LMFDB - Embedded newform 768.3.b.e.127.5

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			127.5
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.3.b.e.127.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -8.29253i q^{5} +8.55583i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} -8.29253i q^{5} +8.55583i q^{7} +3.00000 q^{9} -13.7980 q^{11} -17.0693i q^{13} -14.3631i q^{15} +20.3246 q^{17} -20.4440 q^{19} +14.8191i q^{21} -5.51575i q^{23} -43.7660 q^{25} +5.19615 q^{27} -41.0586i q^{29} -22.2953i q^{31} -23.8988 q^{33} +70.9495 q^{35} -11.6326i q^{37} -29.5649i q^{39} -35.9527 q^{41} -66.8648 q^{43} -24.8776i q^{45} -19.9366i q^{47} -24.2023 q^{49} +35.2032 q^{51} +17.6329i q^{53} +114.420i q^{55} -35.4100 q^{57} +62.1572 q^{59} +47.4836i q^{61} +25.6675i q^{63} -141.548 q^{65} -74.8105 q^{67} -9.55356i q^{69} +16.9150i q^{71} -101.712 q^{73} -75.8050 q^{75} -118.053i q^{77} -0.879320i q^{79} +9.00000 q^{81} -23.2131 q^{83} -168.542i q^{85} -71.1155i q^{87} +16.3089 q^{89} +146.042 q^{91} -38.6167i q^{93} +169.532i q^{95} +188.307 q^{97} -41.3939 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 8.29253i	− 1.65851i	−0.558874	−	0.829253i	$-0.688766\pi$
$5$	− 8.29253i	− 1.65851i	0.558874	−	0.829253i	$-0.311234\pi$
$6$	0	0
$7$	8.55583i	1.22226i	0.791529	+	0.611131i	$0.209285\pi$
$7$	8.55583i	1.22226i	−0.791529	+	0.611131i	$0.790715\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	−13.7980	−1.25436	−0.627180	−	0.778874i	$-0.715791\pi$
$11$	−13.7980	−1.25436	−0.627180	+	0.778874i	$0.715791\pi$
$12$	0	0
$13$	− 17.0693i	− 1.31302i	−0.754316	−	0.656512i	$-0.772031\pi$
$13$	− 17.0693i	− 1.31302i	0.754316	−	0.656512i	$-0.227969\pi$
$14$	0	0
$15$	− 14.3631i	− 0.957539i
$16$	0	0
$17$	20.3246	1.19556	0.597781	−	0.801659i	$-0.296049\pi$
$17$	20.3246	1.19556	0.597781	+	0.801659i	$0.296049\pi$
$18$	0	0
$19$	−20.4440	−1.07600	−0.537999	−	0.842946i	$-0.680819\pi$
$19$	−20.4440	−1.07600	−0.537999	+	0.842946i	$0.680819\pi$
$20$	0	0
$21$	14.8191i	0.705673i
$22$	0	0
$23$	− 5.51575i	− 0.239815i	−0.992785	−	0.119908i	$-0.961740\pi$
$23$	− 5.51575i	− 0.239815i	0.992785	−	0.119908i	$-0.0382598\pi$
$24$	0	0
$25$	−43.7660	−1.75064
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	− 41.0586i	− 1.41581i	−0.706306	−	0.707906i	$-0.749640\pi$
$29$	− 41.0586i	− 1.41581i	0.706306	−	0.707906i	$-0.250360\pi$
$30$	0	0
$31$	− 22.2953i	− 0.719205i	−0.933106	−	0.359602i	$-0.882912\pi$
$31$	− 22.2953i	− 0.719205i	0.933106	−	0.359602i	$-0.117088\pi$
$32$	0	0
$33$	−23.8988	−0.724205
$34$	0	0
$35$	70.9495	2.02713
$36$	0	0
$37$	− 11.6326i	− 0.314396i	−0.987567	−	0.157198i	$-0.949754\pi$
$37$	− 11.6326i	− 0.314396i	0.987567	−	0.157198i	$-0.0502461\pi$
$38$	0	0
$39$	− 29.5649i	− 0.758075i
$40$	0	0
$41$	−35.9527	−0.876894	−0.438447	−	0.898757i	$-0.644471\pi$
$41$	−35.9527	−0.876894	−0.438447	+	0.898757i	$0.644471\pi$
$42$	0	0
$43$	−66.8648	−1.55499	−0.777497	−	0.628886i	$-0.783511\pi$
$43$	−66.8648	−1.55499	−0.777497	+	0.628886i	$0.783511\pi$
$44$	0	0
$45$	− 24.8776i	− 0.552835i
$46$	0	0
$47$	− 19.9366i	− 0.424182i	−0.977250	−	0.212091i	$-0.931973\pi$
$47$	− 19.9366i	− 0.424182i	0.977250	−	0.212091i	$-0.0680274\pi$
$48$	0	0
$49$	−24.2023	−0.493924
$50$	0	0
$51$	35.2032	0.690259
$52$	0	0
$53$	17.6329i	0.332697i	0.986067	+	0.166348i	$0.0531977\pi$
$53$	17.6329i	0.332697i	−0.986067	+	0.166348i	$0.946802\pi$
$54$	0	0
$55$	114.420i	2.08036i
$56$	0	0
$57$	−35.4100	−0.621227
$58$	0	0
$59$	62.1572	1.05351	0.526756	−	0.850017i	$-0.323408\pi$
$59$	62.1572	1.05351	0.526756	+	0.850017i	$0.323408\pi$
$60$	0	0
$61$	47.4836i	0.778419i	0.921149	+	0.389210i	$0.127252\pi$
$61$	47.4836i	0.778419i	−0.921149	+	0.389210i	$0.872748\pi$
$62$	0	0
$63$	25.6675i	0.407421i
$64$	0	0
$65$	−141.548	−2.17766
$66$	0	0
$67$	−74.8105	−1.11657	−0.558287	−	0.829648i	$-0.688541\pi$
$67$	−74.8105	−1.11657	−0.558287	+	0.829648i	$0.688541\pi$
$68$	0	0
$69$	− 9.55356i	− 0.138457i
$70$	0	0
$71$	16.9150i	0.238240i	0.992880	+	0.119120i	$0.0380073\pi$
$71$	16.9150i	0.238240i	−0.992880	+	0.119120i	$0.961993\pi$
$72$	0	0
$73$	−101.712	−1.39331	−0.696657	−	0.717404i	$-0.745330\pi$
$73$	−101.712	−1.39331	−0.696657	+	0.717404i	$0.745330\pi$
$74$	0	0
$75$	−75.8050	−1.01073
$76$	0	0
$77$	− 118.053i	− 1.53316i
$78$	0	0
$79$	− 0.879320i	− 0.0111306i	−0.999985	−	0.00556532i	$-0.998228\pi$
$79$	− 0.879320i	− 0.0111306i	0.999985	−	0.00556532i	$-0.00177151\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−23.2131	−0.279676	−0.139838	−	0.990174i	$-0.544658\pi$
$83$	−23.2131	−0.279676	−0.139838	+	0.990174i	$0.544658\pi$
$84$	0	0
$85$	− 168.542i	− 1.98285i
$86$	0	0
$87$	− 71.1155i	− 0.817420i
$88$	0	0
