511	−	2.00096i
$778$		0			0
$779$		−	11254.1i		−	0.517612i
$780$		0			0
$781$			20381.9i			0.933831i
$782$		0			0
$783$	−10885.8	+	4563.28i	−0.496840	+	0.208274i
$784$		0			0
$785$			31474.8i			1.43106i
$786$		0			0
$787$	−25227.6			−1.14265			−0.571325	−	0.820724i	$-0.693571\pi$
$787$	−25227.6			−1.14265			−0.571325	+	0.820724i	$0.693571\pi$
$788$		0			0
$789$	15802.8	−	38304.4i	0.713045	−	1.72836i
$790$		0			0
$791$	50591.9			2.27413
$792$		0			0
$793$	25569.2			1.14500
$794$		0			0
$795$	3236.30	−	7844.50i	0.144377	−	0.349957i
$796$		0			0
$797$	−11553.8			−0.513497			−0.256748	−	0.966478i	$-0.582651\pi$
$797$	−11553.8			−0.513497			−0.256748	+	0.966478i	$0.582651\pi$
$798$		0			0
$799$			5412.47i			0.239649i
$800$		0			0
$801$	−16256.9	+	16349.2i	−0.717116	+	0.721187i
$802$		0			0
$803$		−	57338.0i		−	2.51982i
$804$		0			0
$805$		−	6721.12i		−	0.294271i
$806$		0			0
$807$	1128.85	−	2736.22i	0.0492408	−	0.119355i
$808$		0			0
$809$			23754.7i			1.03235i	0.856483	+	0.516175i	$0.172645\pi$
$809$			23754.7i			1.03235i	−0.856483	+	0.516175i	$0.827355\pi$
$810$		0			0
$811$	8303.89			0.359543			0.179771	−	0.983708i	$-0.442464\pi$
$811$	8303.89			0.359543			0.179771	+	0.983708i	$0.442464\pi$
$812$		0			0
$813$	32322.7	+	13334.9i	1.39435	+	0.575248i
$814$		0			0
$815$	−11599.1			−0.498526
$816$		0			0
$817$	17470.5			0.748120
$818$		0			0
$819$	31000.9	+	30825.9i	1.32266	+	1.31519i
$820$		0			0
$821$	−38021.6			−1.61628			−0.808139	−	0.588992i	$-0.799525\pi$
$821$	−38021.6			−1.61628			−0.808139	+	0.588992i	$0.799525\pi$
$822$		0			0
$823$			16842.2i			0.713345i	0.934230	+	0.356672i	$0.116089\pi$
$823$			16842.2i			0.713345i	−0.934230	+	0.356672i	$0.883911\pi$
$824$		0			0
$825$	−5543.25	−	2286.90i	−0.233929	−	0.0965088i
$826$		0			0
$827$			17470.7i			0.734600i	0.930102	+	0.367300i	$0.119718\pi$
$827$			17470.7i			0.734600i	−0.930102	+	0.367300i	$0.880282\pi$
$828$		0			0
$829$			19906.7i			0.834001i	0.908906	+	0.417001i	$0.136919\pi$
$829$			19906.7i			0.834001i	−0.908906	+	0.417001i	$0.863081\pi$
$830$		0			0
$831$	14671.0	+	6052.60i	0.612430	+	0.252662i
$832$		0			0
$833$		−	12804.8i		−	0.532607i
$834$		0			0
$835$	793.604			0.0328908
$836$		0			0
$837$	7071.49	+	16869.2i	0.292027	+	0.696635i
$838$		0			0
$839$	−18413.2			−0.757682			−0.378841	−	0.925462i	$-0.623677\pi$
$839$	−18413.2			−0.757682			−0.378841	+	0.925462i	$0.623677\pi$
$840$		0			0
$841$	−17310.6			−0.709772
$842$		0			0
$843$	−3235.94	−	1335.01i	−0.132208	−	0.0545435i
$844$		0			0
$845$	34650.1			1.41065
$846$		0			0
$847$		−	56878.0i		−	2.30738i
$848$		0			0
$849$	3908.39	−	9473.59i	0.157992	−	0.382960i
$850$		0			0
$851$		−	9819.81i		−	0.395557i
