LMFDB - Embedded newform 768.4.c.u.767.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(767,768)]

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, 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("768.767");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1731891456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			767.3
Root			$-1.35234 + 0.780776i$ of defining polynomial
Character	$\chi$	$=$	768.767
Dual form			768.4.c.u.767.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.70469 - 4.43674i) q^{3} +19.2658i q^{5} +13.3693i q^{7} +(-12.3693 + 24.0000i) q^{9} +O(q^{10})$	\(q+(-2.70469 - 4.43674i) q^{3} +19.2658i q^{5} +13.3693i q^{7} +(-12.3693 + 24.0000i) q^{9} -22.3034 q^{11} -39.3845 q^{13} +(85.4773 - 52.1080i) q^{15} +122.955i q^{17} +107.574i q^{19} +(59.3162 - 36.1598i) q^{21} +21.0455 q^{23} -246.170 q^{25} +(139.937 - 10.0331i) q^{27} -72.0277i q^{29} -203.278i q^{31} +(60.3239 + 98.9545i) q^{33} -257.570 q^{35} -205.661 q^{37} +(106.523 + 174.739i) q^{39} +21.0455i q^{41} -276.036i q^{43} +(-462.379 - 238.305i) q^{45} +533.909 q^{47} +164.261 q^{49} +(545.517 - 332.554i) q^{51} -240.384i q^{53} -429.693i q^{55} +(477.278 - 290.955i) q^{57} -478.607 q^{59} +33.5192 q^{61} +(-320.864 - 165.369i) q^{63} -758.773i q^{65} +261.431i q^{67} +(-56.9214 - 93.3732i) q^{69} +470.773 q^{71} +354.250 q^{73} +(665.814 + 1092.19i) q^{75} -298.182i q^{77} +1122.27i q^{79} +(-423.000 - 593.727i) q^{81} -590.042 q^{83} -2368.82 q^{85} +(-319.568 + 194.813i) q^{87} -516.182i q^{89} -526.543i q^{91} +(-901.893 + 549.805i) q^{93} -2072.50 q^{95} -272.989 q^{97} +(275.878 - 535.282i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 288 q^{15} + 960 q^{23} - 584 q^{25} - 408 q^{33} + 1248 q^{39} + 2688 q^{47} + 1512 q^{49} + 2136 q^{57} - 192 q^{63} - 192 q^{71} - 1520 q^{73} - 3384 q^{81} - 3744 q^{87} - 7872 q^{95} + 2368 q^{97}+O(q^{100}) $ 8 * q + 288 * q^15 + 960 * q^23 - 584 * q^25 - 408 * q^33 + 1248 * q^39 + 2688 * q^47 + 1512 * q^49 + 2136 * q^57 - 192 * q^63 - 192 * q^71 - 1520 * q^73 - 3384 * q^81 - 3744 * q^87 - 7872 * q^95 + 2368 * q^97

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
$3$	−2.70469	−	4.43674i	−0.520518	−	0.853851i
$3$	−2.70469	−	4.43674i	−0.520518	−	0.853851i
$4$		0			0
$5$			19.2658i			1.72318i	0.507601	+	0.861592i	$0.330532\pi$
$5$			19.2658i			1.72318i	−0.507601	+	0.861592i	$0.669468\pi$
$6$		0			0
$7$			13.3693i			0.721875i	0.932590	+	0.360938i	$0.117543\pi$
$7$			13.3693i			0.721875i	−0.932590	+	0.360938i	$0.882457\pi$
$8$		0			0
$9$	−12.3693	+	24.0000i	−0.458123	+	0.888889i
$10$		0			0
$11$	−22.3034			−0.611340			−0.305670	−	0.952138i	$-0.598880\pi$
$11$	−22.3034			−0.611340			−0.305670	+	0.952138i	$0.598880\pi$
$12$		0			0
$13$	−39.3845			−0.840253			−0.420126	−	0.907466i	$-0.638014\pi$
$13$	−39.3845			−0.840253			−0.420126	+	0.907466i	$0.638014\pi$
$14$		0			0
$15$	85.4773	−	52.1080i	1.47134	−	0.896948i
$16$		0			0
$17$			122.955i			1.75417i	0.480337	+	0.877084i	$0.340514\pi$
$17$			122.955i			1.75417i	−0.480337	+	0.877084i	$0.659486\pi$
$18$		0			0
$19$			107.574i			1.29891i	0.760402	+	0.649453i	$0.225002\pi$
$19$			107.574i			1.29891i	−0.760402	+	0.649453i	$0.774998\pi$
$20$		0			0
$21$	59.3162	−	36.1598i	0.616374	−	0.375749i
$22$		0			0
$23$	21.0455			0.190795			0.0953975	−	0.995439i	$-0.469588\pi$
$23$	21.0455			0.190795			0.0953975	+	0.995439i	$0.469588\pi$
$24$		0			0
$25$	−246.170			−1.96936
$26$		0			0
$27$	139.937	−	10.0331i	0.997440	−	0.0715137i
$28$		0			0
$29$		−	72.0277i		−	0.461214i	−0.973047	−	0.230607i	$-0.925929\pi$
$29$		−	72.0277i		−	0.461214i	0.973047	−	0.230607i	$-0.0740712\pi$
$30$		0			0
$31$		−	203.278i		−	1.17774i	−0.808229	−	0.588869i	$-0.799573\pi$
$31$		−	203.278i		−	1.17774i	0.808229	−	0.588869i	$-0.200427\pi$
$32$		0			0
$33$	60.3239	+	98.9545i	0.318213	+	0.521993i
$34$		0			0
$35$	−257.570			−1.24392
$36$		0			0
$37$	−205.661			−0.913798			−0.456899	−	0.889519i	$-0.651040\pi$
$37$	−205.661			−0.913798			−0.456899	+	0.889519i	$0.651040\pi$
$38$		0			0
$39$	106.523	+	174.739i	0.437366	+	0.717451i
$40$		0			0
$41$			21.0455i			0.0801646i	0.999196	+	0.0400823i	$0.0127620\pi$
$41$			21.0455i			0.0801646i	−0.999196	+	0.0400823i	$0.987238\pi$
$42$		0			0
$43$		−	276.036i		−	0.978955i	−0.872016	−	0.489477i	$-0.837188\pi$
$43$		−	276.036i		−	0.978955i	0.872016	−	0.489477i	$-0.162812\pi$
$44$		0			0
$45$	−462.379	−	238.305i	−1.53172	−	0.789430i
$46$		0			0
$47$	533.909			1.65699			0.828496	−	0.559995i	$-0.189197\pi$
$47$	533.909			1.65699			0.828496	+	0.559995i	$0.189197\pi$
$48$		0			0
$49$	164.261			0.478896
$50$		0			0
$51$	545.517	−	332.554i	1.49780	−	0.913075i
$52$		0			0
$53$		−	240.384i		−	0.623006i	−0.950245	−	0.311503i	$-0.899168\pi$
$53$		−	240.384i		−	0.623006i	0.950245	−	0.311503i	$-0.100832\pi$
$54$		0			0
$55$		−	429.693i		−	1.05345i
$56$		0			0
$57$	477.278	−	290.955i	1.10907	−	0.676103i
$58$		0			0
$59$	−478.607			−1.05609			−0.528045	−	0.849216i	$-0.677075\pi$
$59$	−478.607			−1.05609			−0.528045	+	0.849216i	$0.677075\pi$
$60$		0			0
$61$	33.5192			0.0703556			0.0351778	−	0.999381i	$-0.488800\pi$
$61$	33.5192			0.0703556			0.0351778	+	0.999381i	$0.488800\pi$
$62$		0			0
$63$	−320.864	−	165.369i	−0.641667	−	0.330708i
$64$		0			0
$65$		−	758.773i		−	1.44791i
$66$		0			0
$67$			261.431i			0.476701i	0.971179	+	0.238350i	$0.0766066\pi$
$67$			261.431i			0.476701i	−0.971179	+	0.238350i	$0.923393\pi$
$68$		0			0
$69$	−56.9214	−	93.3732i	−0.0993121	−	0.162910i
$70$		0			0
$71$	470.773			0.786908			0.393454	−	0.919344i	$-0.371280\pi$
$71$	470.773			0.786908			0.393454	+	0.919344i	$0.371280\pi$
$72$		0			0
$73$	354.250			0.567970			0.283985	−	0.958829i	$-0.408343\pi$
