LMFDB - Embedded newform 768.3.b.f.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(127,768)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("768.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.b (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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.3.b.f.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -2.63567i q^{5} +12.5558i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} -2.63567i q^{5} +12.5558i q^{7} +3.00000 q^{9} -5.79796 q^{11} -8.78710i q^{13} +4.56512i q^{15} -30.1810 q^{17} +17.4125 q^{19} -21.7473i q^{21} +2.48425i q^{23} +18.0532 q^{25} -5.19615 q^{27} +26.4175i q^{29} -38.0082i q^{31} +10.0424 q^{33} +33.0931 q^{35} -47.7930i q^{37} +15.2197i q^{39} -53.3294 q^{41} +30.7044 q^{43} -7.90702i q^{45} -16.2238i q^{47} -108.649 q^{49} +52.2750 q^{51} -49.8432i q^{53} +15.2815i q^{55} -30.1593 q^{57} -107.412 q^{59} -62.6220i q^{61} +37.6675i q^{63} -23.1599 q^{65} -60.9540 q^{67} -4.30285i q^{69} -19.9465i q^{71} -5.13929 q^{73} -31.2691 q^{75} -72.7982i q^{77} +6.83349i q^{79} +9.00000 q^{81} -159.213 q^{83} +79.5472i q^{85} -45.7565i q^{87} -39.4473 q^{89} +110.329 q^{91} +65.8321i q^{93} -45.8936i q^{95} -60.5944 q^{97} -17.3939 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} + 32 q^{11} + 16 q^{17} + 96 q^{19} + 8 q^{25} - 96 q^{35} - 80 q^{41} + 224 q^{43} - 88 q^{49} + 96 q^{51} - 96 q^{57} - 512 q^{59} - 160 q^{65} + 16 q^{73} + 192 q^{75} + 72 q^{81} - 544 q^{83} + 240 q^{89} - 32 q^{91} + 400 q^{97} + 96 q^{99}+O(q^{100}) $ 8 * q + 24 * q^9 + 32 * q^11 + 16 * q^17 + 96 * q^19 + 8 * q^25 - 96 * q^35 - 80 * q^41 + 224 * q^43 - 88 * q^49 + 96 * q^51 - 96 * q^57 - 512 * q^59 - 160 * q^65 + 16 * q^73 + 192 * q^75 + 72 * q^81 - 544 * q^83 + 240 * q^89 - 32 * q^91 + 400 * q^97 + 96 * 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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	− 2.63567i	− 0.527135i	−0.964641	−	0.263567i	$-0.915101\pi$
$5$	− 2.63567i	− 0.527135i	0.964641	−	0.263567i	$-0.0848992\pi$
$6$	0	0
$7$	12.5558i	1.79369i	0.442344	+	0.896845i	$0.354147\pi$
$7$	12.5558i	1.79369i	−0.442344	+	0.896845i	$0.645853\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	−5.79796	−0.527087	−0.263544	−	0.964647i	$-0.584891\pi$
$11$	−5.79796	−0.527087	−0.263544	+	0.964647i	$0.584891\pi$
$12$	0	0
$13$	− 8.78710i	− 0.675931i	−0.941159	−	0.337965i	$-0.890261\pi$
$13$	− 8.78710i	− 0.675931i	0.941159	−	0.337965i	$-0.109739\pi$
$14$	0	0
$15$	4.56512i	0.304341i
$16$	0	0
$17$	−30.1810	−1.77535	−0.887676	−	0.460469i	$-0.847681\pi$
$17$	−30.1810	−1.77535	−0.887676	+	0.460469i	$0.847681\pi$
$18$	0	0
$19$	17.4125	0.916445	0.458222	−	0.888838i	$-0.348486\pi$
$19$	17.4125	0.916445	0.458222	+	0.888838i	$0.348486\pi$
$20$	0	0
$21$	− 21.7473i	− 1.03559i
$22$	0	0
$23$	2.48425i	0.108011i	0.998541	+	0.0540054i	$0.0171988\pi$
$23$	2.48425i	0.108011i	−0.998541	+	0.0540054i	$0.982801\pi$
$24$	0	0
$25$	18.0532	0.722129
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	26.4175i	0.910950i	0.890249	+	0.455475i	$0.150531\pi$
$29$	26.4175i	0.910950i	−0.890249	+	0.455475i	$0.849469\pi$
$30$	0	0
$31$	− 38.0082i	− 1.22607i	−0.790056	−	0.613035i	$-0.789949\pi$
$31$	− 38.0082i	− 1.22607i	0.790056	−	0.613035i	$-0.210051\pi$
$32$	0	0
$33$	10.0424	0.304314
$34$	0	0
$35$	33.0931	0.945517
$36$	0	0
$37$	− 47.7930i	− 1.29170i	−0.763463	−	0.645851i	$-0.776503\pi$
$37$	− 47.7930i	− 1.29170i	0.763463	−	0.645851i	$-0.223497\pi$
$38$	0	0
$39$	15.2197i	0.390249i
$40$	0	0
$41$	−53.3294	−1.30072	−0.650358	−	0.759628i	$-0.725381\pi$
$41$	−53.3294	−1.30072	−0.650358	+	0.759628i	$0.725381\pi$
$42$	0	0
$43$	30.7044	0.714057	0.357028	−	0.934094i	$-0.383790\pi$
$43$	30.7044	0.714057	0.357028	+	0.934094i	$0.383790\pi$
$44$	0	0
$45$	− 7.90702i	− 0.175712i
$46$	0	0
$47$	− 16.2238i	− 0.345186i	−0.984993	−	0.172593i	$-0.944785\pi$
$47$	− 16.2238i	− 0.345186i	0.984993	−	0.172593i	$-0.0552146\pi$
$48$	0	0
$49$	−108.649	−2.21733
$50$	0	0
$51$	52.2750	1.02500
$52$	0	0
$53$	− 49.8432i	− 0.940437i	−0.882550	−	0.470219i	$-0.844175\pi$
$53$	− 49.8432i	− 0.940437i	0.882550	−	0.470219i	$-0.155825\pi$
$54$	0	0
$55$	15.2815i	0.277846i
$56$	0	0
$57$	−30.1593	−0.529110
$58$	0	0
$59$	−107.412	−1.82054	−0.910271	−	0.414012i	$-0.864127\pi$
$59$	−107.412	−1.82054	−0.910271	+	0.414012i	$0.864127\pi$
$60$	0	0
$61$	− 62.6220i	− 1.02659i	−0.858212	−	0.513295i	$-0.828425\pi$
$61$	− 62.6220i	− 1.02659i	0.858212	−	0.513295i	$-0.171575\pi$
$62$	0	0
$63$	37.6675i	0.597897i
$64$	0	0
$65$	−23.1599	−0.356307
$66$	0	0
$67$	−60.9540	−0.909762	−0.454881	−	0.890552i	$-0.650318\pi$
$67$	−60.9540	−0.909762	−0.454881	+	0.890552i	$0.650318\pi$
$68$	0	0
$69$	− 4.30285i	− 0.0623601i
$70$	0	0
$71$	− 19.9465i	− 0.280937i	−0.990085	−	0.140469i	$-0.955139\pi$
$71$	− 19.9465i	− 0.280937i	0.990085	−	0.140469i	$-0.0448609\pi$