$852$		0			0
$853$		−	23193.1i		−	0.930969i	−0.885056	−	0.465485i	$-0.845880\pi$
$853$		−	23193.1i		−	0.930969i	0.885056	−	0.465485i	$-0.154120\pi$
$854$		0			0
$855$	24798.6	+	24658.6i	0.991922	+	0.986323i
$856$		0			0
$857$			16140.3i			0.643341i	0.946852	+	0.321671i	$0.104244\pi$
$857$			16140.3i			0.643341i	−0.946852	+	0.321671i	$0.895756\pi$
$858$		0			0
$859$	−2325.65			−0.0923751			−0.0461875	−	0.998933i	$-0.514707\pi$
$859$	−2325.65			−0.0923751			−0.0461875	+	0.998933i	$0.514707\pi$
$860$		0			0
$861$	4684.19	−	11354.0i	0.185408	−	0.449413i
$862$		0			0
$863$	8816.27			0.347751			0.173876	−	0.984768i	$-0.444371\pi$
$863$	8816.27			0.347751			0.173876	+	0.984768i	$0.444371\pi$
$864$		0			0
$865$	−36690.4			−1.44221
$866$		0			0
$867$	−1212.85	+	2939.83i	−0.0475092	+	0.115158i
$868$		0			0
$869$	−3466.09			−0.135304
$870$		0			0
$871$			32064.2i			1.24736i
$872$		0			0
$873$	13688.9	+	13611.7i	0.530699	+	0.527703i
$874$		0			0
$875$			28937.2i			1.11801i
$876$		0			0
$877$			14921.3i			0.574524i	0.957852	+	0.287262i	$0.0927451\pi$
$877$			14921.3i			0.574524i	−0.957852	+	0.287262i	$0.907255\pi$
$878$		0			0
$879$	−7215.34	+	17489.3i	−0.276869	+	0.671105i
$880$		0			0
$881$			16861.0i			0.644793i	0.946605	+	0.322397i	$0.104488\pi$
$881$			16861.0i			0.644793i	−0.946605	+	0.322397i	$0.895512\pi$
$882$		0			0
$883$	14458.8			0.551049			0.275524	−	0.961294i	$-0.411149\pi$
$883$	14458.8			0.551049			0.275524	+	0.961294i	$0.411149\pi$
$884$		0			0
$885$	−14015.6	−	5782.23i	−0.532350	−	0.219624i
$886$		0			0
$887$	−19882.3			−0.752629			−0.376314	−	0.926492i	$-0.622809\pi$
$887$	−19882.3			−0.752629			−0.376314	+	0.926492i	$0.622809\pi$
$888$		0			0
$889$	−30552.0			−1.15262
$890$		0			0
$891$	45154.3	−	255.598i	1.69778	−	0.00961038i
$892$		0			0
$893$	7869.67			0.294903
$894$		0			0
$895$		−	11376.7i		−	0.424895i
$896$		0			0
$897$	8465.15	+	3492.35i	0.315098	+	0.129996i
$898$		0			0
$899$		−	10969.0i		−	0.406938i
$900$		0			0
$901$		−	10128.8i		−	0.374516i
$902$		0			0
$903$	17625.6	+	7271.57i	0.649550	+	0.267976i
$904$		0			0
$905$		−	10485.6i		−	0.385142i
$906$		0			0
$907$	−29281.5			−1.07197			−0.535985	−	0.844227i	$-0.680060\pi$
$907$	−29281.5			−1.07197			−0.535985	+	0.844227i	$0.680060\pi$
$908$		0			0
$909$	−32599.3	−	32415.3i	−1.18950	−	1.18278i
$910$		0			0
$911$	37513.5			1.36430			0.682150	−	0.731212i	$-0.261045\pi$
$911$	37513.5			1.36430			0.682150	+	0.731212i	$0.261045\pi$
$912$		0			0
$913$	−57493.3			−2.08406
$914$		0			0
$915$	20635.2	+	8513.20i	0.745551	+	0.307582i
$916$		0			0
$917$	−15425.1			−0.555485
$918$		0			0
$919$			22016.4i			0.790266i	0.918624	+	0.395133i	$0.129302\pi$