$73$	354.250			0.567970			0.283985	+	0.958829i	$0.408343\pi$
$74$		0			0
$75$	665.814	+	1092.19i	1.02509	+	1.68154i
$76$		0			0
$77$		−	298.182i		−	0.441311i
$78$		0			0
$79$			1122.27i			1.59829i	0.601138	+	0.799145i	$0.294714\pi$
$79$			1122.27i			1.59829i	−0.601138	+	0.799145i	$0.705286\pi$
$80$		0			0
$81$	−423.000	−	593.727i	−0.580247	−	0.814441i
$82$		0			0
$83$	−590.042			−0.780308			−0.390154	−	0.920750i	$-0.627578\pi$
$83$	−590.042			−0.780308			−0.390154	+	0.920750i	$0.627578\pi$
$84$		0			0
$85$	−2368.82			−3.02275
$86$		0			0
$87$	−319.568	+	194.813i	−0.393808	+	0.240070i
$88$		0			0
$89$		−	516.182i		−	0.614777i	−0.951584	−	0.307389i	$-0.900545\pi$
$89$		−	516.182i		−	0.614777i	0.951584	−	0.307389i	$-0.0994551\pi$
$90$		0			0
$91$		−	526.543i		−	0.606558i
$92$		0			0
$93$	−901.893	+	549.805i	−1.00561	+	0.613033i
$94$		0			0
$95$	−2072.50			−2.23825
$96$		0			0
$97$	−272.989			−0.285750			−0.142875	−	0.989741i	$-0.545635\pi$
$97$	−272.989			−0.285750			−0.142875	+	0.989741i	$0.545635\pi$
$98$		0			0
$99$	275.878	−	535.282i	0.280069	−	0.543413i
$100$		0			0
$101$		−	1351.89i		−	1.33186i	−0.746014	−	0.665930i	$-0.768035\pi$
$101$		−	1351.89i		−	1.33186i	0.746014	−	0.665930i	$-0.231965\pi$
$102$		0			0
$103$		−	1176.54i		−	1.12551i	−0.826622	−	0.562757i	$-0.809741\pi$
$103$		−	1176.54i		−	1.12551i	0.826622	−	0.562757i	$-0.190259\pi$
$104$		0			0
$105$	696.648	+	1142.77i	0.647484	+	1.06213i
$106$		0			0
$107$	−1342.61			−1.21304			−0.606519	−	0.795069i	$-0.707435\pi$
$107$	−1342.61			−1.21304			−0.606519	+	0.795069i	$0.707435\pi$
$108$		0			0
$109$	−660.797			−0.580669			−0.290334	−	0.956925i	$-0.593766\pi$
$109$	−660.797			−0.580669			−0.290334	+	0.956925i	$0.593766\pi$
$110$		0			0
$111$	556.250	+	912.466i	0.475648	+	0.780247i
$112$		0			0
$113$			1355.82i			1.12871i	0.825531	+	0.564357i	$0.190876\pi$
$113$			1355.82i			1.12871i	−0.825531	+	0.564357i	$0.809124\pi$
$114$		0			0
$115$			405.457i			0.328775i
$116$		0			0
$117$	487.159	−	945.227i	0.384939	−	0.746891i
$118$		0			0
$119$	−1643.82			−1.26629
$120$		0			0
$121$	−833.557			−0.626264
$122$		0			0
$123$	93.3732	−	56.9214i	0.0684486	−	0.0417271i
$124$		0			0
$125$		−	2334.44i		−	1.67039i
$126$		0			0
$127$		−	1126.23i		−	0.786906i	−0.919345	−	0.393453i	$-0.871280\pi$
$127$		−	1126.23i		−	0.786906i	0.919345	−	0.393453i	$-0.128720\pi$
$128$		0			0
$129$	−1224.70	+	746.591i	−0.835882	+	0.509563i
$130$		0			0
$131$	677.176			0.451642			0.225821	−	0.974169i	$-0.427493\pi$
$131$	677.176			0.451642			0.225821	+	0.974169i	$0.427493\pi$
$132$		0			0
$133$	−1438.19			−0.937647
$134$		0			0
$135$	193.295	+	2695.99i	0.123231	+	1.71877i
$136$		0			0
$137$			2570.95i			1.60330i	0.597797	+	0.801648i	$0.296043\pi$
$137$			2570.95i			1.60330i	−0.597797	+	0.801648i	$0.703957\pi$
$138$		0			0
$139$			324.218i			0.197840i	0.995095	+	0.0989202i	$0.0315389\pi$
$139$			324.218i			0.197840i	−0.995095	+	0.0989202i	$0.968461\pi$
$140$		0			0
$141$	−1444.06	−	2368.82i	−0.862494	−	1.41482i
$142$		0			0
$143$	878.409			0.513680
$144$		0			0
$145$	1387.67			0.794757
$146$		0			0
$147$	−444.276	−	728.785i	−0.249274	−	0.408906i
$148$		0			0
$149$			270.597i			0.148780i	0.997229	+	0.0743898i	$0.0237009\pi$
$149$			270.597i			0.148780i	−0.997229	+	0.0743898i	$0.976299\pi$
$150$		0			0
$151$			786.551i			0.423898i	0.977281	+	0.211949i	$0.0679811\pi$
$151$			786.551i			0.423898i	−0.977281	+	0.211949i	$0.932019\pi$
$152$		0			0
$153$	−2950.91	−	1520.86i	−1.55926	−	0.803624i
$154$		0			0
$155$	3916.32			2.02946
$156$		0			0
$157$	−2437.41			−1.23902			−0.619511	−	0.784988i	$-0.712669\pi$
$157$	−2437.41			−1.23902			−0.619511	+	0.784988i	$0.712669\pi$
$158$		0			0
$159$	−1066.52	+	650.165i	−0.531954	+	0.324286i
$160$		0			0
$161$			281.363i			0.137730i
$162$		0			0
$163$			1490.46i			0.716207i	0.933682	+	0.358104i	$0.116577\pi$
$163$			1490.46i			0.716207i	−0.933682	+	0.358104i	$0.883423\pi$
$164$		0			0
$165$	−1906.44	+	1162.19i	−0.899490	+	0.548340i
$166$		0			0
$167$	−836.318			−0.387522			−0.193761	−	0.981049i	$-0.562069\pi$
$167$	−836.318			−0.387522			−0.193761	+	0.981049i	$0.562069\pi$
$168$		0			0
$169$	−645.864			−0.293975
$170$		0			0
$171$	−2581.78	−	1330.62i	−1.15458	−	0.595058i
$172$		0			0
$173$			3172.29i			1.39413i	0.717008	+	0.697065i	$0.245511\pi$
$173$			3172.29i			1.39413i	−0.717008	+	0.697065i	$0.754489\pi$
$174$		0			0
$175$		−	3291.13i		−	1.42163i
$176$		0			0
$177$	1294.48	+	2123.45i	0.549714	+	0.901744i
$178$		0			0
$179$	519.218			0.216806			0.108403	−	0.994107i	$-0.465426\pi$
$179$	519.218			0.216806			0.108403	+	0.994107i	$0.465426\pi$
$180$		0			0
$181$	2381.87			0.978136			0.489068	−	0.872246i	$-0.337337\pi$
$181$	2381.87			0.978136			0.489068	+	0.872246i	$0.337337\pi$
$182$		0			0
$183$	−90.6589	−	148.716i	−0.0366213	−	0.0600732i
$184$		0			0
$185$		−	3962.23i		−	1.57464i
$186$		0			0
$187$		−	2742.31i		−	1.07239i
$188$		0			0
$189$	134.136	+	1870.86i	0.0516240	+	0.720027i
$190$		0			0
$191$	4678.91			1.77253			0.886267	−	0.463175i	$-0.153290\pi$
$191$	4678.91			1.77253			0.886267	+	0.463175i	$0.153290\pi$
$192$		0			0
$193$	1217.70			0.454157			0.227078	−	0.973876i	$-0.427083\pi$
$193$	1217.70			0.454157			0.227078	+	0.973876i	$0.427083\pi$
$194$		0			0
$195$	−3366.48	+	2052.24i	−1.23630	+	0.753663i
$196$		0			0
$197$		−	467.414i		−	0.169045i	−0.996422	−	0.0845226i	$-0.973063\pi$
$197$		−	467.414i		−	0.169045i	0.996422	−	0.0845226i	$-0.0269365\pi$
$198$		0			0
$199$		−	1761.04i		−	0.627321i	−0.949535	−	0.313660i	$-0.898445\pi$
$199$		−	1761.04i		−	0.627321i	0.949535	−	0.313660i	$-0.101555\pi$
$200$		0			0
$201$	1159.90	−	707.091i	0.407031	−	0.248131i
$202$		0			0
$203$	962.961			0.332939
$204$		0			0