$72$	0	0
$73$	−5.13929	−0.0704013	−0.0352006	−	0.999380i	$-0.511207\pi$
$73$	−5.13929	−0.0704013	−0.0352006	+	0.999380i	$0.511207\pi$
$74$	0	0
$75$	−31.2691	−0.416921
$76$	0	0
$77$	− 72.7982i	− 0.945431i
$78$	0	0
$79$	6.83349i	0.0864999i	0.999064	+	0.0432500i	$0.0137712\pi$
$79$	6.83349i	0.0864999i	−0.999064	+	0.0432500i	$0.986229\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−159.213	−1.91823	−0.959115	−	0.283016i	$-0.908665\pi$
$83$	−159.213	−1.91823	−0.959115	+	0.283016i	$0.908665\pi$
$84$	0	0
$85$	79.5472i	0.935850i
$86$	0	0
$87$	− 45.7565i	− 0.525937i
$88$	0	0
$89$	−39.4473	−0.443229	−0.221614	−	0.975134i	$-0.571133\pi$
$89$	−39.4473	−0.443229	−0.221614	+	0.975134i	$0.571133\pi$
$90$	0	0
$91$	110.329	1.21241
$92$	0	0
$93$	65.8321i	0.707872i
$94$	0	0
$95$	− 45.8936i	− 0.483090i
$96$	0	0
$97$	−60.5944	−0.624684	−0.312342	−	0.949970i	$-0.601114\pi$
$97$	−60.5944	−0.624684	−0.312342	+	0.949970i	$0.601114\pi$
$98$	0	0
$99$	−17.3939	−0.175696
$100$	0	0
$101$	− 169.076i	− 1.67402i	−0.547186	−	0.837011i	$-0.684301\pi$
$101$	− 169.076i	− 1.67402i	0.547186	−	0.837011i	$-0.315699\pi$
$102$	0	0
$103$	163.390i	1.58631i	0.609020	+	0.793155i	$0.291563\pi$
$103$	163.390i	1.58631i	−0.609020	+	0.793155i	$0.708437\pi$
$104$	0	0
$105$	−57.3189	−0.545894
$106$	0	0
$107$	37.4288	0.349802	0.174901	−	0.984586i	$-0.444040\pi$
$107$	37.4288	0.349802	0.174901	+	0.984586i	$0.444040\pi$
$108$	0	0
$109$	− 210.117i	− 1.92768i	−0.266489	−	0.963838i	$-0.585863\pi$
$109$	− 210.117i	− 1.92768i	0.266489	−	0.963838i	$-0.414137\pi$
$110$	0	0
$111$	82.7799i	0.745765i
$112$	0	0
$113$	190.809	1.68857	0.844286	−	0.535893i	$-0.180025\pi$
$113$	190.809	1.68857	0.844286	+	0.535893i	$0.180025\pi$
$114$	0	0
$115$	6.54768	0.0569363
$116$	0	0
$117$	− 26.3613i	− 0.225310i
$118$	0	0
$119$	− 378.947i	− 3.18443i
$120$	0	0
$121$	−87.3837	−0.722179
$122$	0	0
$123$	92.3692	0.750969
$124$	0	0
$125$	− 113.474i	− 0.907794i
$126$	0	0
$127$	− 34.4377i	− 0.271163i	−0.990766	−	0.135581i	$-0.956710\pi$
$127$	− 34.4377i	− 0.271163i	0.990766	−	0.135581i	$-0.0432902\pi$
$128$	0	0
$129$	−53.1817	−0.412261
$130$	0	0
$131$	92.5491	0.706482	0.353241	−	0.935532i	$-0.385080\pi$
$131$	92.5491	0.706482	0.353241	+	0.935532i	$0.385080\pi$
$132$	0	0
$133$	218.628i	1.64382i
$134$	0	0
$135$	13.6954i	0.101447i
$136$	0	0
$137$	20.0745	0.146529	0.0732647	−	0.997313i	$-0.476658\pi$
$137$	20.0745	0.146529	0.0732647	+	0.997313i	$0.476658\pi$
$138$	0	0
$139$	70.0557	0.503998	0.251999	−	0.967728i	$-0.418912\pi$
$139$	70.0557	0.503998	0.251999	+	0.967728i	$0.418912\pi$
$140$	0	0
$141$	28.1004i	0.199293i
$142$	0	0
$143$	50.9472i	0.356274i
$144$	0	0
$145$	69.6281	0.480193
$146$	0	0
$147$	188.186	1.28017
$148$	0	0
$149$	165.919i	1.11355i	0.830664	+	0.556774i	$0.187961\pi$
$149$	165.919i	1.11355i	−0.830664	+	0.556774i	$0.812039\pi$
$150$	0	0
$151$	− 59.7106i	− 0.395434i	−0.980259	−	0.197717i	$-0.936647\pi$
$151$	− 59.7106i	− 0.395434i	0.980259	−	0.197717i	$-0.0633528\pi$
$152$	0	0
$153$	−90.5429	−0.591784
$154$	0	0
$155$	−100.177	−0.646304
$156$	0	0
$157$	138.133i	0.879829i	0.898040	+	0.439915i	$0.144991\pi$
$157$	138.133i	0.879829i	−0.898040	+	0.439915i	$0.855009\pi$
$158$	0	0
$159$	86.3309i	0.542962i
$160$	0	0
$161$	−31.1918	−0.193738
$162$	0	0
$163$	−178.296	−1.09384	−0.546920	−	0.837185i	$-0.684200\pi$
$163$	−178.296	−1.09384	−0.546920	+	0.837185i	$0.684200\pi$
$164$	0	0
$165$	− 26.4684i	− 0.160414i
$166$	0	0
$167$	98.6284i	0.590589i	0.955406	+	0.295295i	$0.0954178\pi$
$167$	98.6284i	0.590589i	−0.955406	+	0.295295i	$0.904582\pi$
$168$	0	0
$169$	91.7869	0.543118
$170$	0	0
$171$	52.2374	0.305482
$172$	0	0
$173$	166.380i	0.961733i	0.876794	+	0.480867i	$0.159678\pi$
$173$	166.380i	0.961733i	−0.876794	+	0.480867i	$0.840322\pi$
$174$	0	0
$175$	226.673i	1.29528i
$176$	0	0
$177$	186.043	1.05109
$178$	0	0
$179$	−206.496	−1.15361	−0.576805	−	0.816882i	$-0.695701\pi$
$179$	−206.496	−1.15361	−0.576805	+	0.816882i	$0.695701\pi$
$180$	0	0
$181$	− 181.330i	− 1.00182i	−0.865499	−	0.500911i	$-0.832998\pi$
$181$	− 181.330i	− 1.00182i	0.865499	−	0.500911i	$-0.167002\pi$
$182$	0	0
$183$	108.465i	0.592702i
$184$	0	0
$185$	−125.967	−0.680901
$186$	0	0
$187$	174.988	0.935765
$188$	0	0
$189$	− 65.2420i	− 0.345196i
$190$	0	0
$191$	− 185.390i	− 0.970627i	−0.874340	−	0.485314i	$-0.838705\pi$
$191$	− 185.390i	− 0.970627i	0.874340	−	0.485314i	$-0.161295\pi$
$192$	0	0
$193$	15.5439	0.0805382	0.0402691	−	0.999189i	$-0.487178\pi$
$193$	15.5439	0.0805382	0.0402691	+	0.999189i	$0.487178\pi$
$194$	0	0
$195$	40.1142	0.205714
$196$	0	0
$197$	114.464i	0.581034i	0.956870	+	0.290517i	$0.0938273\pi$
$197$	114.464i	0.581034i	−0.956870	+	0.290517i	$0.906173\pi$
$198$	0	0
$199$	299.009i	1.50256i	0.659984	+	0.751280i	$0.270563\pi$
$199$	299.009i	1.50256i	−0.659984	+	0.751280i	$0.729437\pi$
$200$	0	0