$89$	16.3089	0.183246	0.0916231	−	0.995794i	$-0.470795\pi$
$89$	16.3089	0.183246	0.0916231	+	0.995794i	$0.470795\pi$
$90$	0	0
$91$	146.042	1.60486
$92$	0	0
$93$	− 38.6167i	− 0.415233i
$94$	0	0
$95$	169.532i	1.78455i
$96$	0	0
$97$	188.307	1.94131	0.970656	−	0.240474i	$-0.0773028\pi$
$97$	188.307	1.94131	0.970656	+	0.240474i	$0.0773028\pi$
$98$	0	0
$99$	−41.3939	−0.418120
$100$	0	0
$101$	− 66.8468i	− 0.661849i	−0.943657	−	0.330925i	$-0.892639\pi$
$101$	− 66.8468i	− 0.661849i	0.943657	−	0.330925i	$-0.107361\pi$
$102$	0	0
$103$	− 166.313i	− 1.61469i	−0.590083	−	0.807343i	$-0.700905\pi$
$103$	− 166.313i	− 1.61469i	0.590083	−	0.807343i	$-0.299095\pi$
$104$	0	0
$105$	122.888	1.17036
$106$	0	0
$107$	80.1467	0.749035	0.374517	−	0.927220i	$-0.377808\pi$
$107$	80.1467	0.749035	0.374517	+	0.927220i	$0.377808\pi$
$108$	0	0
$109$	− 26.8781i	− 0.246589i	−0.992370	−	0.123294i	$-0.960654\pi$
$109$	− 26.8781i	− 0.246589i	0.992370	−	0.123294i	$-0.0393459\pi$
$110$	0	0
$111$	− 20.1483i	− 0.181516i
$112$	0	0
$113$	−79.0958	−0.699963	−0.349981	−	0.936757i	$-0.613812\pi$
$113$	−79.0958	−0.699963	−0.349981	+	0.936757i	$0.613812\pi$
$114$	0	0
$115$	−45.7395	−0.397735
$116$	0	0
$117$	− 51.2079i	− 0.437675i
$118$	0	0
$119$	173.894i	1.46129i
$120$	0	0
$121$	69.3837	0.573419
$122$	0	0
$123$	−62.2718	−0.506275
$124$	0	0
$125$	155.618i	1.24494i
$126$	0	0
$127$	− 170.725i	− 1.34429i	−0.740419	−	0.672145i	$-0.765373\pi$
$127$	− 170.725i	− 1.34429i	0.740419	−	0.672145i	$-0.234627\pi$
$128$	0	0
$129$	−115.813	−0.897777
$130$	0	0
$131$	178.980	1.36626	0.683129	−	0.730297i	$-0.260619\pi$
$131$	178.980	1.36626	0.683129	+	0.730297i	$0.260619\pi$
$132$	0	0
$133$	− 174.915i	− 1.31515i
$134$	0	0
$135$	− 43.0892i	− 0.319180i
$136$	0	0
$137$	93.2075	0.680347	0.340173	−	0.940363i	$-0.389514\pi$
$137$	93.2075	0.680347	0.340173	+	0.940363i	$0.389514\pi$
$138$	0	0
$139$	222.476	1.60055	0.800274	−	0.599635i	$-0.204687\pi$
$139$	222.476	1.60055	0.800274	+	0.599635i	$0.204687\pi$
$140$	0	0
$141$	− 34.5312i	− 0.244902i
$142$	0	0
$143$	235.522i	1.64700i
$144$	0	0
$145$	−340.479	−2.34813
$146$	0	0
$147$	−41.9196	−0.285167
$148$	0	0
$149$	94.0043i	0.630901i	0.948942	+	0.315451i	$0.102156\pi$
$149$	94.0043i	0.630901i	−0.948942	+	0.315451i	$0.897844\pi$
$150$	0	0
$151$	− 82.0081i	− 0.543100i	−0.962424	−	0.271550i	$-0.912464\pi$
$151$	− 82.0081i	− 0.543100i	0.962424	−	0.271550i	$-0.0875362\pi$
$152$	0	0
$153$	60.9737	0.398521
$154$	0	0
$155$	−184.885	−1.19281
$156$	0	0
$157$	56.7180i	0.361261i	0.983551	+	0.180631i	$0.0578139\pi$
$157$	56.7180i	0.361261i	−0.983551	+	0.180631i	$0.942186\pi$
$158$	0	0
$159$	30.5411i	0.192083i
$160$	0	0
$161$	47.1918	0.293117
$162$	0	0
$163$	−189.588	−1.16312	−0.581559	−	0.813504i	$-0.697557\pi$
$163$	−189.588	−1.16312	−0.581559	+	0.813504i	$0.697557\pi$
$164$	0	0
$165$	198.181i	1.20110i
$166$	0	0
$167$	− 312.777i	− 1.87291i	−0.350783	−	0.936457i	$-0.614084\pi$
$167$	− 312.777i	− 1.87291i	0.350783	−	0.936457i	$-0.385916\pi$
$168$	0	0
$169$	−122.361	−0.724031
$170$	0	0
$171$	−61.3319	−0.358666
$172$	0	0
$173$	182.538i	1.05513i	0.849514	+	0.527567i	$0.176896\pi$
$173$	182.538i	1.05513i	−0.849514	+	0.527567i	$0.823104\pi$
$174$	0	0
$175$	− 374.455i	− 2.13974i
$176$	0	0
$177$	107.659	0.608245
$178$	0	0
$179$	−19.7781	−0.110492	−0.0552461	−	0.998473i	$-0.517594\pi$
$179$	−19.7781	−0.110492	−0.0552461	+	0.998473i	$0.517594\pi$
$180$	0	0
$181$	265.750i	1.46823i	0.679023	+	0.734117i	$0.262403\pi$
$181$	265.750i	1.46823i	−0.679023	+	0.734117i	$0.737597\pi$
$182$	0	0
$183$	82.2440i	0.449421i
$184$	0	0
$185$	−96.4640	−0.521427
$186$	0	0
$187$	−280.438	−1.49967
$188$	0	0
$189$	44.4574i	0.235224i
$190$	0	0
$191$	288.025i	1.50799i	0.656882	+	0.753993i	$0.271875\pi$
$191$	288.025i	1.50799i	−0.656882	+	0.753993i	$0.728125\pi$
$192$	0	0
$193$	281.010	1.45601	0.728005	−	0.685572i	$-0.240448\pi$
$193$	281.010	1.45601	0.728005	+	0.685572i	$0.240448\pi$
$194$	0	0
$195$	−245.168	−1.25727
$196$	0	0
$197$	− 33.8330i	− 0.171741i	−0.996306	−	0.0858705i	$-0.972633\pi$
$197$	− 33.8330i	− 0.171741i	0.996306	−	0.0858705i	$-0.0273671\pi$
$198$	0	0
$199$	− 84.9700i	− 0.426985i	−0.976945	−	0.213493i	$-0.931516\pi$
$199$	− 84.9700i	− 0.426985i	0.976945	−	0.213493i	$-0.0684839\pi$
$200$	0	0
$201$	−129.576	−0.644654
$202$	0	0
$203$	351.290	1.73049
$204$	0	0
$205$	298.138i	1.45433i
$206$	0	0
$207$	− 16.5472i	− 0.0799384i
$208$	0	0
$209$	282.085	1.34969
$210$	0	0
$211$	140.700	0.666826	0.333413	−	0.942781i	$-0.391800\pi$
$211$	140.700	0.666826	0.333413	+	0.942781i	$0.391800\pi$
$212$	0	0
$213$	29.2977i	0.137548i
$214$	0	0
$215$	554.478i	2.57897i
$216$	0	0
$217$	190.755	0.879057
$218$	0	0
$219$	−176.170	−0.804431
$220$	0	0