$205$	−405.457			−0.138138
$206$		0			0
$207$	−260.318	+	505.091i	−0.0874075	+	0.169595i
$208$		0			0
$209$		−	2399.27i		−	0.794073i
$210$		0			0
$211$		−	3262.58i		−	1.06448i	−0.846593	−	0.532241i	$-0.821350\pi$
$211$		−	3262.58i		−	1.06448i	0.846593	−	0.532241i	$-0.178650\pi$
$212$		0			0
$213$	−1273.29	−	2088.70i	−0.409599	−	0.671902i
$214$		0			0
$215$	5318.05			1.68692
$216$		0			0
$217$	2717.69			0.850180
$218$		0			0
$219$	−958.136	−	1571.71i	−0.295638	−	0.484962i
$220$		0			0
$221$		−	4842.50i		−	1.47394i
$222$		0			0
$223$		−	3682.05i		−	1.10569i	−0.833285	−	0.552844i	$-0.813543\pi$
$223$		−	3682.05i		−	1.10569i	0.833285	−	0.552844i	$-0.186457\pi$
$224$		0			0
$225$	3044.96	−	5908.09i	0.902210	−	1.75055i
$226$		0			0
$227$	3149.27			0.920813			0.460406	−	0.887708i	$-0.347704\pi$
$227$	3149.27			0.920813			0.460406	+	0.887708i	$0.347704\pi$
$228$		0			0
$229$	1022.44			0.295043			0.147521	−	0.989059i	$-0.452870\pi$
$229$	1022.44			0.295043			0.147521	+	0.989059i	$0.452870\pi$
$230$		0			0
$231$	−1322.95	+	806.489i	−0.376814	+	0.229710i
$232$		0			0
$233$		−	965.909i		−	0.271583i	−0.990737	−	0.135791i	$-0.956642\pi$
$233$		−	965.909i		−	0.271583i	0.990737	−	0.135791i	$-0.0433577\pi$
$234$		0			0
$235$			10286.2i			2.85530i
$236$		0			0
$237$	4979.21	−	3035.38i	1.36470	−	0.831938i
$238$		0			0
$239$	−4215.91			−1.14102			−0.570511	−	0.821290i	$-0.693255\pi$
$239$	−4215.91			−1.14102			−0.570511	+	0.821290i	$0.693255\pi$
$240$		0			0
$241$	−2468.48			−0.659787			−0.329893	−	0.944018i	$-0.607013\pi$
$241$	−2468.48			−0.659787			−0.329893	+	0.944018i	$0.607013\pi$
$242$		0			0
$243$	−1490.13	+	3482.59i	−0.393382	+	0.919375i
$244$		0			0
$245$			3164.62i			0.825226i
$246$		0			0
$247$		−	4236.75i		−	1.09141i
$248$		0			0
$249$	1595.88	+	2617.86i	0.406164	+	0.666266i
$250$		0			0
$251$	−991.176			−0.249253			−0.124626	−	0.992204i	$-0.539773\pi$
$251$	−991.176			−0.249253			−0.124626	+	0.992204i	$0.539773\pi$
$252$		0			0
$253$	−469.386			−0.116641
$254$		0			0
$255$	6406.91	+	10509.8i	1.57340	+	2.58098i
$256$		0			0
$257$			6.63651i			0.00161079i	1.00000		0.000805397i	$0.000256366\pi$
$257$			6.63651i			0.00161079i	−1.00000		0.000805397i	$0.999744\pi$
$258$		0			0
$259$		−	2749.55i		−	0.659648i
$260$		0			0
$261$	1728.67	+	890.934i	0.409968	+	0.211293i
$262$		0			0
$263$	1454.41			0.340999			0.170499	−	0.985358i	$-0.445462\pi$
$263$	1454.41			0.340999			0.170499	+	0.985358i	$0.445462\pi$
$264$		0			0
$265$	4631.19			1.07355
$266$		0			0
$267$	−2290.16	+	1396.11i	−0.524928	+	0.320002i
$268$		0			0
$269$			6524.09i			1.47874i	0.673299	+	0.739370i	$0.264876\pi$
$269$			6524.09i			1.47874i	−0.673299	+	0.739370i	$0.735124\pi$
$270$		0			0
$271$		−	4945.91i		−	1.10865i	−0.832302	−	0.554323i	$-0.812977\pi$
$271$		−	4945.91i		−	1.10865i	0.832302	−	0.554323i	$-0.187023\pi$
$272$		0			0
$273$	−2336.14	+	1424.14i	−0.517910	+	0.315724i
$274$		0			0
$275$	5490.45			1.20395
$276$		0			0
$277$	3172.55			0.688158			0.344079	−	0.938941i	$-0.388191\pi$
$277$	3172.55			0.688158			0.344079	+	0.938941i	$0.388191\pi$
$278$		0			0
$279$	4878.68	+	2514.41i	1.04688	+	0.539549i
$280$		0			0
$281$			3256.64i			0.691369i	0.938351	+	0.345684i	$0.112353\pi$
$281$			3256.64i			0.691369i	−0.938351	+	0.345684i	$0.887647\pi$
$282$		0			0
$283$		−	5504.68i		−	1.15625i	−0.815947	−	0.578126i	$-0.803784\pi$
$283$		−	5504.68i		−	1.15625i	0.815947	−	0.578126i	$-0.196216\pi$
$284$		0			0
$285$	5605.47	+	9195.14i	1.16505	+	1.91113i
$286$		0			0
$287$	−281.363			−0.0578689
$288$		0			0
$289$	−10204.8			−2.07711
$290$		0			0
$291$	738.349	+	1211.18i	0.148738	+	0.243988i
$292$		0			0
$293$		−	3257.01i		−	0.649409i	−0.945816	−	0.324704i	$-0.894735\pi$
$293$		−	3257.01i		−	0.649409i	0.945816	−	0.324704i	$-0.105265\pi$
$294$		0			0
$295$		−	9220.74i		−	1.81984i
$296$		0			0
$297$	−3121.07	+	223.772i	−0.609775	+	0.0437192i
$298$		0			0
$299$	−828.864			−0.160316
$300$		0			0
$301$	3690.41			0.706683
$302$		0			0
$303$	−5997.98	+	3656.44i	−1.13721	+	0.693257i
$304$		0			0
$305$			645.773i			0.121236i
$306$		0			0
$307$		−	7749.07i		−	1.44059i	−0.693665	−	0.720297i	$-0.744005\pi$
$307$		−	7749.07i		−	1.44059i	0.693665	−	0.720297i	$-0.255995\pi$
$308$		0			0
$309$	−5220.00	+	3182.17i	−0.961021	+	0.585850i
$310$		0			0
$311$	−4910.41			−0.895318			−0.447659	−	0.894204i	$-0.647742\pi$
$311$	−4910.41			−0.895318			−0.447659	+	0.894204i	$0.647742\pi$
$312$		0			0
$313$	−4879.35			−0.881141			−0.440571	−	0.897718i	$-0.645224\pi$
$313$	−4879.35			−0.881141			−0.440571	+	0.897718i	$0.645224\pi$
$314$		0			0
$315$	3185.97	−	6181.69i	0.569870	−	1.10571i
$316$		0			0
$317$			1372.69i			0.243210i	0.992579	+	0.121605i	$0.0388042\pi$
$317$			1372.69i			0.243210i	−0.992579	+	0.121605i	$0.961196\pi$
$318$		0			0
$319$			1606.47i			0.281959i
$320$		0			0
$321$	3631.35	+	5956.82i	0.631408	+	1.03575i
$322$		0			0
$323$	−13226.7			−2.27850
$324$		0			0
$325$	9695.29			1.65476
$326$		0			0
$327$	1787.25	+	2931.78i	0.302248	+	0.495804i
$328$		0			0
$329$			7138.00i			1.19614i
$330$		0			0
$331$			3959.32i			0.657474i	0.944422	+	0.328737i	$0.106623\pi$
$331$			3959.32i			0.657474i	−0.944422	+	0.328737i	$0.893377\pi$
$332$		0			0
$333$	2543.89	−	4935.87i	0.418632	−	0.812265i
$334$		0			0
$335$	−5036.68			−0.821443
$336$		0			0
$337$	1603.63			0.259214			0.129607	−	0.991565i	$-0.458628\pi$
$337$	1603.63			0.259214			0.129607	+	0.991565i	$0.458628\pi$
$338$		0			0
$339$	6015.41	−	3667.07i	0.963753	−	0.587515i
$340$		0			0
$341$			4533.81i			0.719998i
$342$		0			0
$343$			6781.74i			1.06758i
$344$		0			0
$345$	1798.91	−	1096.64i	0.280725	−	0.171133i
$346$		0			0
$347$	5556.34			0.859597			0.429798	−	0.902925i	$-0.358585\pi$