$201$	105.576	0.525251
$202$	0	0
$203$	−331.694	−1.63396
$204$	0	0
$205$	140.559i	0.685653i
$206$	0	0
$207$	7.45275i	0.0360036i
$208$	0	0
$209$	−100.957	−0.483046
$210$	0	0
$211$	139.706	0.662111	0.331056	−	0.943611i	$-0.392595\pi$
$211$	139.706	0.662111	0.331056	+	0.943611i	$0.392595\pi$
$212$	0	0
$213$	34.5484i	0.162199i
$214$	0	0
$215$	− 80.9269i	− 0.376404i
$216$	0	0
$217$	477.224	2.19919
$218$	0	0
$219$	8.90152	0.0406462
$220$	0	0
$221$	265.203i	1.20001i
$222$	0	0
$223$	66.1141i	0.296476i	0.988952	+	0.148238i	$0.0473601\pi$
$223$	66.1141i	0.296476i	−0.988952	+	0.148238i	$0.952640\pi$
$224$	0	0
$225$	54.1597	0.240710
$226$	0	0
$227$	−49.9611	−0.220093	−0.110046	−	0.993926i	$-0.535100\pi$
$227$	−49.9611	−0.220093	−0.110046	+	0.993926i	$0.535100\pi$
$228$	0	0
$229$	129.255i	0.564431i	0.959351	+	0.282215i	$0.0910693\pi$
$229$	129.255i	0.564431i	−0.959351	+	0.282215i	$0.908931\pi$
$230$	0	0
$231$	126.090i	0.545845i
$232$	0	0
$233$	196.318	0.842567	0.421284	−	0.906929i	$-0.361580\pi$
$233$	196.318	0.842567	0.421284	+	0.906929i	$0.361580\pi$
$234$	0	0
$235$	−42.7606	−0.181960
$236$	0	0
$237$	− 11.8360i	− 0.0499407i
$238$	0	0
$239$	− 153.654i	− 0.642905i	−0.946926	−	0.321452i	$-0.895829\pi$
$239$	− 153.654i	− 0.642905i	0.946926	−	0.321452i	$-0.104171\pi$
$240$	0	0
$241$	−41.3543	−0.171595	−0.0857974	−	0.996313i	$-0.527344\pi$
$241$	−41.3543	−0.171595	−0.0857974	+	0.996313i	$0.527344\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	286.363i	1.16883i
$246$	0	0
$247$	− 153.005i	− 0.619453i
$248$	0	0
$249$	275.765	1.10749
$250$	0	0
$251$	−35.4495	−0.141233	−0.0706165	−	0.997504i	$-0.522497\pi$
$251$	−35.4495	−0.141233	−0.0706165	+	0.997504i	$0.522497\pi$
$252$	0	0
$253$	− 14.4036i	− 0.0569312i
$254$	0	0
$255$	− 137.780i	− 0.540313i
$256$	0	0
$257$	66.0626	0.257053	0.128527	−	0.991706i	$-0.458975\pi$
$257$	66.0626	0.257053	0.128527	+	0.991706i	$0.458975\pi$
$258$	0	0
$259$	600.081	2.31691
$260$	0	0
$261$	79.2526i	0.303650i
$262$	0	0
$263$	93.5910i	0.355859i	0.984043	+	0.177930i	$0.0569400\pi$
$263$	93.5910i	0.355859i	−0.984043	+	0.177930i	$0.943060\pi$
$264$	0	0
$265$	−131.370	−0.495737
$266$	0	0
$267$	68.3248	0.255898
$268$	0	0
$269$	305.548i	1.13587i	0.823074	+	0.567933i	$0.192257\pi$
$269$	305.548i	1.13587i	−0.823074	+	0.567933i	$0.807743\pi$
$270$	0	0
$271$	− 191.602i	− 0.707017i	−0.935431	−	0.353509i	$-0.884988\pi$
$271$	− 191.602i	− 0.707017i	0.935431	−	0.353509i	$-0.115012\pi$
$272$	0	0
$273$	−191.096	−0.699985
$274$	0	0
$275$	−104.672	−0.380625
$276$	0	0
$277$	− 188.436i	− 0.680276i	−0.940376	−	0.340138i	$-0.889526\pi$
$277$	− 188.436i	− 0.680276i	0.940376	−	0.340138i	$-0.110474\pi$
$278$	0	0
$279$	− 114.024i	− 0.408690i
$280$	0	0
$281$	296.087	1.05369	0.526844	−	0.849962i	$-0.323375\pi$
$281$	296.087	1.05369	0.526844	+	0.849962i	$0.323375\pi$
$282$	0	0
$283$	−260.681	−0.921136	−0.460568	−	0.887625i	$-0.652354\pi$
$283$	−260.681	−0.921136	−0.460568	+	0.887625i	$0.652354\pi$
$284$	0	0
$285$	79.4900i	0.278912i
$286$	0	0
$287$	− 669.595i	− 2.33308i
$288$	0	0
$289$	621.891	2.15187
$290$	0	0
$291$	104.953	0.360662
$292$	0	0
$293$	− 314.756i	− 1.07425i	−0.843501	−	0.537127i	$-0.819510\pi$
$293$	− 314.756i	− 1.07425i	0.843501	−	0.537127i	$-0.180490\pi$
$294$	0	0
$295$	283.103i	0.959672i
$296$	0	0
$297$	30.1271	0.101438
$298$	0	0
$299$	21.8294	0.0730079
$300$	0	0
$301$	385.520i	1.28080i
$302$	0	0
$303$	292.849i	0.966497i
$304$	0	0
$305$	−165.051	−0.541152
$306$	0	0
$307$	168.120	0.547621	0.273811	−	0.961784i	$-0.411716\pi$
$307$	168.120	0.547621	0.273811	+	0.961784i	$0.411716\pi$
$308$	0	0
$309$	− 283.000i	− 0.915856i
$310$	0	0
$311$	− 468.033i	− 1.50493i	−0.658633	−	0.752464i	$-0.728865\pi$
$311$	− 468.033i	− 1.50493i	0.658633	−	0.752464i	$-0.271135\pi$
$312$	0	0
$313$	−225.370	−0.720033	−0.360017	−	0.932946i	$-0.617229\pi$
$313$	−225.370	−0.720033	−0.360017	+	0.932946i	$0.617229\pi$
$314$	0	0
$315$	99.2793	0.315172
$316$	0	0
$317$	− 162.613i	− 0.512976i	−0.966547	−	0.256488i	$-0.917435\pi$
$317$	− 162.613i	− 0.512976i	0.966547	−	0.256488i	$-0.0825654\pi$
$318$	0	0
$319$	− 153.168i	− 0.480150i
$320$	0	0
$321$	−64.8285	−0.201958
$322$	0	0
$323$	−525.525	−1.62701
$324$	0	0
$325$	− 158.635i	− 0.488109i
$326$	0	0
$327$	363.933i	1.11294i
$328$	0	0
$329$	203.703	0.619158
$330$	0	0
$331$	−254.527	−0.768964	−0.384482	−	0.923133i	$-0.625620\pi$
$331$	−254.527	−0.768964	−0.384482	+	0.923133i	$0.625620\pi$
$332$	0	0
$333$	− 143.379i	− 0.430567i
$334$	0	0
$335$	160.655i	0.479567i
$336$	0	0
$337$	−94.7347	−0.281112	−0.140556	−	0.990073i	$-0.544889\pi$
$337$	−94.7347	−0.281112	−0.140556	+	0.990073i	$0.544889\pi$
$338$	0	0
$339$	−330.490	−0.974897
$340$	0	0
$341$	220.370i	0.646246i
$342$	0	0
$343$	− 748.942i	− 2.18351i
$344$	0	0
$345$	−11.3409	−0.0328722
$346$	0	0
$347$	−173.315	−0.499467	−0.249734	−	0.968315i	$-0.580343\pi$