$919$			22016.4i			0.790266i	−0.918624	+	0.395133i	$0.870698\pi$
$920$		0			0
$921$	5642.69	−	13677.4i	0.201882	−	0.489343i
$922$		0			0
$923$		−	23471.9i		−	0.837041i
$924$		0			0
$925$		−	7405.20i		−	0.263223i
$926$		0			0
$927$	−12487.4	+	12558.3i	−0.442440	+	0.444951i
$928$		0			0
$929$		−	37944.4i		−	1.34006i	−0.742334	−	0.670030i	$-0.766281\pi$
$929$		−	37944.4i		−	1.34006i	0.742334	−	0.670030i	$-0.233719\pi$
$930$		0			0
$931$	−18618.1			−0.655407
$932$		0			0
$933$	−17408.5	+	42196.6i	−0.610856	+	1.48066i
$934$		0			0
$935$	−55178.5			−1.92998
$936$		0			0
$937$	−30773.5			−1.07292			−0.536460	−	0.843925i	$-0.680239\pi$
$937$	−30773.5			−1.07292			−0.536460	+	0.843925i	$0.680239\pi$
$938$		0			0
$939$	9492.69	−	23009.4i	0.329906	−	0.799663i
$940$		0			0
$941$	−4127.42			−0.142986			−0.0714931	−	0.997441i	$-0.522776\pi$
$941$	−4127.42			−0.142986			−0.0714931	+	0.997441i	$0.522776\pi$
$942$		0			0
$943$		−	2572.66i		−	0.0888414i
$944$		0			0
$945$	14755.4	+	35199.3i	0.507929	+	1.21167i
$946$		0			0
$947$		−	11313.0i		−	0.388196i	−0.980982	−	0.194098i	$-0.937822\pi$
$947$		−	11313.0i		−	0.388196i	0.980982	−	0.194098i	$-0.0621780\pi$
$948$		0			0
$949$			66030.9i			2.25864i
$950$		0			0
$951$	12663.7	−	30695.5i	0.431805	−	1.04666i
$952$		0			0
$953$		−	25633.6i		−	0.871304i	−0.900115	−	0.435652i	$-0.856518\pi$
$953$		−	25633.6i		−	0.871304i	0.900115	−	0.435652i	$-0.143482\pi$
$954$		0			0
$955$	−8187.67			−0.277431
$956$		0			0
$957$	−25032.1	−	10327.1i	−0.845529	−	0.348829i
$958$		0			0
$959$	13948.6			0.469679
$960$		0			0
$961$	12792.9			0.429420
$962$		0			0
$963$	3845.78	−	3867.61i	0.128690	−	0.129421i
$964$		0			0
$965$	−36311.7			−1.21131
$966$		0			0
$967$		−	50706.5i		−	1.68626i	−0.537712	−	0.843129i	$-0.680711\pi$
$967$		−	50706.5i		−	1.68626i	0.537712	−	0.843129i	$-0.319289\pi$
$968$		0			0
$969$	38587.5	+	15919.5i	1.27927	+	0.527770i
$970$		0			0
$971$		−	48466.7i		−	1.60182i	−0.598782	−	0.800912i	$-0.704348\pi$
$971$		−	48466.7i		−	1.60182i	0.598782	−	0.800912i	$-0.295652\pi$
$972$		0			0
$973$			71818.8i			2.36630i
$974$		0			0
$975$	6383.64	+	2633.61i	0.209682	+	0.0865058i
$976$		0			0
$977$			33563.4i			1.09907i	0.835472	+	0.549533i	$0.185194\pi$
$977$			33563.4i			1.09907i	−0.835472	+	0.549533i	$0.814806\pi$
$978$		0			0
$979$	−52893.3			−1.72674
$980$		0			0
$981$	11208.0	−	11271.6i	0.364774	−	0.366845i
$982$		0			0
$983$	−25587.6			−0.830231			−0.415115	−	0.909769i	$-0.636259\pi$
$983$	−25587.6			−0.830231			−0.415115	+	0.909769i	$0.636259\pi$
$984$		0			0
$985$	31895.0			1.03173
$986$		0			0
$987$	7939.56	+	3275.52i	0.256048	+	0.105634i
$988$		0			0
$989$	3993.71			0.128405