$347$	5556.34			0.859597			0.429798	+	0.902925i	$0.358585\pi$
$348$		0			0
$349$	10421.8			1.59847			0.799236	−	0.601018i	$-0.205238\pi$
$349$	10421.8			1.59847			0.799236	+	0.601018i	$0.205238\pi$
$350$		0			0
$351$	−5511.34	+	395.148i	−0.838101	+	0.0600896i
$352$		0			0
$353$			5157.45i			0.777631i	0.921316	+	0.388815i	$0.127116\pi$
$353$			5157.45i			0.777631i	−0.921316	+	0.388815i	$0.872884\pi$
$354$		0			0
$355$			9069.80i			1.35599i
$356$		0			0
$357$	4446.02	+	7293.19i	0.659126	+	1.08122i
$358$		0			0
$359$	−9307.95			−1.36840			−0.684199	−	0.729295i	$-0.739848\pi$
$359$	−9307.95			−1.36840			−0.684199	+	0.729295i	$0.739848\pi$
$360$		0			0
$361$	−4713.19			−0.687154
$362$		0			0
$363$	2254.51	+	3698.27i	0.325981	+	0.534736i
$364$		0			0
$365$			6824.90i			0.978717i
$366$		0			0
$367$		−	6516.78i		−	0.926902i	−0.886123	−	0.463451i	$-0.846611\pi$
$367$		−	6516.78i		−	0.926902i	0.886123	−	0.463451i	$-0.153389\pi$
$368$		0			0
$369$	−505.091	−	260.318i	−0.0712574	−	0.0367252i
$370$		0			0
$371$	3213.77			0.449733
$372$		0			0
$373$	8841.15			1.22729			0.613643	−	0.789584i	$-0.289703\pi$
$373$	8841.15			1.22729			0.613643	+	0.789584i	$0.289703\pi$
$374$		0			0
$375$	−10357.3	+	6313.94i	−1.42627	+	0.869468i
$376$		0			0
$377$			2836.77i			0.387536i
$378$		0			0
$379$		−	387.123i		−	0.0524675i	−0.999656	−	0.0262337i	$-0.991649\pi$
$379$		−	387.123i		−	0.0524675i	0.999656	−	0.0262337i	$-0.00835141\pi$
$380$		0			0
$381$	−4996.80	+	3046.11i	−0.671900	+	0.409598i
$382$		0			0
$383$	7603.27			1.01438			0.507192	−	0.861833i	$-0.330684\pi$
$383$	7603.27			1.01438			0.507192	+	0.861833i	$0.330684\pi$
$384$		0			0
$385$	5744.70			0.760460
$386$		0			0
$387$	6624.86	+	3414.37i	0.870182	+	0.448482i
$388$		0			0
$389$			8821.10i			1.14974i	0.818246	+	0.574868i	$0.194947\pi$
$389$			8821.10i			1.14974i	−0.818246	+	0.574868i	$0.805053\pi$
$390$		0			0
$391$			2587.64i			0.334686i
$392$		0			0
$393$	−1831.55	−	3004.45i	−0.235088	−	0.385635i
$394$		0			0
$395$	−21621.4			−2.75415
$396$		0			0
$397$	−5088.86			−0.643332			−0.321666	−	0.946853i	$-0.604243\pi$
$397$	−5088.86			−0.643332			−0.321666	+	0.946853i	$0.604243\pi$
$398$		0			0
$399$	3889.86	+	6380.89i	0.488062	+	0.800611i
$400$		0			0
$401$		−	3521.32i		−	0.438519i	−0.975667	−	0.219260i	$-0.929636\pi$
$401$		−	3521.32i		−	0.438519i	0.975667	−	0.219260i	$-0.0703642\pi$
$402$		0			0
$403$			8006.01i			0.989598i
$404$		0			0
$405$	11438.6	−	8149.43i	1.40343	−	0.999872i
$406$		0			0
$407$	4586.95			0.558641
$408$		0			0
$409$	3794.23			0.458710			0.229355	−	0.973343i	$-0.426338\pi$
$409$	3794.23			0.458710			0.229355	+	0.973343i	$0.426338\pi$
$410$		0			0
$411$	11406.7	−	6953.63i	1.36898	−	0.834543i
$412$		0			0
$413$		−	6398.65i		−	0.762365i
$414$		0			0
$415$		−	11367.6i		−	1.34461i
$416$		0			0
$417$	1438.47	−	876.909i	0.168926	−	0.102979i
$418$		0			0
$419$	−8082.13			−0.942334			−0.471167	−	0.882044i	$-0.656167\pi$
$419$	−8082.13			−0.942334			−0.471167	+	0.882044i	$0.656167\pi$
$420$		0			0
$421$	−6007.64			−0.695474			−0.347737	−	0.937592i	$-0.613050\pi$
$421$	−6007.64			−0.695474			−0.347737	+	0.937592i	$0.613050\pi$
$422$		0			0
$423$	−6604.09	+	12813.8i	−0.759106	+	1.47288i
$424$		0			0
$425$		−	30267.8i		−	3.45459i
$426$		0			0
$427$			448.128i			0.0507879i
$428$		0			0
$429$	−2375.82	−	3897.27i	−0.267380	−	0.438606i
$430$		0			0
$431$	3148.09			0.351829			0.175914	−	0.984405i	$-0.443712\pi$
$431$	3148.09			0.351829			0.175914	+	0.984405i	$0.443712\pi$
$432$		0			0
$433$	15245.4			1.69202			0.846011	−	0.533165i	$-0.178997\pi$
$433$	15245.4			1.69202			0.846011	+	0.533165i	$0.178997\pi$
$434$		0			0
$435$	−3753.22	−	6156.73i	−0.413685	−	0.678604i
$436$		0			0
$437$			2263.95i			0.247824i
$438$		0			0
$439$		−	16138.5i		−	1.75456i	−0.479982	−	0.877278i	$-0.659357\pi$
$439$		−	16138.5i		−	1.75456i	0.479982	−	0.877278i	$-0.340643\pi$
$440$		0			0
$441$	−2031.80	+	3942.27i	−0.219393	+	0.425685i
$442$		0			0
$443$	5666.35			0.607712			0.303856	−	0.952718i	$-0.401726\pi$
$443$	5666.35			0.607712			0.303856	+	0.952718i	$0.401726\pi$
$444$		0			0
$445$	9944.65			1.05937
$446$		0			0
$447$	1200.57	−	731.880i	0.127036	−	0.0774424i
$448$		0			0
$449$		−	3251.14i		−	0.341716i	−0.985296	−	0.170858i	$-0.945346\pi$
$449$		−	3251.14i		−	0.341716i	0.985296	−	0.170858i	$-0.0546540\pi$
$450$		0			0
$451$		−	469.386i		−	0.0490078i
$452$		0			0
$453$	3489.72	−	2127.38i	0.361946	−	0.220646i
$454$		0			0
$455$	10144.3			1.04521
$456$		0			0
$457$	−10391.7			−1.06368			−0.531841	−	0.846844i	$-0.678499\pi$
$457$	−10391.7			−1.06368			−0.531841	+	0.846844i	$0.678499\pi$
$458$		0			0
$459$	1233.61	+	17205.9i	0.125447	+	1.74968i
$460$		0			0
$461$		−	1589.65i		−	0.160601i	−0.996771	−	0.0803006i	$-0.974412\pi$
$461$		−	1589.65i		−	0.160601i	0.996771	−	0.0803006i	$-0.0255880\pi$
$462$		0			0
$463$		−	967.097i		−	0.0970730i	−0.998821	−	0.0485365i	$-0.984544\pi$
$463$		−	967.097i		−	0.0970730i	0.998821	−	0.0485365i	$-0.0154557\pi$
$464$		0			0
$465$	−10592.4	−	17375.7i	−1.05637	−	1.73286i
$466$		0			0
$467$	−14504.9			−1.43727			−0.718635	−	0.695387i	$-0.755233\pi$
$467$	−14504.9			−1.43727			−0.718635	+	0.695387i	$0.755233\pi$
$468$		0			0
$469$	−3495.16			−0.344118
$470$		0			0
$471$	6592.43	+	10814.1i	0.644933	+	1.05794i
$472$		0			0
$473$			6156.55i			0.598474i
$474$		0			0
$475$		−	26481.6i		−	2.55802i
$476$		0			0
$477$	5769.22	+	2973.39i	0.553783	+	0.285413i
$478$		0			0
$479$	−20740.5			−1.97841			−0.989206	−	0.146530i	$-0.953190\pi$
$479$	−20740.5			−1.97841			−0.989206	+	0.146530i	$0.953190\pi$
$480$		0			0
$481$	8099.86			0.767821
$482$		0			0
$483$	1248.34	−	761.001i	0.117601	−	0.0716910i
$484$		0			0
$485$		−	5259.34i		−	0.492401i
$486$		0			0