$221$	− 346.926i	− 1.56980i
$222$	0	0
$223$	247.529i	1.11000i	0.831851	+	0.554999i	$0.187281\pi$
$223$	247.529i	1.11000i	−0.831851	+	0.554999i	$0.812719\pi$
$224$	0	0
$225$	−131.298	−0.583547
$226$	0	0
$227$	129.752	0.571593	0.285797	−	0.958290i	$-0.407742\pi$
$227$	129.752	0.571593	0.285797	+	0.958290i	$0.407742\pi$
$228$	0	0
$229$	294.304i	1.28517i	0.766214	+	0.642586i	$0.222138\pi$
$229$	294.304i	1.28517i	−0.766214	+	0.642586i	$0.777862\pi$
$230$	0	0
$231$	− 204.474i	− 0.885168i
$232$	0	0
$233$	239.948	1.02982	0.514911	−	0.857244i	$-0.327825\pi$
$233$	239.948	1.02982	0.514911	+	0.857244i	$0.327825\pi$
$234$	0	0
$235$	−165.325	−0.703509
$236$	0	0
$237$	− 1.52303i	− 0.00642628i
$238$	0	0
$239$	− 425.346i	− 1.77969i	−0.456261	−	0.889846i	$-0.650812\pi$
$239$	− 425.346i	− 1.77969i	0.456261	−	0.889846i	$-0.349188\pi$
$240$	0	0
$241$	264.493	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$241$	264.493	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	200.698i	0.819176i
$246$	0	0
$247$	348.964i	1.41281i
$248$	0	0
$249$	−40.2063	−0.161471
$250$	0	0
$251$	221.125	0.880976	0.440488	−	0.897759i	$-0.354805\pi$
$251$	221.125	0.880976	0.440488	+	0.897759i	$0.354805\pi$
$252$	0	0
$253$	76.1061i	0.300815i
$254$	0	0
$255$	− 291.923i	− 1.14480i
$256$	0	0
$257$	87.0655	0.338776	0.169388	−	0.985549i	$-0.445821\pi$
$257$	87.0655	0.338776	0.169388	+	0.985549i	$0.445821\pi$
$258$	0	0
$259$	99.5270	0.384274
$260$	0	0
$261$	− 123.176i	− 0.471937i
$262$	0	0
$263$	277.324i	1.05447i	0.849721	+	0.527233i	$0.176770\pi$
$263$	277.324i	1.05447i	−0.849721	+	0.527233i	$0.823230\pi$
$264$	0	0
$265$	146.222	0.551780
$266$	0	0
$267$	28.2479	0.105797
$268$	0	0
$269$	49.8005i	0.185132i	0.995707	+	0.0925659i	$0.0295069\pi$
$269$	49.8005i	0.185132i	−0.995707	+	0.0925659i	$0.970493\pi$
$270$	0	0
$271$	− 31.3145i	− 0.115552i	−0.998330	−	0.0577758i	$-0.981599\pi$
$271$	− 31.3145i	− 0.115552i	0.998330	−	0.0577758i	$-0.0184009\pi$
$272$	0	0
$273$	252.952	0.926566
$274$	0	0
$275$	603.882	2.19593
$276$	0	0
$277$	− 79.1431i	− 0.285715i	−0.989743	−	0.142858i	$-0.954371\pi$
$277$	− 79.1431i	− 0.285715i	0.989743	−	0.142858i	$-0.0456291\pi$
$278$	0	0
$279$	− 66.8860i	− 0.239735i
$280$	0	0
$281$	106.190	0.377902	0.188951	−	0.981987i	$-0.439491\pi$
$281$	106.190	0.377902	0.188951	+	0.981987i	$0.439491\pi$
$282$	0	0
$283$	−270.251	−0.954949	−0.477475	−	0.878646i	$-0.658448\pi$
$283$	−270.251	−0.954949	−0.477475	+	0.878646i	$0.658448\pi$
$284$	0	0
$285$	293.638i	1.03031i
$286$	0	0
$287$	− 307.605i	− 1.07179i
$288$	0	0
$289$	124.088	0.429371
$290$	0	0
$291$	326.158	1.12082
$292$	0	0
$293$	24.2487i	0.0827600i	0.999143	+	0.0413800i	$0.0131754\pi$
$293$	24.2487i	0.0827600i	−0.999143	+	0.0413800i	$0.986825\pi$
$294$	0	0
$295$	− 515.440i	− 1.74726i
$296$	0	0
$297$	−71.6963	−0.241402
$298$	0	0
$299$	−94.1500	−0.314883
$300$	0	0
$301$	− 572.084i	− 1.90061i
$302$	0	0
$303$	− 115.782i	− 0.382119i
$304$	0	0
$305$	393.759	1.29101
$306$	0	0
$307$	−122.865	−0.400211	−0.200106	−	0.979774i	$-0.564129\pi$
$307$	−122.865	−0.400211	−0.200106	+	0.979774i	$0.564129\pi$
$308$	0	0
$309$	− 288.062i	− 0.932239i
$310$	0	0
$311$	108.808i	0.349865i	0.984580	+	0.174933i	$0.0559707\pi$
$311$	108.808i	0.349865i	−0.984580	+	0.174933i	$0.944029\pi$
$312$	0	0
$313$	52.2216	0.166842	0.0834211	−	0.996514i	$-0.473415\pi$
$313$	52.2216	0.166842	0.0834211	+	0.996514i	$0.473415\pi$
$314$	0	0
$315$	212.848	0.675709
$316$	0	0
$317$	− 94.3251i	− 0.297555i	−0.988871	−	0.148778i	$-0.952466\pi$
$317$	− 94.3251i	− 0.297555i	0.988871	−	0.148778i	$-0.0475339\pi$
$318$	0	0
$319$	566.524i	1.77594i
$320$	0	0
$321$	138.818	0.432456
$322$	0	0
$323$	−415.515	−1.28642
$324$	0	0
$325$	747.056i	2.29863i
$326$	0	0
$327$	− 46.5543i	− 0.142368i
$328$	0	0
$329$	170.574	0.518462
$330$	0	0
$331$	−406.107	−1.22691	−0.613454	−	0.789730i	$-0.710220\pi$
$331$	−406.107	−1.22691	−0.613454	+	0.789730i	$0.710220\pi$
$332$	0	0
$333$	− 34.8979i	− 0.104799i
$334$	0	0
$335$	620.368i	1.85184i
$336$	0	0
$337$	−22.4140	−0.0665105	−0.0332552	−	0.999447i	$-0.510587\pi$
$337$	−22.4140	−0.0665105	−0.0332552	+	0.999447i	$0.510587\pi$
$338$	0	0
$339$	−136.998	−0.404124
$340$	0	0
$341$	307.630i	0.902142i
$342$	0	0
$343$	212.165i	0.618557i
$344$	0	0
$345$	−79.2232	−0.229632
$346$	0	0
$347$	458.377	1.32097	0.660486	−	0.750839i	$-0.270350\pi$
$347$	458.377	1.32097	0.660486	+	0.750839i	$0.270350\pi$
$348$	0	0
$349$	− 282.960i	− 0.810774i	−0.914145	−	0.405387i	$-0.867137\pi$
$349$	− 282.960i	− 0.810774i	0.914145	−	0.405387i	$-0.132863\pi$
$350$	0	0
$351$	− 88.6947i	− 0.252692i
$352$	0	0
$353$	−175.820	−0.498073	−0.249036	−	0.968494i	$-0.580114\pi$