$347$	−173.315	−0.499467	−0.249734	+	0.968315i	$0.580343\pi$
$348$	0	0
$349$	195.247i	0.559448i	0.960081	+	0.279724i	$0.0902429\pi$
$349$	195.247i	0.559448i	−0.960081	+	0.279724i	$0.909757\pi$
$350$	0	0
$351$	45.6591i	0.130083i
$352$	0	0
$353$	296.107	0.838830	0.419415	−	0.907795i	$-0.362235\pi$
$353$	296.107	0.838830	0.419415	+	0.907795i	$0.362235\pi$
$354$	0	0
$355$	−52.5726	−0.148092
$356$	0	0
$357$	656.356i	1.83853i
$358$	0	0
$359$	448.964i	1.25060i	0.780386	+	0.625298i	$0.215023\pi$
$359$	448.964i	1.25060i	−0.780386	+	0.625298i	$0.784977\pi$
$360$	0	0
$361$	−57.8065	−0.160129
$362$	0	0
$363$	151.353	0.416950
$364$	0	0
$365$	13.5455i	0.0371110i
$366$	0	0
$367$	623.564i	1.69908i	0.527521	+	0.849542i	$0.323122\pi$
$367$	623.564i	1.69908i	−0.527521	+	0.849542i	$0.676878\pi$
$368$	0	0
$369$	−159.988	−0.433572
$370$	0	0
$371$	625.823	1.68685
$372$	0	0
$373$	297.116i	0.796558i	0.917264	+	0.398279i	$0.130392\pi$
$373$	297.116i	0.796558i	−0.917264	+	0.398279i	$0.869608\pi$
$374$	0	0
$375$	196.543i	0.524115i
$376$	0	0
$377$	232.134	0.615739
$378$	0	0
$379$	−224.021	−0.591084	−0.295542	−	0.955330i	$-0.595500\pi$
$379$	−224.021	−0.591084	−0.295542	+	0.955330i	$0.595500\pi$
$380$	0	0
$381$	59.6478i	0.156556i
$382$	0	0
$383$	573.847i	1.49830i	0.662403	+	0.749148i	$0.269537\pi$
$383$	573.847i	1.49830i	−0.662403	+	0.749148i	$0.730463\pi$
$384$	0	0
$385$	−191.872	−0.498370
$386$	0	0
$387$	92.1133	0.238019
$388$	0	0
$389$	− 148.849i	− 0.382646i	−0.981527	−	0.191323i	$-0.938722\pi$
$389$	− 148.849i	− 0.382646i	0.981527	−	0.191323i	$-0.0612779\pi$
$390$	0	0
$391$	− 74.9771i	− 0.191757i
$392$	0	0
$393$	−160.300	−0.407887
$394$	0	0
$395$	18.0109	0.0455971
$396$	0	0
$397$	245.952i	0.619526i	0.950814	+	0.309763i	$0.100250\pi$
$397$	245.952i	0.619526i	−0.950814	+	0.309763i	$0.899750\pi$
$398$	0	0
$399$	− 378.675i	− 0.949059i
$400$	0	0
$401$	64.0509	0.159728	0.0798640	−	0.996806i	$-0.474551\pi$
$401$	64.0509	0.159728	0.0798640	+	0.996806i	$0.474551\pi$
$402$	0	0
$403$	−333.981	−0.828738
$404$	0	0
$405$	− 23.7211i	− 0.0585705i
$406$	0	0
$407$	277.102i	0.680840i
$408$	0	0
$409$	−555.403	−1.35795	−0.678977	−	0.734160i	$-0.737576\pi$
$409$	−555.403	−1.35795	−0.678977	+	0.734160i	$0.737576\pi$
$410$	0	0
$411$	−34.7701	−0.0845988
$412$	0	0
$413$	− 1348.65i	− 3.26549i
$414$	0	0
$415$	419.634i	1.01117i
$416$	0	0
$417$	−121.340	−0.290983
$418$	0	0
$419$	−793.486	−1.89376	−0.946881	−	0.321583i	$-0.895785\pi$
$419$	−793.486	−1.89376	−0.946881	+	0.321583i	$0.895785\pi$
$420$	0	0
$421$	− 87.1175i	− 0.206930i	−0.994633	−	0.103465i	$-0.967007\pi$
$421$	− 87.1175i	− 0.206930i	0.994633	−	0.103465i	$-0.0329930\pi$
$422$	0	0
$423$	− 48.6713i	− 0.115062i
$424$	0	0
$425$	−544.864	−1.28203
$426$	0	0
$427$	786.272	1.84139
$428$	0	0
$429$	− 88.2432i	− 0.205695i
$430$	0	0
$431$	− 432.932i	− 1.00448i	−0.864727	−	0.502242i	$-0.832509\pi$
$431$	− 432.932i	− 1.00448i	0.864727	−	0.502242i	$-0.167491\pi$
$432$	0	0
$433$	391.344	0.903798	0.451899	−	0.892069i	$-0.350747\pi$
$433$	391.344	0.903798	0.451899	+	0.892069i	$0.350747\pi$
$434$	0	0
$435$	−120.599	−0.277240
$436$	0	0
$437$	43.2569i	0.0989860i
$438$	0	0
$439$	12.3066i	0.0280332i	0.999902	+	0.0140166i	$0.00446177\pi$
$439$	12.3066i	0.0280332i	−0.999902	+	0.0140166i	$0.995538\pi$
$440$	0	0
$441$	−325.947	−0.739109
$442$	0	0
$443$	567.554	1.28116	0.640580	−	0.767892i	$-0.278694\pi$
$443$	567.554	1.28116	0.640580	+	0.767892i	$0.278694\pi$
$444$	0	0
$445$	103.970i	0.233641i
$446$	0	0
$447$	− 287.380i	− 0.642908i
$448$	0	0
$449$	−407.897	−0.908457	−0.454229	−	0.890885i	$-0.650085\pi$
$449$	−407.897	−0.908457	−0.454229	+	0.890885i	$0.650085\pi$
$450$	0	0
$451$	309.201	0.685591
$452$	0	0
$453$	103.422i	0.228304i
$454$	0	0
$455$	− 290.792i	− 0.639104i
$456$	0	0
$457$	−231.789	−0.507197	−0.253598	−	0.967310i	$-0.581614\pi$
$457$	−231.789	−0.507197	−0.253598	+	0.967310i	$0.581614\pi$
$458$	0	0
$459$	156.825	0.341667
$460$	0	0
$461$	− 438.217i	− 0.950578i	−0.879830	−	0.475289i	$-0.842343\pi$
$461$	− 438.217i	− 0.950578i	0.879830	−	0.475289i	$-0.157657\pi$
$462$	0	0
$463$	185.141i	0.399873i	0.979809	+	0.199936i	$0.0640735\pi$
$463$	185.141i	0.399873i	−0.979809	+	0.199936i	$0.935926\pi$
$464$	0	0
$465$	173.512	0.373144
$466$	0	0
$467$	82.9765	0.177680	0.0888399	−	0.996046i	$-0.471684\pi$
$467$	82.9765	0.177680	0.0888399	+	0.996046i	$0.471684\pi$
$468$	0	0
$469$	− 765.329i	− 1.63183i
$470$	0	0
$471$	− 239.254i	− 0.507970i
$472$	0	0
$473$	−178.023	−0.376370
$474$	0	0
$475$	314.351	0.661791
$476$	0	0
$477$	− 149.530i	− 0.313479i
$478$	0	0
$479$	− 173.847i	− 0.362936i	−0.983397	−	0.181468i	$-0.941915\pi$
$479$	− 173.847i	− 0.362936i	0.983397	−	0.181468i	$-0.0580849\pi$
$480$	0	0
$481$	−419.962	−0.873101
$482$	0	0
$483$	54.0258	0.111855
$484$	0	0
$485$	159.707i	0.329293i
$486$	0	0
$487$	− 250.667i	− 0.514716i	−0.966316	−	0.257358i	$-0.917148\pi$