$487$		−	4168.95i		−	0.387912i	−0.981010	−	0.193956i	$-0.937868\pi$
$487$		−	4168.95i		−	0.387912i	0.981010	−	0.193956i	$-0.0621319\pi$
$488$		0			0
$489$	6612.78	−	4031.23i	0.611534	−	0.372798i
$490$		0			0
$491$	2041.71			0.187660			0.0938301	−	0.995588i	$-0.470089\pi$
$491$	2041.71			0.187660			0.0938301	+	0.995588i	$0.470089\pi$
$492$		0			0
$493$	8856.13			0.809047
$494$		0			0
$495$	10312.6	+	5315.01i	0.936401	+	0.482610i
$496$		0			0
$497$			6293.91i			0.568049i
$498$		0			0
$499$			1171.00i			0.105052i	0.998620	+	0.0525260i	$0.0167272\pi$
$499$			1171.00i			0.105052i	−0.998620	+	0.0525260i	$0.983273\pi$
$500$		0			0
$501$	2261.98	+	3710.53i	0.201712	+	0.330886i
$502$		0			0
$503$	−9323.50			−0.826470			−0.413235	−	0.910624i	$-0.635601\pi$
$503$	−9323.50			−0.826470			−0.413235	+	0.910624i	$0.635601\pi$
$504$		0			0
$505$	26045.2			2.29504
$506$		0			0
$507$	1746.86	+	2865.53i	0.153019	+	0.251011i
$508$		0			0
$509$			12461.5i			1.08516i	0.840005	+	0.542578i	$0.182552\pi$
$509$			12461.5i			1.08516i	−0.840005	+	0.542578i	$0.817448\pi$
$510$		0			0
$511$			4736.08i			0.410004i
$512$		0			0
$513$	1079.30	+	15053.6i	0.0928895	+	1.29558i
$514$		0			0
$515$	22667.0			1.93947
$516$		0			0
$517$	−11908.0			−1.01299
$518$		0			0
$519$	14074.6	−	8580.05i	1.19038	−	0.725669i
$520$		0			0
$521$			11915.5i			1.00197i	0.865455	+	0.500986i	$0.167029\pi$
$521$			11915.5i			1.00197i	−0.865455	+	0.500986i	$0.832971\pi$
$522$		0			0
$523$			14724.3i			1.23107i	0.788110	+	0.615535i	$0.211060\pi$
$523$			14724.3i			1.23107i	−0.788110	+	0.615535i	$0.788940\pi$
$524$		0			0
$525$	−14601.9	+	8901.48i	−1.21386	+	0.739986i
$526$		0			0
$527$	24994.0			2.06595
$528$		0			0
$529$	−11724.1			−0.963597
$530$		0			0
$531$	5920.04	−	11486.6i	0.483819	−	0.938747i
$532$		0			0
$533$		−	828.864i		−	0.0673585i
$534$		0			0
$535$		−	25866.5i		−	2.09029i
$536$		0			0
$537$	−1404.32	−	2303.64i	−0.112851	−	0.185120i
$538$		0			0
$539$	−3663.59			−0.292768
$540$		0			0
$541$	−18315.8			−1.45556			−0.727781	−	0.685810i	$-0.759448\pi$
$541$	−18315.8			−1.45556			−0.727781	+	0.685810i	$0.759448\pi$
$542$		0			0
$543$	−6442.20	−	10567.7i	−0.509137	−	0.835182i
$544$		0			0
$545$		−	12730.8i		−	1.00060i
$546$		0			0
$547$			11239.4i			0.878540i	0.898355	+	0.439270i	$0.144763\pi$
$547$			11239.4i			0.878540i	−0.898355	+	0.439270i	$0.855237\pi$
$548$		0			0
$549$	−414.609	+	804.460i	−0.0322315	+	0.0625383i
$550$		0			0
$551$	7748.32			0.599073
$552$		0			0
$553$	−15003.9			−1.15377
$554$		0			0
$555$	−17579.4	+	10716.6i	−1.34451	+	0.819629i
$556$		0			0
$557$		−	2635.25i		−	0.200465i	−0.994964	−	0.100233i	$-0.968041\pi$
$557$		−	2635.25i		−	0.200465i	0.994964	−	0.100233i	$-0.0319587\pi$
$558$		0			0
$559$			10871.5i			0.822570i
$560$		0			0
$561$	−12166.9	+	7417.09i	−0.915664	+	0.558199i
$562$		0			0
$563$	11200.3			0.838430			0.419215	−	0.907887i	$-0.362305\pi$
$563$	11200.3			0.838430			0.419215	+	0.907887i	$0.362305\pi$
$564$		0			0
$565$	−26120.9			−1.94498
$566$		0			0
$567$	7937.73	−	5655.22i	0.587925	−	0.418866i
$568$		0			0
$569$			26964.7i			1.98668i	0.115241	+	0.993338i	$0.463236\pi$
$569$			26964.7i			1.98668i	−0.115241	+	0.993338i	$0.536764\pi$
$570$		0			0
$571$		−	9308.11i		−	0.682193i	−0.940028	−	0.341097i	$-0.889202\pi$
$571$		−	9308.11i		−	0.682193i	0.940028	−	0.341097i	$-0.110798\pi$
$572$		0			0
$573$	−12655.0	−	20759.1i	−0.922635	−	1.51348i
$574$		0			0
$575$	−5180.77			−0.375745
$576$		0			0
$577$	−24600.5			−1.77492			−0.887462	−	0.460882i	$-0.847533\pi$
$577$	−24600.5			−1.77492			−0.887462	+	0.460882i	$0.847533\pi$
$578$		0			0
$579$	−3293.51	−	5402.64i	−0.236397	−	0.387782i
$580$		0			0
$581$		−	7888.46i		−	0.563285i
$582$		0			0
$583$			5361.40i			0.380869i
$584$		0			0
$585$	18210.5	+	9385.50i	1.28703	+	0.663321i
$586$		0			0
$587$	9549.78			0.671485			0.335743	−	0.941954i	$-0.391013\pi$
$587$	9549.78			0.671485			0.335743	+	0.941954i	$0.391013\pi$
$588$		0			0
$589$	21867.5			1.52977
$590$		0			0
$591$	−2073.80	+	1264.21i	−0.144339	+	0.0879910i
$592$		0			0
$593$		−	2049.18i		−	0.141905i	−0.997480	−	0.0709526i	$-0.977396\pi$
$593$		−	2049.18i		−	0.141905i	0.997480	−	0.0709526i	$-0.0226039\pi$
$594$		0			0
$595$		−	31669.4i		−	2.18205i
$596$		0			0
$597$	−7813.27	+	4763.06i	−0.535638	+	0.326531i
$598$		0			0
$599$	6206.50			0.423357			0.211678	−	0.977339i	$-0.432107\pi$
$599$	6206.50			0.423357			0.211678	+	0.977339i	$0.432107\pi$
$600$		0			0
$601$	−16193.0			−1.09905			−0.549523	−	0.835478i	$-0.685191\pi$
$601$	−16193.0			−1.09905			−0.549523	+	0.835478i	$0.685191\pi$
$602$		0			0
$603$	−6274.35	−	3233.73i	−0.423734	−	0.218387i
$604$		0			0
$605$		−	16059.1i		−	1.07917i
$606$		0			0
$607$		−	6680.96i		−	0.446741i	−0.974734	−	0.223371i	$-0.928294\pi$
$607$		−	6680.96i		−	0.446741i	0.974734	−	0.223371i	$-0.0717060\pi$
$608$		0			0
$609$	−2604.51	−	4272.41i	−0.173301	−	0.284280i
$610$		0			0
$611$	−21027.7			−1.39229
$612$		0			0
$613$	11042.7			0.727590			0.363795	−	0.931479i	$-0.381481\pi$
$613$	11042.7			0.727590			0.363795	+	0.931479i	$0.381481\pi$
$614$		0			0
$615$	1096.64	+	1798.91i	0.0719035	+	0.117950i
$616$		0			0
$617$		−	183.819i		−	0.0119940i	−0.999982	−	0.00599698i	$-0.998091\pi$
$617$		−	183.819i		−	0.0119940i	0.999982	−	0.00599698i	$-0.00190891\pi$
$618$		0			0
$619$			8335.90i			0.541273i	0.962682	+	0.270636i	$0.0872341\pi$
$619$			8335.90i			0.541273i	−0.962682	+	0.270636i	$0.912766\pi$
$620$		0			0
$621$	2945.04	−	211.151i	0.190306	−	0.0136444i
$622$		0			0
$623$	6901.00			0.443792
$624$		0			0
$625$	14203.6			0.909029
$626$		0			0
$627$	−10644.9	+	6489.29i	−0.678020	+	0.413329i
$628$		0			0
$629$		−	25287.0i		−	1.60295i
$630$		0			0
$631$			7144.64i			0.450751i	0.974272	+	0.225375i	$0.0723609\pi$