$353$	−175.820	−0.498073	−0.249036	+	0.968494i	$0.580114\pi$
$354$	0	0
$355$	140.268	0.395122
$356$	0	0
$357$	301.193i	0.843677i
$358$	0	0
$359$	− 53.5791i	− 0.149246i	−0.997212	−	0.0746228i	$-0.976225\pi$
$359$	− 53.5791i	− 0.149246i	0.997212	−	0.0746228i	$-0.0237753\pi$
$360$	0	0
$361$	56.9552	0.157771
$362$	0	0
$363$	120.176	0.331063
$364$	0	0
$365$	843.449i	2.31082i
$366$	0	0
$367$	− 62.4258i	− 0.170097i	−0.996377	−	0.0850487i	$-0.972895\pi$
$367$	− 62.4258i	− 0.170097i	0.996377	−	0.0850487i	$-0.0271046\pi$
$368$	0	0
$369$	−107.858	−0.292298
$370$	0	0
$371$	−150.864	−0.406643
$372$	0	0
$373$	− 370.552i	− 0.993437i	−0.867912	−	0.496719i	$-0.834538\pi$
$373$	− 370.552i	− 0.993437i	0.867912	−	0.496719i	$-0.165462\pi$
$374$	0	0
$375$	269.538i	0.718768i
$376$	0	0
$377$	−700.841	−1.85900
$378$	0	0
$379$	−68.7286	−0.181342	−0.0906709	−	0.995881i	$-0.528901\pi$
$379$	−68.7286	−0.181342	−0.0906709	+	0.995881i	$0.528901\pi$
$380$	0	0
$381$	− 295.704i	− 0.776126i
$382$	0	0
$383$	155.858i	0.406939i	0.979081	+	0.203470i	$0.0652218\pi$
$383$	155.858i	0.406939i	−0.979081	+	0.203470i	$0.934778\pi$
$384$	0	0
$385$	−978.958	−2.54275
$386$	0	0
$387$	−200.594	−0.518332
$388$	0	0
$389$	− 481.792i	− 1.23854i	−0.785179	−	0.619269i	$-0.787429\pi$
$389$	− 481.792i	− 1.23854i	0.785179	−	0.619269i	$-0.212571\pi$
$390$	0	0
$391$	− 112.105i	− 0.286714i
$392$	0	0
$393$	310.002	0.788810
$394$	0	0
$395$	−7.29179	−0.0184602
$396$	0	0
$397$	422.315i	1.06377i	0.846818	+	0.531883i	$0.178515\pi$
$397$	422.315i	1.06377i	−0.846818	+	0.531883i	$0.821485\pi$
$398$	0	0
$399$	− 302.962i	− 0.759303i
$400$	0	0
$401$	−641.025	−1.59857	−0.799283	−	0.600955i	$-0.794787\pi$
$401$	−641.025	−1.59857	−0.799283	+	0.600955i	$0.794787\pi$
$402$	0	0
$403$	−380.566	−0.944333
$404$	0	0
$405$	− 74.6328i	− 0.184278i
$406$	0	0
$407$	160.507i	0.394365i
$408$	0	0
$409$	278.562	0.681081	0.340541	−	0.940230i	$-0.389390\pi$
$409$	278.562	0.681081	0.340541	+	0.940230i	$0.389390\pi$
$410$	0	0
$411$	161.440	0.392798
$412$	0	0
$413$	531.807i	1.28767i
$414$	0	0
$415$	192.496i	0.463845i
$416$	0	0
$417$	385.340	0.924077
$418$	0	0
$419$	505.611	1.20671	0.603354	−	0.797473i	$-0.293831\pi$
$419$	505.611	1.20671	0.603354	+	0.797473i	$0.293831\pi$
$420$	0	0
$421$	− 179.846i	− 0.427189i	−0.976922	−	0.213594i	$-0.931483\pi$
$421$	− 179.846i	− 0.427189i	0.976922	−	0.213594i	$-0.0685171\pi$
$422$	0	0
$423$	− 59.8097i	− 0.141394i
$424$	0	0
$425$	−889.526	−2.09300
$426$	0	0
$427$	−406.262	−0.951432
$428$	0	0
$429$	407.935i	0.950898i
$430$	0	0
$431$	− 580.645i	− 1.34720i	−0.739094	−	0.673602i	$-0.764746\pi$
$431$	− 580.645i	− 1.34720i	0.739094	−	0.673602i	$-0.235254\pi$
$432$	0	0
$433$	−425.621	−0.982959	−0.491479	−	0.870889i	$-0.663544\pi$
$433$	−425.621	−0.982959	−0.491479	+	0.870889i	$0.663544\pi$
$434$	0	0
$435$	−589.727	−1.35570
$436$	0	0
$437$	112.764i	0.258041i
$438$	0	0
$439$	− 129.991i	− 0.296107i	−0.988979	−	0.148053i	$-0.952699\pi$
$439$	− 129.991i	− 0.296107i	0.988979	−	0.148053i	$-0.0473008\pi$
$440$	0	0
$441$	−72.6069	−0.164641
$442$	0	0
$443$	205.564	0.464027	0.232014	−	0.972713i	$-0.425469\pi$
$443$	205.564	0.464027	0.232014	+	0.972713i	$0.425469\pi$
$444$	0	0
$445$	− 135.242i	− 0.303915i
$446$	0	0
$447$	162.820i	0.364251i
$448$	0	0
$449$	415.190	0.924698	0.462349	−	0.886698i	$-0.347007\pi$
$449$	415.190	0.924698	0.462349	+	0.886698i	$0.347007\pi$
$450$	0	0
$451$	496.073	1.09994
$452$	0	0
$453$	− 142.042i	− 0.313559i
$454$	0	0
$455$	− 1211.06i	− 2.66167i
$456$	0	0
$457$	598.927	1.31056	0.655281	−	0.755385i	$-0.272550\pi$
$457$	598.927	1.31056	0.655281	+	0.755385i	$0.272550\pi$
$458$	0	0
$459$	105.610	0.230086
$460$	0	0
$461$	− 376.804i	− 0.817361i	−0.912677	−	0.408681i	$-0.865989\pi$
$461$	− 376.804i	− 0.817361i	0.912677	−	0.408681i	$-0.134011\pi$
$462$	0	0
$463$	218.577i	0.472088i	0.971742	+	0.236044i	$0.0758510\pi$
$463$	218.577i	0.472088i	−0.971742	+	0.236044i	$0.924149\pi$
$464$	0	0
$465$	−320.230	−0.688666
$466$	0	0
$467$	−443.608	−0.949911	−0.474955	−	0.880010i	$-0.657536\pi$
$467$	−443.608	−0.949911	−0.474955	+	0.880010i	$0.657536\pi$
$468$	0	0
$469$	− 640.066i	− 1.36475i
$470$	0	0
$471$	98.2385i	0.208574i
$472$	0	0
$473$	922.597	1.95052
$474$	0	0
$475$	894.751	1.88369
$476$	0	0
$477$	52.8988i	0.110899i
$478$	0	0
$479$	222.133i	0.463743i	0.972746	+	0.231871i	$0.0744849\pi$
$479$	222.133i	0.463743i	−0.972746	+	0.231871i	$0.925515\pi$
$480$	0	0
$481$	−198.561	−0.412809
$482$	0	0
$483$	81.7387	0.169231
$484$	0	0
$485$	− 1561.54i	− 3.21968i
$486$	0	0
$487$	− 403.795i	− 0.829148i	−0.910016	−	0.414574i	$-0.863931\pi$
$487$	− 403.795i	− 0.829148i	0.910016	−	0.414574i	$-0.136069\pi$
$488$	0	0
$489$	−328.376	−0.671526
$490$	0	0