$487$	− 250.667i	− 0.514716i	0.966316	−	0.257358i	$-0.0828520\pi$
$488$	0	0
$489$	308.817	0.631529
$490$	0	0
$491$	283.478	0.577349	0.288674	−	0.957427i	$-0.406786\pi$
$491$	283.478	0.577349	0.288674	+	0.957427i	$0.406786\pi$
$492$	0	0
$493$	− 797.307i	− 1.61726i
$494$	0	0
$495$	45.8446i	0.0926153i
$496$	0	0
$497$	250.445	0.503914
$498$	0	0
$499$	192.802	0.386377	0.193188	−	0.981162i	$-0.438117\pi$
$499$	192.802	0.386377	0.193188	+	0.981162i	$0.438117\pi$
$500$	0	0
$501$	− 170.829i	− 0.340977i
$502$	0	0
$503$	− 398.251i	− 0.791752i	−0.918304	−	0.395876i	$-0.870441\pi$
$503$	− 398.251i	− 0.791752i	0.918304	−	0.395876i	$-0.129559\pi$
$504$	0	0
$505$	−445.630	−0.882436
$506$	0	0
$507$	−158.980	−0.313569
$508$	0	0
$509$	550.212i	1.08097i	0.841355	+	0.540483i	$0.181758\pi$
$509$	550.212i	1.08097i	−0.841355	+	0.540483i	$0.818242\pi$
$510$	0	0
$511$	− 64.5281i	− 0.126278i
$512$	0	0
$513$	−90.4778	−0.176370
$514$	0	0
$515$	430.643	0.836199
$516$	0	0
$517$	94.0647i	0.181943i
$518$	0	0
$519$	− 288.178i	− 0.555257i
$520$	0	0
$521$	164.181	0.315126	0.157563	−	0.987509i	$-0.449636\pi$
$521$	164.181	0.315126	0.157563	+	0.987509i	$0.449636\pi$
$522$	0	0
$523$	−643.556	−1.23051	−0.615255	−	0.788328i	$-0.710947\pi$
$523$	−643.556	−1.23051	−0.615255	+	0.788328i	$0.710947\pi$
$524$	0	0
$525$	− 392.610i	− 0.747828i
$526$	0	0
$527$	1147.12i	2.17670i
$528$	0	0
$529$	522.828	0.988334
$530$	0	0
$531$	−322.236	−0.606848
$532$	0	0
$533$	468.610i	0.879194i
$534$	0	0
$535$	− 98.6501i	− 0.184393i
$536$	0	0
$537$	357.662	0.666037
$538$	0	0
$539$	629.942	1.16872
$540$	0	0
$541$	− 727.713i	− 1.34513i	−0.740040	−	0.672563i	$-0.765193\pi$
$541$	− 727.713i	− 1.34513i	0.740040	−	0.672563i	$-0.234807\pi$
$542$	0	0
$543$	314.072i	0.578402i
$544$	0	0
$545$	−553.799	−1.01615
$546$	0	0
$547$	−97.2958	−0.177872	−0.0889359	−	0.996037i	$-0.528347\pi$
$547$	−97.2958	−0.177872	−0.0889359	+	0.996037i	$0.528347\pi$
$548$	0	0
$549$	− 187.866i	− 0.342197i
$550$	0	0
$551$	459.994i	0.834835i
$552$	0	0
$553$	−85.8002	−0.155154
$554$	0	0
$555$	218.181	0.393119
$556$	0	0
$557$	520.548i	0.934556i	0.884110	+	0.467278i	$0.154765\pi$
$557$	520.548i	0.934556i	−0.884110	+	0.467278i	$0.845235\pi$
$558$	0	0
$559$	− 269.803i	− 0.482653i
$560$	0	0
$561$	−303.088	−0.540264
$562$	0	0
$563$	525.371	0.933163	0.466581	−	0.884478i	$-0.345485\pi$
$563$	525.371	0.933163	0.466581	+	0.884478i	$0.345485\pi$
$564$	0	0
$565$	− 502.909i	− 0.890105i
$566$	0	0
$567$	113.003i	0.199299i
$568$	0	0
$569$	−824.774	−1.44951	−0.724757	−	0.689004i	$-0.758048\pi$
$569$	−824.774	−1.44951	−0.724757	+	0.689004i	$0.758048\pi$
$570$	0	0
$571$	−1088.90	−1.90701	−0.953504	−	0.301381i	$-0.902552\pi$
$571$	−1088.90	−1.90701	−0.953504	+	0.301381i	$0.902552\pi$
$572$	0	0
$573$	321.105i	0.560392i
$574$	0	0
$575$	44.8487i	0.0779978i
$576$	0	0
$577$	−688.463	−1.19318	−0.596589	−	0.802547i	$-0.703478\pi$
$577$	−688.463	−1.19318	−0.596589	+	0.802547i	$0.703478\pi$
$578$	0	0
$579$	−26.9228	−0.0464988
$580$	0	0
$581$	− 1999.05i	− 3.44071i
$582$	0	0
$583$	288.989i	0.495692i
$584$	0	0
$585$	−69.4798	−0.118769
$586$	0	0
$587$	−441.108	−0.751462	−0.375731	−	0.926729i	$-0.622608\pi$
$587$	−441.108	−0.751462	−0.375731	+	0.926729i	$0.622608\pi$
$588$	0	0
$589$	− 661.815i	− 1.12363i
$590$	0	0
$591$	− 198.257i	− 0.335460i
$592$	0	0
$593$	−481.317	−0.811664	−0.405832	−	0.913948i	$-0.633018\pi$
$593$	−481.317	−0.811664	−0.405832	+	0.913948i	$0.633018\pi$
$594$	0	0
$595$	−998.782	−1.67862
$596$	0	0
$597$	− 517.899i	− 0.867503i
$598$	0	0
$599$	541.322i	0.903709i	0.892092	+	0.451854i	$0.149237\pi$
$599$	541.322i	0.903709i	−0.892092	+	0.451854i	$0.850763\pi$
$600$	0	0
$601$	−64.7772	−0.107782	−0.0538912	−	0.998547i	$-0.517162\pi$
$601$	−64.7772	−0.107782	−0.0538912	+	0.998547i	$0.517162\pi$
$602$	0	0
$603$	−182.862	−0.303254
$604$	0	0
$605$	230.315i	0.380686i
$606$	0	0
$607$	− 608.260i	− 1.00208i	−0.865426	−	0.501038i	$-0.832952\pi$
$607$	− 608.260i	− 1.00208i	0.865426	−	0.501038i	$-0.167048\pi$
$608$	0	0
$609$	574.511	0.943368
$610$	0	0
$611$	−142.560	−0.233322
$612$	0	0
$613$	924.407i	1.50800i	0.656872	+	0.754002i	$0.271879\pi$
$613$	924.407i	1.50800i	−0.656872	+	0.754002i	$0.728121\pi$
$614$	0	0
$615$	− 243.455i	− 0.395862i
$616$	0	0
$617$	−652.781	−1.05799	−0.528996	−	0.848624i	$-0.677431\pi$
$617$	−652.781	−1.05799	−0.528996	+	0.848624i	$0.677431\pi$
$618$	0	0
$619$	−246.098	−0.397574	−0.198787	−	0.980043i	$-0.563700\pi$
$619$	−246.098	−0.397574	−0.198787	+	0.980043i	$0.563700\pi$
$620$	0	0
$621$	− 12.9085i	− 0.0207867i
$622$	0	0
$623$	− 495.294i	− 0.795015i
$624$	0	0
$625$	152.249	0.243599
$626$	0	0
$627$	174.862	0.278887
$628$	0	0
$629$	1442.44i	2.29323i
$630$	0	0
$631$	563.715i	0.893368i	0.894692	+	0.446684i	$0.147395\pi$
$631$	563.715i	0.893368i	−0.894692	+	0.446684i	$0.852605\pi$
$632$	0	0
$633$	−241.977	−0.382270