$631$			7144.64i			0.450751i	−0.974272	+	0.225375i	$0.927639\pi$
$632$		0			0
$633$	−14475.2	+	8824.27i	−0.908908	+	0.554081i
$634$		0			0
$635$	21697.8			1.35598
$636$		0			0
$637$	−6469.35			−0.402394
$638$		0			0
$639$	−5823.14	+	11298.5i	−0.360500	+	0.699473i
$640$		0			0
$641$			20686.2i			1.27466i	0.770591	+	0.637330i	$0.219961\pi$
$641$			20686.2i			1.27466i	−0.770591	+	0.637330i	$0.780039\pi$
$642$		0			0
$643$			18859.6i			1.15669i	0.815793	+	0.578344i	$0.196301\pi$
$643$			18859.6i			1.15669i	−0.815793	+	0.578344i	$0.803699\pi$
$644$		0			0
$645$	−14383.7	−	23594.8i	−0.878071	−	1.44038i
$646$		0			0
$647$	7958.77			0.483604			0.241802	−	0.970326i	$-0.422262\pi$
$647$	7958.77			0.483604			0.241802	+	0.970326i	$0.422262\pi$
$648$		0			0
$649$	10674.6			0.645630
$650$		0			0
$651$	−7350.51	−	12057.7i	−0.442534	−	0.725927i
$652$		0			0
$653$		−	24419.6i		−	1.46342i	−0.681617	−	0.731709i	$-0.738723\pi$
$653$		−	24419.6i		−	1.46342i	0.681617	−	0.731709i	$-0.261277\pi$
$654$		0			0
$655$			13046.3i			0.778263i
$656$		0			0
$657$	−4381.83	+	8502.00i	−0.260200	+	0.504862i
$658$		0			0
$659$	28858.1			1.70584			0.852922	−	0.522038i	$-0.174828\pi$
$659$	28858.1			1.70584			0.852922	+	0.522038i	$0.174828\pi$
$660$		0			0
$661$	−17587.0			−1.03488			−0.517440	−	0.855719i	$-0.673115\pi$
$661$	−17587.0			−1.03488			−0.517440	+	0.855719i	$0.673115\pi$
$662$		0			0
$663$	−21484.9	+	13097.5i	−1.25853	+	0.767214i
$664$		0			0
$665$		−	27707.9i		−	1.61574i
$666$		0			0
$667$		−	1515.86i		−	0.0879973i
$668$		0			0
$669$	−16336.3	+	9958.80i	−0.944093	+	0.575530i
$670$		0			0
$671$	−747.592			−0.0430112
$672$		0			0
$673$	−991.989			−0.0568178			−0.0284089	−	0.999596i	$-0.509044\pi$
$673$	−991.989			−0.0568178			−0.0284089	+	0.999596i	$0.509044\pi$
$674$		0			0
$675$	−34448.3	+	2469.85i	−1.96432	+	0.140836i
$676$		0			0
$677$		−	7937.72i		−	0.450623i	−0.974287	−	0.225311i	$-0.927660\pi$
$677$		−	7937.72i		−	0.450623i	0.974287	−	0.225311i	$-0.0723399\pi$
$678$		0			0
$679$		−	3649.67i		−	0.206276i
$680$		0			0
$681$	−8517.80	−	13972.5i	−0.479299	−	0.786237i
$682$		0			0
$683$	−18277.0			−1.02394			−0.511968	−	0.859004i	$-0.671083\pi$
$683$	−18277.0			−1.02394			−0.511968	+	0.859004i	$0.671083\pi$
$684$		0			0
$685$	−49531.5			−2.76277
$686$		0			0
$687$	−2765.39	−	4536.31i	−0.153575	−	0.251923i
$688$		0			0
$689$			9467.41i			0.523483i
$690$		0			0
$691$			35599.3i			1.95986i	0.199344	+	0.979930i	$0.436119\pi$
$691$			35599.3i			1.95986i	−0.199344	+	0.979930i	$0.563881\pi$
$692$		0			0
$693$	7156.36	+	3688.30i	0.392277	+	0.202175i
$694$		0			0
$695$	−6246.32			−0.340916
$696$		0			0
$697$	−2587.64			−0.140622
$698$		0			0
$699$	−4285.49	+	2612.48i	−0.231891	+	0.141364i
$700$		0			0
$701$			13138.6i			0.707901i	0.935264	+	0.353950i	$0.115162\pi$
$701$			13138.6i			0.707901i	−0.935264	+	0.353950i	$0.884838\pi$
$702$		0			0
$703$		−	22123.8i		−	1.18694i
$704$		0			0
$705$	45637.1	−	27820.9i	2.43800	−	1.48624i
$706$		0			0
$707$	18073.8			0.961437
$708$		0			0
$709$	−16973.6			−0.899095			−0.449548	−	0.893256i	$-0.648415\pi$
$709$	−16973.6			−0.899095			−0.449548	+	0.893256i	$0.648415\pi$
$710$		0			0
$711$	−26934.4	−	13881.7i	−1.42070	−	0.732213i
$712$		0			0
$713$		−	4278.09i		−	0.224706i
$714$		0			0
$715$			16923.2i			0.885165i
$716$		0			0
$717$	11402.7	+	18704.9i	0.593922	+	0.974263i
$718$		0			0
$719$	−29715.0			−1.54128			−0.770642	−	0.637269i	$-0.780064\pi$
$719$	−29715.0			−1.54128			−0.770642	+	0.637269i	$0.780064\pi$
$720$		0			0
$721$	15729.5			0.812480
$722$		0			0
$723$	6676.46	+	10952.0i	0.343431	+	0.563359i
$724$		0			0
$725$			17731.1i			0.908298i
$726$		0			0
$727$			27975.7i			1.42718i	0.700563	+	0.713591i	$0.252932\pi$
$727$			27975.7i			1.42718i	−0.700563	+	0.713591i	$0.747068\pi$
$728$		0			0
$729$	19481.7	−	2808.00i	0.989772	−	0.142661i
$730$		0			0
$731$	33939.8			1.71725
$732$		0			0
$733$	−29924.4			−1.50789			−0.753945	−	0.656937i	$-0.771852\pi$
$733$	−29924.4			−1.50789			−0.753945	+	0.656937i	$0.771852\pi$
$734$		0			0
$735$	14040.6	−	8559.32i	0.704620	−	0.429545i
$736$		0			0
$737$		−	5830.82i		−	0.291426i
$738$		0			0
$739$		−	23186.6i		−	1.15417i	−0.816683	−	0.577086i	$-0.804190\pi$
$739$		−	23186.6i		−	1.15417i	0.816683	−	0.577086i	$-0.195810\pi$
$740$		0			0
$741$	−18797.4	+	11459.1i	−0.931900	+	0.568097i
$742$		0			0
$743$	−20872.3			−1.03059			−0.515297	−	0.857012i	$-0.672318\pi$
$743$	−20872.3			−1.03059			−0.515297	+	0.857012i	$0.672318\pi$
$744$		0			0
$745$	−5213.26			−0.256375
$746$		0			0
$747$	7298.42	−	14161.0i	0.357477	−	0.693607i
$748$		0			0
$749$		−	17949.8i		−	0.875663i
$750$		0			0
$751$		−	23052.5i		−	1.12010i	−0.828458	−	0.560052i	$-0.810781\pi$
$751$		−	23052.5i		−	1.12010i	0.828458	−	0.560052i	$-0.189219\pi$
$752$		0			0
$753$	2680.82	+	4397.59i	0.129741	+	0.212825i
$754$		0			0
$755$	−15153.5			−0.730454
$756$		0			0
$757$	34394.4			1.65137			0.825684	−	0.564132i	$-0.190789\pi$
$757$	34394.4			1.65137			0.825684	+	0.564132i	$0.190789\pi$
$758$		0			0
$759$	1269.54	+	2082.54i	0.0607135	+	0.0995936i
$760$		0			0
$761$		−	36681.4i		−	1.74731i	−0.486549	−	0.873653i	$-0.661745\pi$
$761$		−	36681.4i		−	1.74731i	0.486549	−	0.873653i	$-0.338255\pi$
$762$		0			0
$763$		−	8834.40i		−	0.419170i
$764$		0			0
$765$	29300.6	−	56851.6i	1.38479	−	2.68689i
$766$		0			0
$767$	18849.7			0.887383
$768$		0			0
$769$	−15052.1			−0.705843			−0.352921	−	0.935653i	$-0.614812\pi$
$769$	−15052.1			−0.705843			−0.352921	+	0.935653i	$0.614812\pi$
$770$		0			0
$771$	29.4445	−	17.9497i	0.00137538	−	0.000838447i
$772$		0			0
$773$			41461.9i			1.92921i	0.263692	+	0.964607i	$0.415060\pi$
$773$			41461.9i			1.92921i	−0.263692	+	0.964607i	$0.584940\pi$