$491$	−808.347	−1.64633	−0.823164	−	0.567803i	$-0.807793\pi$
$491$	−808.347	−1.64633	−0.823164	+	0.567803i	$0.807793\pi$
$492$	0	0
$493$	− 834.498i	− 1.69269i
$494$	0	0
$495$	343.260i	0.693454i
$496$	0	0
$497$	−144.722	−0.291192
$498$	0	0
$499$	−132.172	−0.264874	−0.132437	−	0.991191i	$-0.542280\pi$
$499$	−132.172	−0.264874	−0.132437	+	0.991191i	$0.542280\pi$
$500$	0	0
$501$	− 541.745i	− 1.08133i
$502$	0	0
$503$	− 92.8360i	− 0.184565i	−0.995733	−	0.0922823i	$-0.970584\pi$
$503$	− 92.8360i	− 0.184565i	0.995733	−	0.0922823i	$-0.0294162\pi$
$504$	0	0
$505$	−554.329	−1.09768
$506$	0	0
$507$	−211.936	−0.418020
$508$	0	0
$509$	− 927.417i	− 1.82204i	−0.412366	−	0.911018i	$-0.635297\pi$
$509$	− 927.417i	− 1.82204i	0.412366	−	0.911018i	$-0.364703\pi$
$510$	0	0
$511$	− 870.231i	− 1.70300i
$512$	0	0
$513$	−106.230	−0.207076
$514$	0	0
$515$	−1379.15	−2.67796
$516$	0	0
$517$	275.084i	0.532077i
$518$	0	0
$519$	316.165i	0.609182i
$520$	0	0
$521$	−527.145	−1.01179	−0.505897	−	0.862594i	$-0.668838\pi$
$521$	−527.145	−1.01179	−0.505897	+	0.862594i	$0.668838\pi$
$522$	0	0
$523$	972.249	1.85898	0.929492	−	0.368843i	$-0.120246\pi$
$523$	972.249	1.85898	0.929492	+	0.368843i	$0.120246\pi$
$524$	0	0
$525$	− 648.575i	− 1.23538i
$526$	0	0
$527$	− 453.143i	− 0.859854i
$528$	0	0
$529$	498.577	0.942489
$530$	0	0
$531$	186.472	0.351171
$532$	0	0
$533$	613.687i	1.15138i
$534$	0	0
$535$	− 664.619i	− 1.24228i
$536$	0	0
$537$	−34.2567	−0.0637927
$538$	0	0
$539$	333.942	0.619559
$540$	0	0
$541$	− 4.66591i	− 0.00862461i	−0.999991	−	0.00431231i	$-0.998627\pi$
$541$	− 4.66591i	− 0.00862461i	0.999991	−	0.00431231i	$-0.00137265\pi$
$542$	0	0
$543$	460.293i	0.847685i
$544$	0	0
$545$	−222.888	−0.408968
$546$	0	0
$547$	386.796	0.707123	0.353561	−	0.935411i	$-0.384971\pi$
$547$	386.796	0.707123	0.353561	+	0.935411i	$0.384971\pi$
$548$	0	0
$549$	142.451i	0.259473i
$550$	0	0
$551$	839.399i	1.52341i
$552$	0	0
$553$	7.52332	0.0136046
$554$	0	0
$555$	−167.081	−0.301046
$556$	0	0
$557$	− 219.630i	− 0.394308i	−0.980373	−	0.197154i	$-0.936830\pi$
$557$	− 219.630i	− 0.394308i	0.980373	−	0.197154i	$-0.0631699\pi$
$558$	0	0
$559$	1141.34i	2.04175i
$560$	0	0
$561$	−485.732	−0.865833
$562$	0	0
$563$	266.807	0.473902	0.236951	−	0.971522i	$-0.423852\pi$
$563$	266.807	0.473902	0.236951	+	0.971522i	$0.423852\pi$
$564$	0	0
$565$	655.904i	1.16089i
$566$	0	0
$567$	77.0025i	0.135807i
$568$	0	0
$569$	972.046	1.70834	0.854170	−	0.519993i	$-0.174066\pi$
$569$	972.046	1.70834	0.854170	+	0.519993i	$0.174066\pi$
$570$	0	0
$571$	−340.194	−0.595786	−0.297893	−	0.954599i	$-0.596284\pi$
$571$	−340.194	−0.595786	−0.297893	+	0.954599i	$0.596284\pi$
$572$	0	0
$573$	498.875i	0.870637i
$574$	0	0
$575$	241.402i	0.419830i
$576$	0	0
$577$	65.0584	0.112753	0.0563764	−	0.998410i	$-0.482045\pi$
$577$	65.0584	0.112753	0.0563764	+	0.998410i	$0.482045\pi$
$578$	0	0
$579$	486.723	0.840628
$580$	0	0
$581$	− 198.608i	− 0.341838i
$582$	0	0
$583$	− 243.299i	− 0.417322i
$584$	0	0
$585$	−424.643	−0.725886
$586$	0	0
$587$	−162.144	−0.276226	−0.138113	−	0.990417i	$-0.544104\pi$
$587$	−162.144	−0.276226	−0.138113	+	0.990417i	$0.544104\pi$
$588$	0	0
$589$	455.805i	0.773862i
$590$	0	0
$591$	− 58.6004i	− 0.0991547i
$592$	0	0
$593$	−793.227	−1.33765	−0.668825	−	0.743420i	$-0.733203\pi$
$593$	−793.227	−1.33765	−0.668825	+	0.743420i	$0.733203\pi$
$594$	0	0
$595$	1442.02	2.42356
$596$	0	0
$597$	− 147.172i	− 0.246520i
$598$	0	0
$599$	432.194i	0.721525i	0.932658	+	0.360763i	$0.117484\pi$
$599$	432.194i	0.721525i	−0.932658	+	0.360763i	$0.882516\pi$
$600$	0	0
$601$	−936.310	−1.55792	−0.778960	−	0.627074i	$-0.784252\pi$
$601$	−936.310	−1.55792	−0.778960	+	0.627074i	$0.784252\pi$
$602$	0	0
$603$	−224.431	−0.372191
$604$	0	0
$605$	− 575.366i	− 0.951018i
$606$	0	0
$607$	− 223.972i	− 0.368982i	−0.982834	−	0.184491i	$-0.940936\pi$
$607$	− 223.972i	− 0.368982i	0.982834	−	0.184491i	$-0.0590637\pi$
$608$	0	0
$609$	608.453	0.999101
$610$	0	0
$611$	−340.304	−0.556962
$612$	0	0
$613$	− 1010.40i	− 1.64828i	−0.566385	−	0.824141i	$-0.691658\pi$
$613$	− 1010.40i	− 1.64828i	0.566385	−	0.824141i	$-0.308342\pi$
$614$	0	0
$615$	516.391i	0.839660i
$616$	0	0
$617$	400.206	0.648633	0.324316	−	0.945949i	$-0.394866\pi$
$617$	400.206	0.648633	0.324316	+	0.945949i	$0.394866\pi$
$618$	0	0
$619$	−417.349	−0.674231	−0.337116	−	0.941463i	$-0.609451\pi$
$619$	−417.349	−0.674231	−0.337116	+	0.941463i	$0.609451\pi$
$620$	0	0
$621$	− 28.6607i	− 0.0461525i
$622$	0	0
$623$	139.536i	0.223975i
$624$	0	0
$625$	196.315	0.314104
$626$	0	0
$627$	488.585	0.779243
$628$	0	0
$629$	− 236.428i	− 0.375880i
$630$	0	0
$631$	718.033i	1.13793i	0.822362	+	0.568965i	$0.192656\pi$