$634$	0	0
$635$	−90.7665	−0.142939
$636$	0	0
$637$	954.709i	1.49876i
$638$	0	0
$639$	− 59.8396i	− 0.0936457i
$640$	0	0
$641$	−163.485	−0.255046	−0.127523	−	0.991836i	$-0.540703\pi$
$641$	−163.485	−0.255046	−0.127523	+	0.991836i	$0.540703\pi$
$642$	0	0
$643$	−307.908	−0.478862	−0.239431	−	0.970913i	$-0.576961\pi$
$643$	−307.908	−0.478862	−0.239431	+	0.970913i	$0.576961\pi$
$644$	0	0
$645$	140.170i	0.217317i
$646$	0	0
$647$	909.931i	1.40638i	0.711000	+	0.703192i	$0.248243\pi$
$647$	909.931i	1.40638i	−0.711000	+	0.703192i	$0.751757\pi$
$648$	0	0
$649$	622.771	0.959585
$650$	0	0
$651$	−826.576	−1.26970
$652$	0	0
$653$	299.487i	0.458632i	0.973352	+	0.229316i	$0.0736489\pi$
$653$	299.487i	0.458632i	−0.973352	+	0.229316i	$0.926351\pi$
$654$	0	0
$655$	− 243.929i	− 0.372411i
$656$	0	0
$657$	−15.4179	−0.0234671
$658$	0	0
$659$	141.105	0.214120	0.107060	−	0.994253i	$-0.465856\pi$
$659$	141.105	0.214120	0.107060	+	0.994253i	$0.465856\pi$
$660$	0	0
$661$	− 1176.74i	− 1.78024i	−0.455725	−	0.890121i	$-0.650620\pi$
$661$	− 1176.74i	− 1.78024i	0.455725	−	0.890121i	$-0.349380\pi$
$662$	0	0
$663$	− 459.345i	− 0.692829i
$664$	0	0
$665$	576.232	0.866514
$666$	0	0
$667$	−65.6278	−0.0983925
$668$	0	0
$669$	− 114.513i	− 0.171170i
$670$	0	0
$671$	363.080i	0.541103i
$672$	0	0
$673$	−230.780	−0.342912	−0.171456	−	0.985192i	$-0.554847\pi$
$673$	−230.780	−0.342912	−0.171456	+	0.985192i	$0.554847\pi$
$674$	0	0
$675$	−93.8073	−0.138974
$676$	0	0
$677$	− 768.499i	− 1.13515i	−0.823320	−	0.567577i	$-0.807881\pi$
$677$	− 768.499i	− 1.13515i	0.823320	−	0.567577i	$-0.192119\pi$
$678$	0	0
$679$	− 760.813i	− 1.12049i
$680$	0	0
$681$	86.5352	0.127071
$682$	0	0
$683$	963.083	1.41008	0.705039	−	0.709169i	$-0.250930\pi$
$683$	963.083	1.41008	0.705039	+	0.709169i	$0.250930\pi$
$684$	0	0
$685$	− 52.9099i	− 0.0772408i
$686$	0	0
$687$	− 223.876i	− 0.325874i
$688$	0	0
$689$	−437.977	−0.635670
$690$	0	0
$691$	376.440	0.544776	0.272388	−	0.962188i	$-0.412187\pi$
$691$	376.440	0.544776	0.272388	+	0.962188i	$0.412187\pi$
$692$	0	0
$693$	− 218.395i	− 0.315144i
$694$	0	0
$695$	− 184.644i	− 0.265675i
$696$	0	0
$697$	1609.53	2.30923
$698$	0	0
$699$	−340.033	−0.486456
$700$	0	0
$701$	− 543.266i	− 0.774988i	−0.921872	−	0.387494i	$-0.873341\pi$
$701$	− 543.266i	− 0.774988i	0.921872	−	0.387494i	$-0.126659\pi$
$702$	0	0
$703$	− 832.193i	− 1.18377i
$704$	0	0
$705$	74.0635	0.105055
$706$	0	0
$707$	2122.89	3.00268
$708$	0	0
$709$	403.852i	0.569608i	0.958586	+	0.284804i	$0.0919286\pi$
$709$	403.852i	0.569608i	−0.958586	+	0.284804i	$0.908071\pi$
$710$	0	0
$711$	20.5005i	0.0288333i
$712$	0	0
$713$	94.4218	0.132429
$714$	0	0
$715$	134.280	0.187805
$716$	0	0
$717$	266.137i	0.371181i
$718$	0	0
$719$	− 183.791i	− 0.255620i	−0.991799	−	0.127810i	$-0.959205\pi$
$719$	− 183.791i	− 0.255620i	0.991799	−	0.127810i	$-0.0407948\pi$
$720$	0	0
$721$	−2051.50	−2.84535
$722$	0	0
$723$	71.6278	0.0990703
$724$	0	0
$725$	476.922i	0.657823i
$726$	0	0
$727$	843.226i	1.15987i	0.814663	+	0.579935i	$0.196922\pi$
$727$	843.226i	1.15987i	−0.814663	+	0.579935i	$0.803078\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−926.690	−1.26770
$732$	0	0
$733$	286.176i	0.390418i	0.980762	+	0.195209i	$0.0625385\pi$
$733$	286.176i	0.390418i	−0.980762	+	0.195209i	$0.937461\pi$
$734$	0	0
$735$	− 495.996i	− 0.674824i
$736$	0	0
$737$	353.409	0.479524
$738$	0	0
$739$	−821.243	−1.11129	−0.555645	−	0.831420i	$-0.687529\pi$
$739$	−821.243	−1.11129	−0.555645	+	0.831420i	$0.687529\pi$
$740$	0	0
$741$	265.012i	0.357641i
$742$	0	0
$743$	− 1227.08i	− 1.65153i	−0.564018	−	0.825763i	$-0.690745\pi$
$743$	− 1227.08i	− 1.65153i	0.564018	−	0.825763i	$-0.309255\pi$
$744$	0	0
$745$	437.308	0.586990
$746$	0	0
$747$	−477.639	−0.639410
$748$	0	0
$749$	469.949i	0.627436i
$750$	0	0
$751$	− 966.507i	− 1.28696i	−0.765463	−	0.643480i	$-0.777490\pi$
$751$	− 966.507i	− 1.28696i	0.765463	−	0.643480i	$-0.222510\pi$
$752$	0	0
$753$	61.4003	0.0815409
$754$	0	0
$755$	−157.378	−0.208447
$756$	0	0
$757$	876.359i	1.15767i	0.815443	+	0.578837i	$0.196493\pi$
$757$	876.359i	1.15767i	−0.815443	+	0.578837i	$0.803507\pi$
$758$	0	0
$759$	24.9477i	0.0328692i
$760$	0	0
$761$	−217.575	−0.285907	−0.142953	−	0.989729i	$-0.545660\pi$
$761$	−217.575	−0.285907	−0.142953	+	0.989729i	$0.545660\pi$
$762$	0	0
$763$	2638.19	3.45765
$764$	0	0
$765$	238.642i	0.311950i
$766$	0	0
$767$	943.840i	1.23056i
$768$	0	0
$769$	497.583	0.647053	0.323526	−	0.946219i	$-0.395132\pi$
$769$	497.583	0.647053	0.323526	+	0.946219i	$0.395132\pi$
$770$	0	0
$771$	−114.424	−0.148410
$772$	0	0
$773$	− 664.490i	− 0.859624i	−0.902918	−	0.429812i	$-0.858580\pi$
$773$	− 664.490i	− 0.859624i	0.902918	−	0.429812i	$-0.141420\pi$
$774$	0	0
$775$	− 686.170i	− 0.885380i
$776$	0	0
$777$	−1039.37	−1.33767
$778$	0	0
$779$	−928.595	−1.19203
$780$	0	0
$781$	115.649i	0.148078i
$782$	0	0
$783$	− 137.270i	− 0.175312i