$774$		0			0
$775$			50041.1i			2.31939i
$776$		0			0
$777$	−12199.0	+	7436.68i	−0.563241	+	0.343358i
$778$		0			0
$779$	−2263.95			−0.104126
$780$		0			0
$781$	−10499.8			−0.481068
$782$		0			0
$783$	−722.661	−	10079.3i	−0.0329831	−	0.460033i
$784$		0			0
$785$		−	46958.6i		−	2.13506i
$786$		0			0
$787$		−	31416.1i		−	1.42295i	−0.702711	−	0.711475i	$-0.748027\pi$
$787$		−	31416.1i		−	1.42295i	0.702711	−	0.711475i	$-0.251973\pi$
$788$		0			0
$789$	−3933.72	−	6452.83i	−0.177496	−	0.291162i
$790$		0			0
$791$	−18126.4			−0.814790
$792$		0			0
$793$	−1320.13			−0.0591164
$794$		0			0
$795$	−12525.9	−	20547.4i	−0.558804	−	0.916655i
$796$		0			0
$797$			10187.9i			0.452790i	0.974036	+	0.226395i	$0.0726939\pi$
$797$			10187.9i			0.452790i	−0.974036	+	0.226395i	$0.927306\pi$
$798$		0			0
$799$			65646.5i			2.90664i
$800$		0			0
$801$	12388.4	+	6384.82i	0.546469	+	0.281643i
$802$		0			0
$803$	−7900.99			−0.347223
$804$		0			0
$805$	−5420.69			−0.237334
$806$		0			0
$807$	28945.7	−	17645.6i	1.26262	−	0.769710i
$808$		0			0
$809$			21311.0i			0.926151i	0.886319	+	0.463076i	$0.153254\pi$
$809$			21311.0i			0.926151i	−0.886319	+	0.463076i	$0.846746\pi$
$810$		0			0
$811$			10020.2i			0.433856i	0.976188	+	0.216928i	$0.0696038\pi$
$811$			10020.2i			0.433856i	−0.976188	+	0.216928i	$0.930396\pi$
$812$		0			0
$813$	−21943.7	+	13377.2i	−0.946618	+	0.577070i
$814$		0			0
$815$	−28714.9			−1.23416
$816$		0			0
$817$	29694.3			1.27157
$818$		0			0
$819$	12637.0	+	6512.98i	0.539162	+	0.277878i
$820$		0			0
$821$		−	46720.6i		−	1.98607i	−0.117833	−	0.993033i	$-0.537595\pi$
$821$		−	46720.6i		−	1.98607i	0.117833	−	0.993033i	$-0.462405\pi$
$822$		0			0
$823$			16458.3i			0.697083i	0.937293	+	0.348542i	$0.113323\pi$
$823$			16458.3i			0.697083i	−0.937293	+	0.348542i	$0.886677\pi$
$824$		0			0
$825$	−14849.9	−	24359.7i	−0.626677	−	1.02799i
$826$		0			0
$827$	−37242.6			−1.56597			−0.782983	−	0.622043i	$-0.786303\pi$
$827$	−37242.6			−1.56597			−0.782983	+	0.622043i	$0.786303\pi$
$828$		0			0
$829$	644.852			0.0270164			0.0135082	−	0.999909i	$-0.495700\pi$
$829$	644.852			0.0270164			0.0135082	+	0.999909i	$0.495700\pi$
$830$		0			0
$831$	−8580.75	−	14075.8i	−0.358198	−	0.587585i
$832$		0			0
$833$			20196.7i			0.840064i
$834$		0			0
$835$		−	16112.3i		−	0.667772i
$836$		0			0
$837$	−2039.51	−	28446.1i	−0.0842244	−	1.17472i
$838$		0			0
$839$	−40974.7			−1.68606			−0.843029	−	0.537868i	$-0.819230\pi$
$839$	−40974.7			−1.68606			−0.843029	+	0.537868i	$0.819230\pi$
$840$		0			0
$841$	19201.0			0.787282
$842$		0			0
$843$	14448.8	−	8808.19i	0.590326	−	0.359870i
$844$		0			0
$845$		−	12443.1i		−	0.506573i
$846$		0			0
$847$		−	11144.1i		−	0.452084i
$848$		0			0
$849$	−24422.8	+	14888.5i	−0.987267	+	0.601850i
$850$		0			0
$851$	−4328.24			−0.174348
$852$		0			0
$853$	4704.70			0.188846			0.0944232	−	0.995532i	$-0.469899\pi$
$853$	4704.70			0.188846			0.0944232	+	0.995532i	$0.469899\pi$
$854$		0			0
$855$	25635.4	−	49740.0i	1.02539	−	1.98956i
$856$		0			0
$857$			15670.7i			0.624621i	0.949980	+	0.312311i	$0.101103\pi$
$857$			15670.7i			0.624621i	−0.949980	+	0.312311i	$0.898897\pi$
$858$		0			0
$859$		−	9348.60i		−	0.371327i	−0.982613	−	0.185664i	$-0.940557\pi$
$859$		−	9348.60i		−	0.371327i	0.982613	−	0.185664i	$-0.0594435\pi$
$860$		0			0
$861$	761.001	+	1248.34i	0.0301218	+	0.0494114i
$862$		0			0
$863$	35282.4			1.39169			0.695843	−	0.718194i	$-0.255031\pi$
$863$	35282.4			1.39169			0.695843	+	0.718194i	$0.255031\pi$
$864$		0			0
$865$	−61116.6			−2.40234
$866$		0			0
$867$	27600.9	+	45276.1i	1.08117	+	1.77354i
$868$		0			0
$869$		−	25030.4i		−	0.977098i
$870$		0			0
$871$		−	10296.3i		−	0.400549i
$872$		0			0
$873$	3376.68	−	6551.73i	0.130909	−	0.254000i
$874$		0			0
$875$	31209.9			1.20581
$876$		0			0
$877$	6689.00			0.257550			0.128775	−	0.991674i	$-0.458895\pi$
$877$	6689.00			0.257550			0.128775	+	0.991674i	$0.458895\pi$
$878$		0			0
$879$	−14450.5	+	8809.21i	−0.554498	+	0.338029i
$880$		0			0
$881$			27165.1i			1.03884i	0.854520	+	0.519418i	$0.173851\pi$
$881$			27165.1i			1.03884i	−0.854520	+	0.519418i	$0.826149\pi$
$882$		0			0
$883$		−	19495.6i		−	0.743012i	−0.928431	−	0.371506i	$-0.878841\pi$
$883$		−	19495.6i		−	0.743012i	0.928431	−	0.371506i	$-0.121159\pi$
$884$		0			0
$885$	−40910.0	+	24939.2i	−1.55387	+	0.947258i
$886$		0			0
$887$	49012.1			1.85532			0.927659	−	0.373429i	$-0.121818\pi$
$887$	49012.1			1.85532			0.927659	+	0.373429i	$0.121818\pi$
$888$		0			0
$889$	15057.0			0.568048
$890$		0			0
$891$	9434.35	+	13242.2i	0.354728	+	0.497900i
$892$		0			0
$893$			57434.8i			2.15228i
$894$		0			0
$895$			10003.1i			0.373596i
$896$		0			0
$897$	2241.82	+	3677.46i	0.0834473	+	0.136886i
$898$		0			0
$899$	−14641.7			−0.543189
$900$		0			0
$901$	29556.3			1.09286
$902$		0			0
$903$	−9981.41	−	16373.4i	−0.367841	−	0.603402i
$904$		0			0
$905$			45888.5i			1.68551i
$906$		0			0
$907$			5103.24i			0.186825i	0.995627	+	0.0934125i	$0.0297775\pi$
$907$			5103.24i			0.186825i	−0.995627	+	0.0934125i	$0.970222\pi$
$908$		0			0
$909$	32445.3	+	16721.9i	1.18388	+	0.610156i
$910$		0			0
$911$	1849.73			0.0672714			0.0336357	−	0.999434i	$-0.489291\pi$
$911$	1849.73			0.0672714			0.0336357	+	0.999434i	$0.489291\pi$
$912$		0			0
$913$	13160.0			0.477033
$914$		0			0
$915$	2865.13	−	1746.61i	0.103517	−	0.0631052i
$916$		0			0
$917$			9053.38i			0.326029i
$918$		0			0
$919$			3990.55i			0.143239i	0.997432	+	0.0716193i	$0.0228167\pi$
$919$			3990.55i			0.143239i	−0.997432	+	0.0716193i	$0.977183\pi$
$920$		0			0
$921$	−34380.6	+	20958.8i	−1.23005	+	0.749855i
$922$		0			0
$923$	−18541.1			−0.661201
$924$		0			0
$925$	50627.7			1.79960
$926$		0			0
$927$	28237.0	+	14553.0i	1.00046	+	0.515624i