$631$	718.033i	1.13793i	−0.822362	+	0.568965i	$0.807344\pi$
$632$	0	0
$633$	243.700	0.384992
$634$	0	0
$635$	−1415.74	−2.22951
$636$	0	0
$637$	413.116i	0.648534i
$638$	0	0
$639$	50.7451i	0.0794133i
$640$	0	0
$641$	139.926	0.218293	0.109146	−	0.994026i	$-0.465188\pi$
$641$	139.926	0.218293	0.109146	+	0.994026i	$0.465188\pi$
$642$	0	0
$643$	−777.990	−1.20994	−0.604969	−	0.796249i	$-0.706814\pi$
$643$	−777.990	−1.20994	−0.604969	+	0.796249i	$0.706814\pi$
$644$	0	0
$645$	960.384i	1.48897i
$646$	0	0
$647$	673.336i	1.04070i	0.853952	+	0.520352i	$0.174199\pi$
$647$	673.336i	1.04070i	−0.853952	+	0.520352i	$0.825801\pi$
$648$	0	0
$649$	−857.642	−1.32148
$650$	0	0
$651$	330.398	0.507524
$652$	0	0
$653$	− 223.410i	− 0.342128i	−0.985260	−	0.171064i	$-0.945279\pi$
$653$	− 223.410i	− 0.342128i	0.985260	−	0.171064i	$-0.0547206\pi$
$654$	0	0
$655$	− 1484.20i	− 2.26595i
$656$	0	0
$657$	−305.136	−0.464438
$658$	0	0
$659$	166.367	0.252453	0.126227	−	0.992001i	$-0.459713\pi$
$659$	166.367	0.252453	0.126227	+	0.992001i	$0.459713\pi$
$660$	0	0
$661$	− 655.773i	− 0.992092i	−0.868296	−	0.496046i	$-0.834785\pi$
$661$	− 655.773i	− 0.992092i	0.868296	−	0.496046i	$-0.165215\pi$
$662$	0	0
$663$	− 600.894i	− 0.906326i
$664$	0	0
$665$	−1450.49	−2.18119
$666$	0	0
$667$	−226.469	−0.339533
$668$	0	0
$669$	428.734i	0.640857i
$670$	0	0
$671$	− 655.176i	− 0.976418i
$672$	0	0
$673$	−863.477	−1.28303	−0.641513	−	0.767112i	$-0.721693\pi$
$673$	−863.477	−1.28303	−0.641513	+	0.767112i	$0.721693\pi$
$674$	0	0
$675$	−227.415	−0.336911
$676$	0	0
$677$	831.520i	1.22824i	0.789212	+	0.614121i	$0.210489\pi$
$677$	831.520i	1.22824i	−0.789212	+	0.614121i	$0.789511\pi$
$678$	0	0
$679$	1611.13i	2.37279i
$680$	0	0
$681$	224.737	0.330010
$682$	0	0
$683$	308.273	0.451351	0.225676	−	0.974202i	$-0.427541\pi$
$683$	308.273	0.451351	0.225676	+	0.974202i	$0.427541\pi$
$684$	0	0
$685$	− 772.926i	− 1.12836i
$686$	0	0
$687$	509.750i	0.741994i
$688$	0	0
$689$	300.982	0.436839
$690$	0	0
$691$	−666.811	−0.964994	−0.482497	−	0.875898i	$-0.660270\pi$
$691$	−666.811	−0.964994	−0.482497	+	0.875898i	$0.660270\pi$
$692$	0	0
$693$	− 354.159i	− 0.511052i
$694$	0	0
$695$	− 1844.89i	− 2.65452i
$696$	0	0
$697$	−730.722	−1.04838
$698$	0	0
$699$	415.603	0.594568
$700$	0	0
$701$	− 242.205i	− 0.345514i	−0.984964	−	0.172757i	$-0.944732\pi$
$701$	− 242.205i	− 0.345514i	0.984964	−	0.172757i	$-0.0552675\pi$
$702$	0	0
$703$	237.817i	0.338289i
$704$	0	0
$705$	−286.351	−0.406171
$706$	0	0
$707$	571.930	0.808953
$708$	0	0
$709$	1380.24i	1.94674i	0.229237	+	0.973371i	$0.426377\pi$
$709$	1380.24i	1.94674i	−0.229237	+	0.973371i	$0.573623\pi$
$710$	0	0
$711$	− 2.63796i	− 0.00371021i
$712$	0	0
$713$	−122.976	−0.172476
$714$	0	0
$715$	1953.07	2.73157
$716$	0	0
$717$	− 736.722i	− 1.02751i
$718$	0	0
$719$	− 968.365i	− 1.34682i	−0.739268	−	0.673411i	$-0.764828\pi$
$719$	− 968.365i	− 1.34682i	0.739268	−	0.673411i	$-0.235172\pi$
$720$	0	0
$721$	1422.94	1.97357
$722$	0	0
$723$	458.115	0.633631
$724$	0	0
$725$	1796.97i	2.47858i
$726$	0	0
$727$	1017.52i	1.39962i	0.714329	+	0.699810i	$0.246732\pi$
$727$	1017.52i	1.39962i	−0.714329	+	0.699810i	$0.753268\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−1359.00	−1.85909
$732$	0	0
$733$	1026.22i	1.40003i	0.714127	+	0.700016i	$0.246824\pi$
$733$	1026.22i	1.40003i	−0.714127	+	0.700016i	$0.753176\pi$
$734$	0	0
$735$	347.619i	0.472952i
$736$	0	0
$737$	1032.23	1.40059
$738$	0	0
$739$	923.444	1.24959	0.624793	−	0.780791i	$-0.285183\pi$
$739$	923.444	1.24959	0.624793	+	0.780791i	$0.285183\pi$
$740$	0	0
$741$	604.424i	0.815686i
$742$	0	0
$743$	− 1267.12i	− 1.70542i	−0.522387	−	0.852708i	$-0.674958\pi$
$743$	− 1267.12i	− 1.70542i	0.522387	−	0.852708i	$-0.325042\pi$
$744$	0	0
$745$	779.533	1.04635
$746$	0	0
$747$	−69.6394	−0.0932254
$748$	0	0
$749$	685.722i	0.915517i
$750$	0	0
$751$	791.442i	1.05385i	0.849912	+	0.526925i	$0.176655\pi$
$751$	791.442i	1.05385i	−0.849912	+	0.526925i	$0.823345\pi$
$752$	0	0
$753$	383.000	0.508632
$754$	0	0
$755$	−680.054	−0.900734
$756$	0	0
$757$	− 333.990i	− 0.441202i	−0.975364	−	0.220601i	$-0.929198\pi$
$757$	− 333.990i	− 0.441202i	0.975364	−	0.220601i	$-0.0708019\pi$
$758$	0	0
$759$	131.820i	0.173675i
$760$	0	0
$761$	−617.122	−0.810936	−0.405468	−	0.914109i	$-0.632891\pi$
$761$	−617.122	−0.810936	−0.405468	+	0.914109i	$0.632891\pi$
$762$	0	0
$763$	229.965	0.301396
$764$	0	0
$765$	− 505.626i	− 0.660949i
$766$	0	0
$767$	− 1060.98i	− 1.38329i
$768$	0	0
$769$	−657.604	−0.855142	−0.427571	−	0.903982i	$-0.640631\pi$
$769$	−657.604	−0.855142	−0.427571	+	0.903982i	$0.640631\pi$
$770$	0	0
$771$	150.802	0.195593
$772$	0	0
$773$	− 1314.59i	− 1.70064i	−0.526268	−	0.850319i	$-0.676409\pi$