$784$	0	0
$785$	364.074	0.463789
$786$	0	0
$787$	214.321	0.272326	0.136163	−	0.990686i	$-0.456523\pi$
$787$	214.321	0.272326	0.136163	+	0.990686i	$0.456523\pi$
$788$	0	0
$789$	− 162.104i	− 0.205456i
$790$	0	0
$791$	2395.76i	3.02878i
$792$	0	0
$793$	−550.266	−0.693904
$794$	0	0
$795$	227.540	0.286214
$796$	0	0
$797$	487.535i	0.611713i	0.952078	+	0.305857i	$0.0989428\pi$
$797$	487.535i	0.611713i	−0.952078	+	0.305857i	$0.901057\pi$
$798$	0	0
$799$	489.649i	0.612827i
$800$	0	0
$801$	−118.342	−0.147743
$802$	0	0
$803$	29.7974	0.0371076
$804$	0	0
$805$	82.2115i	0.102126i
$806$	0	0
$807$	− 529.225i	− 0.655793i
$808$	0	0
$809$	1232.31	1.52325	0.761623	−	0.648021i	$-0.224403\pi$
$809$	1232.31	1.52325	0.761623	+	0.648021i	$0.224403\pi$
$810$	0	0
$811$	−38.1721	−0.0470679	−0.0235340	−	0.999723i	$-0.507492\pi$
$811$	−38.1721	−0.0470679	−0.0235340	+	0.999723i	$0.507492\pi$
$812$	0	0
$813$	331.864i	0.408197i
$814$	0	0
$815$	469.930i	0.576601i
$816$	0	0
$817$	534.640	0.654394
$818$	0	0
$819$	330.988	0.404137
$820$	0	0
$821$	− 490.147i	− 0.597012i	−0.954408	−	0.298506i	$-0.903512\pi$
$821$	− 490.147i	− 0.597012i	0.954408	−	0.298506i	$-0.0964883\pi$
$822$	0	0
$823$	582.258i	0.707482i	0.935343	+	0.353741i	$0.115091\pi$
$823$	582.258i	0.707482i	−0.935343	+	0.353741i	$0.884909\pi$
$824$	0	0
$825$	181.297	0.219754
$826$	0	0
$827$	852.873	1.03128	0.515642	−	0.856804i	$-0.327553\pi$
$827$	852.873	1.03128	0.515642	+	0.856804i	$0.327553\pi$
$828$	0	0
$829$	692.476i	0.835315i	0.908605	+	0.417657i	$0.137149\pi$
$829$	692.476i	0.835315i	−0.908605	+	0.417657i	$0.862851\pi$
$830$	0	0
$831$	326.381i	0.392757i
$832$	0	0
$833$	3279.13	3.93653
$834$	0	0
$835$	259.952	0.311320
$836$	0	0
$837$	197.496i	0.235957i
$838$	0	0
$839$	− 761.022i	− 0.907059i	−0.891241	−	0.453529i	$-0.850165\pi$
$839$	− 761.022i	− 0.907059i	0.891241	−	0.453529i	$-0.149835\pi$
$840$	0	0
$841$	143.113	0.170170
$842$	0	0
$843$	−512.837	−0.608347
$844$	0	0
$845$	− 241.920i	− 0.286296i
$846$	0	0
$847$	− 1097.17i	− 1.29537i
$848$	0	0
$849$	451.513	0.531818
$850$	0	0
$851$	118.730	0.139518
$852$	0	0
$853$	1245.38i	1.46000i	0.683448	+	0.729999i	$0.260480\pi$
$853$	1245.38i	1.46000i	−0.683448	+	0.729999i	$0.739520\pi$
$854$	0	0
$855$	− 137.681i	− 0.161030i
$856$	0	0
$857$	1517.00	1.77013	0.885064	−	0.465469i	$-0.154114\pi$
$857$	1517.00	1.77013	0.885064	+	0.465469i	$0.154114\pi$
$858$	0	0
$859$	−1311.75	−1.52706	−0.763532	−	0.645770i	$-0.776537\pi$
$859$	−1311.75	−1.52706	−0.763532	+	0.645770i	$0.776537\pi$
$860$	0	0
$861$	1159.77i	1.34701i
$862$	0	0
$863$	1304.34i	1.51140i	0.654920	+	0.755698i	$0.272702\pi$
$863$	1304.34i	1.51140i	−0.654920	+	0.755698i	$0.727298\pi$
$864$	0	0
$865$	438.523	0.506963
$866$	0	0
$867$	−1077.15	−1.24238
$868$	0	0
$869$	− 39.6203i	− 0.0455930i
$870$	0	0
$871$	535.609i	0.614936i
$872$	0	0
$873$	−181.783	−0.208228
$874$	0	0
$875$	1424.76	1.62830
$876$	0	0
$877$	− 1207.07i	− 1.37637i	−0.725537	−	0.688183i	$-0.758409\pi$
$877$	− 1207.07i	− 1.37637i	0.725537	−	0.688183i	$-0.241591\pi$
$878$	0	0
$879$	545.174i	0.620221i
$880$	0	0
$881$	−375.974	−0.426758	−0.213379	−	0.976970i	$-0.568447\pi$
$881$	−375.974	−0.426758	−0.213379	+	0.976970i	$0.568447\pi$
$882$	0	0
$883$	−395.955	−0.448420	−0.224210	−	0.974541i	$-0.571980\pi$
$883$	−395.955	−0.448420	−0.224210	+	0.974541i	$0.571980\pi$
$884$	0	0
$885$	− 490.349i	− 0.554067i
$886$	0	0
$887$	960.050i	1.08236i	0.840908	+	0.541178i	$0.182021\pi$
$887$	960.050i	1.08236i	−0.840908	+	0.541178i	$0.817979\pi$
$888$	0	0
$889$	432.394	0.486382
$890$	0	0
$891$	−52.1816	−0.0585652
$892$	0	0
$893$	− 282.496i	− 0.316344i
$894$	0	0
$895$	544.256i	0.608108i
$896$	0	0
$897$	−37.8095	−0.0421511
$898$	0	0
$899$	1004.08	1.11689
$900$	0	0
$901$	1504.32i	1.66961i
$902$	0	0
$903$	− 667.740i	− 0.739468i
$904$	0	0
$905$	−477.926	−0.528095
$906$	0	0
$907$	−1301.18	−1.43460	−0.717298	−	0.696766i	$-0.754622\pi$
$907$	−1301.18	−1.43460	−0.717298	+	0.696766i	$0.754622\pi$
$908$	0	0
$909$	− 507.229i	− 0.558008i
$910$	0	0
$911$	1274.60i	1.39912i	0.714573	+	0.699561i	$0.246621\pi$
$911$	1274.60i	1.39912i	−0.714573	+	0.699561i	$0.753379\pi$
$912$	0	0
$913$	923.111	1.01107
$914$	0	0
$915$	285.877	0.312434
$916$	0	0
$917$	1162.03i	1.26721i
$918$	0	0
$919$	− 957.109i	− 1.04147i	−0.853719	−	0.520734i	$-0.825658\pi$
$919$	− 957.109i	− 1.04147i	0.853719	−	0.520734i	$-0.174342\pi$
$920$	0	0
$921$	−291.192	−0.316169
$922$	0	0
$923$	−175.272	−0.189894
$924$	0	0
$925$	− 862.817i	− 0.932775i
$926$	0	0
$927$	490.170i	0.528770i
$928$	0	0
$929$	−912.989	−0.982765	−0.491383	−	0.870944i	$-0.663508\pi$
$929$	−912.989	−0.982765	−0.491383	+	0.870944i	$0.663508\pi$
$930$	0	0
$931$	−1891.84	−2.03206
$932$	0	0
$933$	810.657i	0.868871i
$934$	0	0
$935$	− 461.212i	− 0.493274i
$936$	0	0
$937$	−751.958	−0.802516	−0.401258	−	0.915965i	$-0.631427\pi$