$928$		0			0
$929$		−	27144.1i		−	0.958633i	−0.877642	−	0.479317i	$-0.840885\pi$
$929$		−	27144.1i		−	0.958633i	0.877642	−	0.479317i	$-0.159115\pi$
$930$		0			0
$931$			17670.3i			0.622041i
$932$		0			0
$933$	13281.1	+	21786.2i	0.466029	+	0.764468i
$934$		0			0
$935$	52832.7			1.84793
$936$		0			0
$937$	13931.0			0.485705			0.242852	−	0.970063i	$-0.421917\pi$
$937$	13931.0			0.485705			0.242852	+	0.970063i	$0.421917\pi$
$938$		0			0
$939$	13197.1	+	21648.4i	0.458650	+	0.752363i
$940$		0			0
$941$		−	28631.4i		−	0.991877i	−0.868358	−	0.495938i	$-0.834824\pi$
$941$		−	28631.4i		−	0.991877i	0.868358	−	0.495938i	$-0.165176\pi$
$942$		0			0
$943$			442.912i			0.0152950i
$944$		0			0
$945$	−36043.6	+	2584.23i	−1.24074	+	0.0889576i
$946$		0			0
$947$	−39593.3			−1.35862			−0.679309	−	0.733853i	$-0.737720\pi$
$947$	−39593.3			−1.35862			−0.679309	+	0.733853i	$0.737720\pi$
$948$		0			0
$949$	−13951.9			−0.477238
$950$		0			0
$951$	6090.25	−	3712.69i	0.207666	−	0.126595i
$952$		0			0
$953$		−	2974.05i		−	0.101090i	−0.998722	−	0.0505450i	$-0.983904\pi$
$953$		−	2974.05i		−	0.101090i	0.998722	−	0.0505450i	$-0.0160958\pi$
$954$		0			0
$955$			90142.8i			3.05440i
$956$		0			0
$957$	7127.47	−	4344.99i	0.240751	−	0.146764i
$958$		0			0
$959$	−34371.9			−1.15738
$960$		0			0
$961$	−11531.1			−0.387067
$962$		0			0
$963$	16607.2	−	32222.7i	0.555721	−	1.07826i
$964$		0			0
$965$			23460.0i			0.782596i
$966$		0			0
$967$		−	47947.7i		−	1.59451i	−0.603641	−	0.797256i	$-0.706284\pi$
$967$		−	47947.7i		−	1.59451i	0.603641	−	0.797256i	$-0.293716\pi$
$968$		0			0
$969$	35774.2	+	58683.5i	1.18600	+	1.94550i
$970$		0			0
$971$	−49829.9			−1.64688			−0.823439	−	0.567404i	$-0.807948\pi$
$971$	−49829.9			−1.64688			−0.823439	+	0.567404i	$0.807948\pi$
$972$		0			0
$973$	−4334.58			−0.142816
$974$		0			0
$975$	−26222.7	−	43015.5i	−0.861333	−	1.41292i
$976$		0			0
$977$			32986.0i			1.08016i	0.841613	+	0.540080i	$0.181606\pi$
$977$			32986.0i			1.08016i	−0.841613	+	0.540080i	$0.818394\pi$
$978$		0			0
$979$			11512.6i			0.375838i
$980$		0			0
$981$	8173.61	−	15859.1i	0.266018	−	0.516150i
$982$		0			0
$983$	12569.0			0.407823			0.203912	−	0.978989i	$-0.434634\pi$
$983$	12569.0			0.407823			0.203912	+	0.978989i	$0.434634\pi$
$984$		0			0
$985$	9005.10			0.291296
$986$		0			0
$987$	31669.4	−	19306.1i	1.02133	−	0.622613i
$988$		0			0
$989$		−	5809.30i		−	0.186780i
$990$		0			0
$991$			29938.5i			0.959666i	0.877360	+	0.479833i	$0.159303\pi$
$991$			29938.5i			0.959666i	−0.877360	+	0.479833i	$0.840697\pi$
$992$		0			0
$993$	17566.5	−	10708.7i	0.561385	−	0.342227i
$994$		0			0
$995$	33927.8			1.08099
$996$		0			0
$997$	20279.3			0.644185			0.322092	−	0.946708i	$-0.395614\pi$
$997$	20279.3			0.644185			0.322092	+	0.946708i	$0.395614\pi$
$998$		0			0
$999$	−28779.6	+	2063.42i	−0.911458	+	0.0653491i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.c.u.767.3		8
3.2	odd	2		768.4.c.t.767.5		8
4.3	odd	2		768.4.c.t.767.6		8
8.3	odd	2		768.4.c.t.767.3		8
8.5	even	2	inner	768.4.c.u.767.6		8
12.11	even	2	inner	768.4.c.u.767.4		8
16.3	odd	4		192.4.f.c.95.2	yes	8
16.5	even	4		192.4.f.d.95.2	yes	8
16.11	odd	4		192.4.f.c.95.7	yes	8
16.13	even	4		192.4.f.d.95.7	yes	8
24.5	odd	2		768.4.c.t.767.4		8
24.11	even	2	inner	768.4.c.u.767.5		8
48.5	odd	4		192.4.f.c.95.1	✓	8
48.11	even	4		192.4.f.d.95.8	yes	8
48.29	odd	4		192.4.f.c.95.8	yes	8
48.35	even	4		192.4.f.d.95.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.f.c.95.1	✓	8	48.5	odd	4
192.4.f.c.95.2	yes	8	16.3	odd	4
192.4.f.c.95.7	yes	8	16.11	odd	4
192.4.f.c.95.8	yes	8	48.29	odd	4
192.4.f.d.95.1	yes	8	48.35	even	4
192.4.f.d.95.2	yes	8	16.5	even	4
192.4.f.d.95.7	yes	8	16.13	even	4
192.4.f.d.95.8	yes	8	48.11	even	4
768.4.c.t.767.3		8	8.3	odd	2
768.4.c.t.767.4		8	24.5	odd	2
768.4.c.t.767.5		8	3.2	odd	2
768.4.c.t.767.6		8	4.3	odd	2
768.4.c.u.767.3		8	1.1	even	1	trivial
768.4.c.u.767.4		8	12.11	even	2	inner
768.4.c.u.767.5		8	24.11	even	2	inner
768.4.c.u.767.6		8	8.5	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.70469 - 4.43674i) q^{3} +19.2658i q^{5} +13.3693i q^{7} +(-12.3693 + 24.0000i) q^{9} +O(q^{10})\)	\(q+(-2.70469 - 4.43674i) q^{3} +19.2658i q^{5} +13.3693i q^{7} +(-12.3693 + 24.0000i) q^{9} -22.3034 q^{11} -39.3845 q^{13} +(85.4773 - 52.1080i) q^{15} +122.955i q^{17} +107.574i q^{19} +(59.3162 - 36.1598i) q^{21} +21.0455 q^{23} -246.170 q^{25} +(139.937 - 10.0331i) q^{27} -72.0277i q^{29} -203.278i q^{31} +(60.3239 + 98.9545i) q^{33} -257.570 q^{35} -205.661 q^{37} +(106.523 + 174.739i) q^{39} +21.0455i q^{41} -276.036i q^{43} +(-462.379 - 238.305i) q^{45} +533.909 q^{47} +164.261 q^{49} +(545.517 - 332.554i) q^{51} -240.384i q^{53} -429.693i q^{55} +(477.278 - 290.955i) q^{57} -478.607 q^{59} +33.5192 q^{61} +(-320.864 - 165.369i) q^{63} -758.773i q^{65} +261.431i q^{67} +(-56.9214 - 93.3732i) q^{69} +470.773 q^{71} +354.250 q^{73} +(665.814 + 1092.19i) q^{75} -298.182i q^{77} +1122.27i q^{79} +(-423.000 - 593.727i) q^{81} -590.042 q^{83} -2368.82 q^{85} +(-319.568 + 194.813i) q^{87} -516.182i q^{89} -526.543i q^{91} +(-901.893 + 549.805i) q^{93} -2072.50 q^{95} -272.989 q^{97} +(275.878 - 535.282i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 288 q^{15} + 960 q^{23} - 584 q^{25} - 408 q^{33} + 1248 q^{39} + 2688 q^{47} + 1512 q^{49} + 2136 q^{57} - 192 q^{63} - 192 q^{71} - 1520 q^{73} - 3384 q^{81} - 3744 q^{87} - 7872 q^{95} + 2368 q^{97}+O(q^{100}) \) 8 * q + 288 * q^15 + 960 * q^23 - 584 * q^25 - 408 * q^33 + 1248 * q^39 + 2688 * q^47 + 1512 * q^49 + 2136 * q^57 - 192 * q^63 - 192 * q^71 - 1520 * q^73 - 3384 * q^81 - 3744 * q^87 - 7872 * q^95 + 2368 * q^97

Introduction

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