$990$		0			0
$991$			22594.0i			0.724239i	0.932132	+	0.362119i	$0.117947\pi$
$991$			22594.0i			0.724239i	−0.932132	+	0.362119i	$0.882053\pi$
$992$		0			0
$993$	−6881.35	+	16679.8i	−0.219912	+	0.533047i
$994$		0			0
$995$		−	35658.7i		−	1.13614i
$996$		0			0
$997$			5714.27i			0.181517i	0.995873	+	0.0907586i	$0.0289292\pi$
$997$			5714.27i			0.181517i	−0.995873	+	0.0907586i	$0.971071\pi$
$998$		0			0
$999$	21558.2	+	51427.5i	0.682754	+	1.62872i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.98169 - 4.80343i) q^{3} -11.9846 q^{5} +22.6995i q^{7} +(-19.1458 - 19.0378i) q^{9} +O(q^{10})\)	\(q+(1.98169 - 4.80343i) q^{3} -11.9846 q^{5} +22.6995i q^{7} +(-19.1458 - 19.0378i) q^{9} -61.9410i q^{11} +71.3317i q^{13} +(-23.7497 + 57.5672i) q^{15} +74.3305i q^{17} +108.076 q^{19} +(109.036 + 44.9834i) q^{21} +24.7059 q^{23} +18.6309 q^{25} +(-129.388 + 54.2388i) q^{27} +84.1331 q^{29} -130.377i q^{31} +(-297.529 - 122.748i) q^{33} -272.045i q^{35} -397.468i q^{37} +(342.637 + 141.357i) q^{39} -104.132i q^{41} +161.650 q^{43} +(229.455 + 228.160i) q^{45} +72.8162 q^{47} -172.269 q^{49} +(357.041 + 147.300i) q^{51} -136.267 q^{53} +742.339i q^{55} +(214.172 - 519.134i) q^{57} +243.465i q^{59} -358.454i q^{61} +(432.149 - 434.602i) q^{63} -854.883i q^{65} +449.508 q^{67} +(48.9593 - 118.673i) q^{69} -329.053 q^{71} +925.687 q^{73} +(36.9207 - 89.4923i) q^{75} +1406.03 q^{77} -55.9579i q^{79} +(4.12647 + 728.988i) q^{81} -928.194i q^{83} -890.823i q^{85} +(166.725 - 404.127i) q^{87} -853.930i q^{89} -1619.20 q^{91} +(-626.256 - 258.366i) q^{93} -1295.25 q^{95} -714.982 q^{97} +(-1179.22 + 1185.91i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} + 12 q^{5}+O(q^{10}) \) 12 * q + 2 * q^3 + 12 * q^5	\( 12 q + 2 q^{3} + 12 q^{5} - 84 q^{15} + 180 q^{19} + 156 q^{21} - 120 q^{23} + 300 q^{25} - 130 q^{27} + 588 q^{29} - 116 q^{33} + 620 q^{39} - 372 q^{43} - 740 q^{45} + 1248 q^{47} - 948 q^{49} - 360 q^{51} - 948 q^{53} + 172 q^{57} - 2744 q^{63} + 2292 q^{67} + 3280 q^{69} - 2040 q^{71} + 216 q^{73} - 2522 q^{75} + 4824 q^{77} - 1076 q^{81} + 4156 q^{87} + 3480 q^{91} - 4180 q^{93} + 5448 q^{95} - 48 q^{97} - 3048 q^{99}+O(q^{100}) \) 12 * q + 2 * q^3 + 12 * q^5 - 84 * q^15 + 180 * q^19 + 156 * q^21 - 120 * q^23 + 300 * q^25 - 130 * q^27 + 588 * q^29 - 116 * q^33 + 620 * q^39 - 372 * q^43 - 740 * q^45 + 1248 * q^47 - 948 * q^49 - 360 * q^51 - 948 * q^53 + 172 * q^57 - 2744 * q^63 + 2292 * q^67 + 3280 * q^69 - 2040 * q^71 + 216 * q^73 - 2522 * q^75 + 4824 * q^77 - 1076 * q^81 + 4156 * q^87 + 3480 * q^91 - 4180 * q^93 + 5448 * q^95 - 48 * q^97 - 3048 * q^99

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