$773$	− 1314.59i	− 1.70064i	0.526268	−	0.850319i	$-0.323591\pi$
$774$	0	0
$775$	975.779i	1.25907i
$776$	0	0
$777$	172.386	0.221861
$778$	0	0
$779$	735.015	0.943536
$780$	0	0
$781$	− 233.393i	− 0.298839i
$782$	0	0
$783$	− 213.347i	− 0.272473i
$784$	0	0
$785$	470.336	0.599154
$786$	0	0
$787$	301.018	0.382488	0.191244	−	0.981542i	$-0.438748\pi$
$787$	301.018	0.382488	0.191244	+	0.981542i	$0.438748\pi$
$788$	0	0
$789$	480.340i	0.608796i
$790$	0	0
$791$	− 676.731i	− 0.855538i
$792$	0	0
$793$	810.512	1.02208
$794$	0	0
$795$	253.263	0.318570
$796$	0	0
$797$	476.193i	0.597481i	0.954334	+	0.298741i	$0.0965665\pi$
$797$	476.193i	0.597481i	−0.954334	+	0.298741i	$0.903433\pi$
$798$	0	0
$799$	− 405.202i	− 0.507137i
$800$	0	0
$801$	48.9267	0.0610821
$802$	0	0
$803$	1403.42	1.74772
$804$	0	0
$805$	− 391.340i	− 0.486136i
$806$	0	0
$807$	86.2570i	0.106886i
$808$	0	0
$809$	692.874	0.856457	0.428228	−	0.903671i	$-0.359138\pi$
$809$	692.874	0.856457	0.428228	+	0.903671i	$0.359138\pi$
$810$	0	0
$811$	−785.074	−0.968032	−0.484016	−	0.875059i	$-0.660822\pi$
$811$	−785.074	−0.968032	−0.484016	+	0.875059i	$0.660822\pi$
$812$	0	0
$813$	− 54.2383i	− 0.0667138i
$814$	0	0
$815$	1572.17i	1.92904i
$816$	0	0
$817$	1366.98	1.67317
$818$	0	0
$819$	438.126	0.534953
$820$	0	0
$821$	527.681i	0.642729i	0.946956	+	0.321365i	$0.104141\pi$
$821$	527.681i	0.642729i	−0.946956	+	0.321365i	$0.895859\pi$
$822$	0	0
$823$	− 228.552i	− 0.277706i	−0.990313	−	0.138853i	$-0.955658\pi$
$823$	− 228.552i	− 0.277706i	0.990313	−	0.138853i	$-0.0443416\pi$
$824$	0	0
$825$	1045.95	1.26782
$826$	0	0
$827$	−608.409	−0.735683	−0.367841	−	0.929889i	$-0.619903\pi$
$827$	−608.409	−0.735683	−0.367841	+	0.929889i	$0.619903\pi$
$828$	0	0
$829$	151.637i	0.182915i	0.995809	+	0.0914577i	$0.0291526\pi$
$829$	151.637i	0.182915i	−0.995809	+	0.0914577i	$0.970847\pi$
$830$	0	0
$831$	− 137.080i	− 0.164958i
$832$	0	0
$833$	−491.901	−0.590518
$834$	0	0
$835$	−2593.71	−3.10624
$836$	0	0
$837$	− 115.850i	− 0.138411i
$838$	0	0
$839$	− 653.351i	− 0.778725i	−0.921085	−	0.389363i	$-0.872695\pi$
$839$	− 653.351i	− 0.778725i	0.921085	−	0.389363i	$-0.127305\pi$
$840$	0	0
$841$	−844.805	−1.00452
$842$	0	0
$843$	183.927	0.218182
$844$	0	0
$845$	1014.68i	1.20081i
$846$	0	0
$847$	593.635i	0.700868i
$848$	0	0
$849$	−468.088	−0.551340
$850$	0	0
$851$	−64.1628	−0.0753969
$852$	0	0
$853$	− 1372.25i	− 1.60873i	−0.594132	−	0.804367i	$-0.702505\pi$
$853$	− 1372.25i	− 1.60873i	0.594132	−	0.804367i	$-0.297495\pi$
$854$	0	0
$855$	508.596i	0.594849i
$856$	0	0
$857$	1453.61	1.69617	0.848083	−	0.529863i	$-0.177757\pi$
$857$	1453.61	1.69617	0.848083	+	0.529863i	$0.177757\pi$
$858$	0	0
$859$	−962.466	−1.12045	−0.560225	−	0.828341i	$-0.689285\pi$
$859$	−962.466	−1.12045	−0.560225	+	0.828341i	$0.689285\pi$
$860$	0	0
$861$	− 532.787i	− 0.618801i
$862$	0	0
$863$	− 695.890i	− 0.806362i	−0.915120	−	0.403181i	$-0.867905\pi$
$863$	− 695.890i	− 0.806362i	0.915120	−	0.403181i	$-0.132095\pi$
$864$	0	0
$865$	1513.70	1.74994
$866$	0	0
$867$	214.927	0.247897
$868$	0	0
$869$	12.1328i	0.0139618i
$870$	0	0
$871$	1276.96i	1.46609i
$872$	0	0
$873$	564.922	0.647104
$874$	0	0
$875$	−1331.44	−1.52165
$876$	0	0
$877$	338.191i	0.385622i	0.981236	+	0.192811i	$0.0617605\pi$
$877$	338.191i	0.385622i	−0.981236	+	0.192811i	$0.938240\pi$
$878$	0	0
$879$	42.0000i	0.0477815i
$880$	0	0
$881$	−459.985	−0.522117	−0.261058	−	0.965323i	$-0.584072\pi$
$881$	−459.985	−0.522117	−0.261058	+	0.965323i	$0.584072\pi$
$882$	0	0
$883$	−1208.31	−1.36842	−0.684209	−	0.729286i	$-0.739853\pi$
$883$	−1208.31	−1.36842	−0.684209	+	0.729286i	$0.739853\pi$
$884$	0	0
$885$	− 892.769i	− 1.00878i
$886$	0	0
$887$	− 1500.73i	− 1.69192i	−0.533250	−	0.845958i	$-0.679029\pi$
$887$	− 1500.73i	− 1.69192i	0.533250	−	0.845958i	$-0.320971\pi$
$888$	0	0
$889$	1460.69	1.64307
$890$	0	0
$891$	−124.182	−0.139373
$892$	0	0
$893$	407.582i	0.456419i
$894$	0	0
$895$	164.010i	0.183252i
$896$	0	0
$897$	−163.073	−0.181798
$898$	0	0
$899$	−915.415	−1.01826
$900$	0	0
$901$	358.382i	0.397760i
$902$	0	0
$903$	− 990.878i	− 1.09732i
$904$	0	0
$905$	2203.74	2.43507
$906$	0	0
$907$	−665.128	−0.733328	−0.366664	−	0.930353i	$-0.619500\pi$
$907$	−665.128	−0.733328	−0.366664	+	0.930353i	$0.619500\pi$
$908$	0	0
$909$	− 200.540i	− 0.220616i
$910$	0	0
$911$	− 345.687i	− 0.379459i	−0.981836	−	0.189729i	$-0.939239\pi$
$911$	− 345.687i	− 0.379459i	0.981836	−	0.189729i	$-0.0607611\pi$
$912$	0	0
$913$	320.294	0.350815
$914$	0	0
$915$	682.010	0.745367
$916$	0	0
$917$	1531.32i	1.66993i
$918$	0	0
$919$	− 1102.20i	− 1.19934i	−0.800246	−	0.599671i	$-0.795298\pi$
$919$	− 1102.20i	− 1.19934i	0.800246	−	0.599671i	$-0.204702\pi$
$920$	0	0
$921$	−212.808	−0.231062
$922$	0	0