$937$	−751.958	−0.802516	−0.401258	+	0.915965i	$0.631427\pi$
$938$	0	0
$939$	390.353	0.415711
$940$	0	0
$941$	417.550i	0.443730i	0.975077	+	0.221865i	$0.0712144\pi$
$941$	417.550i	0.443730i	−0.975077	+	0.221865i	$0.928786\pi$
$942$	0	0
$943$	− 132.484i	− 0.140492i
$944$	0	0
$945$	−171.957	−0.181965
$946$	0	0
$947$	−928.909	−0.980897	−0.490448	−	0.871470i	$-0.663167\pi$
$947$	−928.909	−0.980897	−0.490448	+	0.871470i	$0.663167\pi$
$948$	0	0
$949$	45.1595i	0.0475864i
$950$	0	0
$951$	281.655i	0.296167i
$952$	0	0
$953$	−676.685	−0.710058	−0.355029	−	0.934855i	$-0.615529\pi$
$953$	−676.685	−0.710058	−0.355029	+	0.934855i	$0.615529\pi$
$954$	0	0
$955$	−488.627	−0.511652
$956$	0	0
$957$	265.294i	0.277215i
$958$	0	0
$959$	252.053i	0.262828i
$960$	0	0
$961$	−483.620	−0.503247
$962$	0	0
$963$	112.286	0.116601
$964$	0	0
$965$	− 40.9686i	− 0.0424545i
$966$	0	0
$967$	− 208.530i	− 0.215646i	−0.994170	−	0.107823i	$-0.965612\pi$
$967$	− 208.530i	− 0.215646i	0.994170	−	0.107823i	$-0.0343880\pi$
$968$	0	0
$969$	910.236	0.939356
$970$	0	0
$971$	932.159	0.959999	0.480000	−	0.877269i	$-0.340637\pi$
$971$	932.159	0.959999	0.480000	+	0.877269i	$0.340637\pi$
$972$	0	0
$973$	879.608i	0.904016i
$974$	0	0
$975$	274.765i	0.281810i
$976$	0	0
$977$	1567.22	1.60412	0.802059	−	0.597245i	$-0.203738\pi$
$977$	1567.22	1.60412	0.802059	+	0.597245i	$0.203738\pi$
$978$	0	0
$979$	228.714	0.233620
$980$	0	0
$981$	− 630.350i	− 0.642559i
$982$	0	0
$983$	− 1201.11i	− 1.22188i	−0.791676	−	0.610941i	$-0.790791\pi$
$983$	− 1201.11i	− 1.22188i	0.791676	−	0.610941i	$-0.209209\pi$
$984$	0	0
$985$	301.689	0.306283
$986$	0	0
$987$	−352.824	−0.357471
$988$	0	0
$989$	76.2775i	0.0771259i
$990$	0	0
$991$	− 568.813i	− 0.573979i	−0.957934	−	0.286989i	$-0.907346\pi$
$991$	− 568.813i	− 0.573979i	0.957934	−	0.286989i	$-0.0926545\pi$
$992$	0	0
$993$	440.854	0.443961
$994$	0	0
$995$	788.091	0.792052
$996$	0	0
$997$	− 66.4659i	− 0.0666659i	−0.999444	−	0.0333330i	$-0.989388\pi$
$997$	− 66.4659i	− 0.0666659i	0.999444	−	0.0333330i	$-0.0106122\pi$
$998$	0	0
$999$	248.340i	0.248588i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.b.f.127.2		8
3.2	odd	2		2304.3.b.q.127.6		8
4.3	odd	2		768.3.b.e.127.6		8
8.3	odd	2	inner	768.3.b.f.127.3		8
8.5	even	2		768.3.b.e.127.7		8
12.11	even	2		2304.3.b.t.127.6		8
16.3	odd	4		384.3.g.a.127.6	yes	8
16.5	even	4		384.3.g.b.127.7	yes	8
16.11	odd	4		384.3.g.b.127.3	yes	8
16.13	even	4		384.3.g.a.127.2	✓	8
24.5	odd	2		2304.3.b.t.127.3		8
24.11	even	2		2304.3.b.q.127.3		8
48.5	odd	4		1152.3.g.c.127.3		8
48.11	even	4		1152.3.g.c.127.4		8
48.29	odd	4		1152.3.g.f.127.5		8
48.35	even	4		1152.3.g.f.127.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.g.a.127.2	✓	8	16.13	even	4
384.3.g.a.127.6	yes	8	16.3	odd	4
384.3.g.b.127.3	yes	8	16.11	odd	4
384.3.g.b.127.7	yes	8	16.5	even	4
768.3.b.e.127.6		8	4.3	odd	2
768.3.b.e.127.7		8	8.5	even	2
768.3.b.f.127.2		8	1.1	even	1	trivial
768.3.b.f.127.3		8	8.3	odd	2	inner
1152.3.g.c.127.3		8	48.5	odd	4
1152.3.g.c.127.4		8	48.11	even	4
1152.3.g.f.127.5		8	48.29	odd	4
1152.3.g.f.127.6		8	48.35	even	4
2304.3.b.q.127.3		8	24.11	even	2
2304.3.b.q.127.6		8	3.2	odd	2
2304.3.b.t.127.3		8	24.5	odd	2
2304.3.b.t.127.6		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -2.63567i q^{5} +12.5558i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} -2.63567i q^{5} +12.5558i q^{7} +3.00000 q^{9} -5.79796 q^{11} -8.78710i q^{13} +4.56512i q^{15} -30.1810 q^{17} +17.4125 q^{19} -21.7473i q^{21} +2.48425i q^{23} +18.0532 q^{25} -5.19615 q^{27} +26.4175i q^{29} -38.0082i q^{31} +10.0424 q^{33} +33.0931 q^{35} -47.7930i q^{37} +15.2197i q^{39} -53.3294 q^{41} +30.7044 q^{43} -7.90702i q^{45} -16.2238i q^{47} -108.649 q^{49} +52.2750 q^{51} -49.8432i q^{53} +15.2815i q^{55} -30.1593 q^{57} -107.412 q^{59} -62.6220i q^{61} +37.6675i q^{63} -23.1599 q^{65} -60.9540 q^{67} -4.30285i q^{69} -19.9465i q^{71} -5.13929 q^{73} -31.2691 q^{75} -72.7982i q^{77} +6.83349i q^{79} +9.00000 q^{81} -159.213 q^{83} +79.5472i q^{85} -45.7565i q^{87} -39.4473 q^{89} +110.329 q^{91} +65.8321i q^{93} -45.8936i q^{95} -60.5944 q^{97} -17.3939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} + 32 q^{11} + 16 q^{17} + 96 q^{19} + 8 q^{25} - 96 q^{35} - 80 q^{41} + 224 q^{43} - 88 q^{49} + 96 q^{51} - 96 q^{57} - 512 q^{59} - 160 q^{65} + 16 q^{73} + 192 q^{75} + 72 q^{81} - 544 q^{83} + 240 q^{89} - 32 q^{91} + 400 q^{97} + 96 q^{99}+O(q^{100}) \) 8 * q + 24 * q^9 + 32 * q^11 + 16 * q^17 + 96 * q^19 + 8 * q^25 - 96 * q^35 - 80 * q^41 + 224 * q^43 - 88 * q^49 + 96 * q^51 - 96 * q^57 - 512 * q^59 - 160 * q^65 + 16 * q^73 + 192 * q^75 + 72 * q^81 - 544 * q^83 + 240 * q^89 - 32 * q^91 + 400 * q^97 + 96 * q^99

Introduction

L-functions

Modular forms

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Database

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