$923$	288.728	0.312815
$924$	0	0
$925$	509.115i	0.550394i
$926$	0	0
$927$	− 498.938i	− 0.538228i
$928$	0	0
$929$	−1256.03	−1.35202	−0.676010	−	0.736893i	$-0.736292\pi$
$929$	−1256.03	−1.35202	−0.676010	+	0.736893i	$0.736292\pi$
$930$	0	0
$931$	494.790	0.531461
$932$	0	0
$933$	188.461i	0.201995i
$934$	0	0
$935$	2325.54i	2.48720i
$936$	0	0
$937$	822.214	0.877496	0.438748	−	0.898610i	$-0.355422\pi$
$937$	822.214	0.877496	0.438748	+	0.898610i	$0.355422\pi$
$938$	0	0
$939$	90.4505	0.0963264
$940$	0	0
$941$	− 1185.72i	− 1.26006i	−0.776570	−	0.630031i	$-0.783042\pi$
$941$	− 1185.72i	− 1.26006i	0.776570	−	0.630031i	$-0.216958\pi$
$942$	0	0
$943$	198.306i	0.210293i
$944$	0	0
$945$	368.664	0.390121
$946$	0	0
$947$	−1335.34	−1.41007	−0.705037	−	0.709170i	$-0.749070\pi$
$947$	−1335.34	−1.41007	−0.705037	+	0.709170i	$0.749070\pi$
$948$	0	0
$949$	1736.15i	1.82945i
$950$	0	0
$951$	− 163.376i	− 0.171794i
$952$	0	0
$953$	−663.428	−0.696147	−0.348073	−	0.937467i	$-0.613164\pi$
$953$	−663.428	−0.696147	−0.348073	+	0.937467i	$0.613164\pi$
$954$	0	0
$955$	2388.46	2.50100
$956$	0	0
$957$	981.249i	1.02534i
$958$	0	0
$959$	797.468i	0.831562i
$960$	0	0
$961$	463.918	0.482745
$962$	0	0
$963$	240.440	0.249678
$964$	0	0
$965$	− 2330.28i	− 2.41480i
$966$	0	0
$967$	− 155.996i	− 0.161320i	−0.996742	−	0.0806600i	$-0.974297\pi$
$967$	− 155.996i	− 0.161320i	0.996742	−	0.0806600i	$-0.0257028\pi$
$968$	0	0
$969$	−719.692	−0.742716
$970$	0	0
$971$	1280.67	1.31892	0.659460	−	0.751740i	$-0.270785\pi$
$971$	1280.67	1.31892	0.659460	+	0.751740i	$0.270785\pi$
$972$	0	0
$973$	1903.47i	1.95629i
$974$	0	0
$975$	1293.94i	1.32712i
$976$	0	0
$977$	−217.470	−0.222590	−0.111295	−	0.993787i	$-0.535500\pi$
$977$	−217.470	−0.222590	−0.111295	+	0.993787i	$0.535500\pi$
$978$	0	0
$979$	−225.030	−0.229857
$980$	0	0
$981$	− 80.6344i	− 0.0821962i
$982$	0	0
$983$	− 1027.63i	− 1.04540i	−0.852515	−	0.522702i	$-0.824924\pi$
$983$	− 1027.63i	− 1.04540i	0.852515	−	0.522702i	$-0.175076\pi$
$984$	0	0
$985$	−280.561	−0.284833
$986$	0	0
$987$	295.443	0.299334
$988$	0	0
$989$	368.809i	0.372911i
$990$	0	0
$991$	797.792i	0.805038i	0.915412	+	0.402519i	$0.131865\pi$
$991$	797.792i	0.805038i	−0.915412	+	0.402519i	$0.868135\pi$
$992$	0	0
$993$	−703.397	−0.708356
$994$	0	0
$995$	−704.616	−0.708157
$996$	0	0
$997$	1237.58i	1.24131i	0.784085	+	0.620654i	$0.213133\pi$
$997$	1237.58i	1.24131i	−0.784085	+	0.620654i	$0.786867\pi$
$998$	0	0
$999$	− 60.4450i	− 0.0605055i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.b.e.127.5		8
3.2	odd	2		2304.3.b.t.127.8		8
4.3	odd	2		768.3.b.f.127.1		8
8.3	odd	2	inner	768.3.b.e.127.8		8
8.5	even	2		768.3.b.f.127.4		8
12.11	even	2		2304.3.b.q.127.8		8
16.3	odd	4		384.3.g.a.127.1	✓	8
16.5	even	4		384.3.g.b.127.4	yes	8
16.11	odd	4		384.3.g.b.127.8	yes	8
16.13	even	4		384.3.g.a.127.5	yes	8
24.5	odd	2		2304.3.b.q.127.1		8
24.11	even	2		2304.3.b.t.127.1		8
48.5	odd	4		1152.3.g.c.127.1		8
48.11	even	4		1152.3.g.c.127.2		8
48.29	odd	4		1152.3.g.f.127.7		8
48.35	even	4		1152.3.g.f.127.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.g.a.127.1	✓	8	16.3	odd	4
384.3.g.a.127.5	yes	8	16.13	even	4
384.3.g.b.127.4	yes	8	16.5	even	4
384.3.g.b.127.8	yes	8	16.11	odd	4
768.3.b.e.127.5		8	1.1	even	1	trivial
768.3.b.e.127.8		8	8.3	odd	2	inner
768.3.b.f.127.1		8	4.3	odd	2
768.3.b.f.127.4		8	8.5	even	2
1152.3.g.c.127.1		8	48.5	odd	4
1152.3.g.c.127.2		8	48.11	even	4
1152.3.g.f.127.7		8	48.29	odd	4
1152.3.g.f.127.8		8	48.35	even	4
2304.3.b.q.127.1		8	24.5	odd	2
2304.3.b.q.127.8		8	12.11	even	2
2304.3.b.t.127.1		8	24.11	even	2
2304.3.b.t.127.8		8	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -8.29253i q^{5} +8.55583i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} -8.29253i q^{5} +8.55583i q^{7} +3.00000 q^{9} -13.7980 q^{11} -17.0693i q^{13} -14.3631i q^{15} +20.3246 q^{17} -20.4440 q^{19} +14.8191i q^{21} -5.51575i q^{23} -43.7660 q^{25} +5.19615 q^{27} -41.0586i q^{29} -22.2953i q^{31} -23.8988 q^{33} +70.9495 q^{35} -11.6326i q^{37} -29.5649i q^{39} -35.9527 q^{41} -66.8648 q^{43} -24.8776i q^{45} -19.9366i q^{47} -24.2023 q^{49} +35.2032 q^{51} +17.6329i q^{53} +114.420i q^{55} -35.4100 q^{57} +62.1572 q^{59} +47.4836i q^{61} +25.6675i q^{63} -141.548 q^{65} -74.8105 q^{67} -9.55356i q^{69} +16.9150i q^{71} -101.712 q^{73} -75.8050 q^{75} -118.053i q^{77} -0.879320i q^{79} +9.00000 q^{81} -23.2131 q^{83} -168.542i q^{85} -71.1155i q^{87} +16.3089 q^{89} +146.042 q^{91} -38.6167i q^{93} +169.532i q^{95} +188.307 q^{97} -41.3939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 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

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