LMFDB - Embedded newform 768.3.b.f.127.6

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			127.6
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.3.b.f.127.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -1.36433i q^{5} +1.24213i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} -1.36433i q^{5} +1.24213i q^{7} +3.00000 q^{9} -5.79796 q^{11} +16.3830i q^{13} -2.36308i q^{15} -5.01086 q^{17} +26.1835 q^{19} +2.15142i q^{21} +25.1117i q^{23} +23.1386 q^{25} +5.19615 q^{27} +32.7743i q^{29} +1.01836i q^{31} -10.0424 q^{33} +1.69466 q^{35} -14.9948i q^{37} +28.3762i q^{39} +72.5212 q^{41} -33.4922 q^{43} -4.09298i q^{45} -66.5640i q^{47} +47.4571 q^{49} -8.67906 q^{51} +54.6513i q^{53} +7.91030i q^{55} +45.3511 q^{57} -20.5880 q^{59} +111.026i q^{61} +3.72638i q^{63} +22.3518 q^{65} +60.9540 q^{67} +43.4947i q^{69} -80.4576i q^{71} -30.0525 q^{73} +40.0773 q^{75} -7.20179i q^{77} +80.9441i q^{79} +9.00000 q^{81} -113.958 q^{83} +6.83644i q^{85} +56.7667i q^{87} +21.0637 q^{89} -20.3498 q^{91} +1.76386i q^{93} -35.7228i q^{95} +160.594 q^{97} -17.3939 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 1.36433i	− 0.272865i	−0.990649	−	0.136433i	$-0.956436\pi$
$5$	− 1.36433i	− 0.272865i	0.990649	−	0.136433i	$-0.0435637\pi$
$6$	0	0
$7$	1.24213i	0.177446i	0.996056	+	0.0887232i	$0.0282787\pi$
$7$	1.24213i	0.177446i	−0.996056	+	0.0887232i	$0.971721\pi$
$8$	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
$13$	16.3830i	1.26023i	0.776501	+	0.630116i	$0.216993\pi$
$13$	16.3830i	1.26023i	−0.776501	+	0.630116i	$0.783007\pi$
$14$	0	0
$15$	− 2.36308i	− 0.157539i
$16$	0	0
$17$	−5.01086	−0.294756	−0.147378	−	0.989080i	$-0.547083\pi$
$17$	−5.01086	−0.294756	−0.147378	+	0.989080i	$0.547083\pi$
$18$	0	0
$19$	26.1835	1.37808	0.689039	−	0.724725i	$-0.258033\pi$
$19$	26.1835	1.37808	0.689039	+	0.724725i	$0.258033\pi$
$20$	0	0
$21$	2.15142i	0.102449i
$22$	0	0
$23$	25.1117i	1.09181i	0.837847	+	0.545906i	$0.183814\pi$
$23$	25.1117i	1.09181i	−0.837847	+	0.545906i	$0.816186\pi$
$24$	0	0
$25$	23.1386	0.925545
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	32.7743i	1.13015i	0.825040	+	0.565074i	$0.191152\pi$
$29$	32.7743i	1.13015i	−0.825040	+	0.565074i	$0.808848\pi$
$30$	0	0
$31$	1.01836i	0.0328504i	0.999865	+	0.0164252i	$0.00522854\pi$
$31$	1.01836i	0.0328504i	−0.999865	+	0.0164252i	$0.994771\pi$
$32$	0	0
$33$	−10.0424	−0.304314
$34$	0	0
$35$	1.69466	0.0484189
$36$	0	0
$37$	− 14.9948i	− 0.405264i	−0.979255	−	0.202632i	$-0.935050\pi$
$37$	− 14.9948i	− 0.405264i	0.979255	−	0.202632i	$-0.0649495\pi$
$38$	0	0
$39$	28.3762i	0.727595i
$40$	0	0
$41$	72.5212	1.76881	0.884405	−	0.466720i	$-0.154565\pi$
$41$	72.5212	1.76881	0.884405	+	0.466720i	$0.154565\pi$
$42$	0	0
$43$	−33.4922	−0.778888	−0.389444	−	0.921050i	$-0.627333\pi$
$43$	−33.4922	−0.778888	−0.389444	+	0.921050i	$0.627333\pi$
$44$	0	0
$45$	− 4.09298i	− 0.0909550i
$46$	0	0
$47$	− 66.5640i	− 1.41626i	−0.706085	−	0.708128i	$-0.749540\pi$
$47$	− 66.5640i	− 1.41626i	0.706085	−	0.708128i	$-0.250460\pi$
$48$	0	0
$49$	47.4571	0.968513
$50$	0	0
$51$	−8.67906	−0.170178
$52$	0	0
$53$	54.6513i	1.03116i	0.856842	+	0.515579i	$0.172423\pi$
$53$	54.6513i	1.03116i	−0.856842	+	0.515579i	$0.827577\pi$
$54$	0	0
$55$	7.91030i	0.143824i
$56$	0	0
$57$	45.3511	0.795633
$58$	0	0
$59$	−20.5880	−0.348949	−0.174474	−	0.984662i	$-0.555823\pi$
$59$	−20.5880	−0.348949	−0.174474	+	0.984662i	$0.555823\pi$
$60$	0	0
$61$	111.026i	1.82010i	0.414499	+	0.910050i	$0.363957\pi$
$61$	111.026i	1.82010i	−0.414499	+	0.910050i	$0.636043\pi$
$62$	0	0
$63$	3.72638i	0.0591488i
$64$	0	0
$65$	22.3518	0.343873
$66$	0	0
$67$	60.9540	0.909762	0.454881	−	0.890552i	$-0.349682\pi$
$67$	60.9540	0.909762	0.454881	+	0.890552i	$0.349682\pi$
$68$	0	0
$69$	43.4947i	0.630358i
$70$	0	0
$71$	− 80.4576i	− 1.13320i	−0.823991	−	0.566602i	$-0.808258\pi$
$71$	− 80.4576i	− 1.13320i	0.823991	−	0.566602i	$-0.191742\pi$
$72$	0	0
$73$	−30.0525	−0.411679	−0.205839	−	0.978586i	$-0.565992\pi$
$73$	−30.0525	−0.411679	−0.205839	+	0.978586i	$0.565992\pi$
$74$	0	0
$75$	40.0773	0.534363
$76$	0	0
$77$	− 7.20179i	− 0.0935298i
$78$	0	0
$79$	80.9441i	1.02461i	0.858804	+	0.512304i	$0.171208\pi$
$79$	80.9441i	1.02461i	−0.858804	+	0.512304i	$0.828792\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−113.958	−1.37299	−0.686496	−	0.727134i	$-0.740852\pi$
$83$	−113.958	−1.37299	−0.686496	+	0.727134i	$0.740852\pi$
$84$	0	0
$85$	6.83644i	0.0804288i
$86$	0	0
$87$	56.7667i	0.652491i
$88$	0	0
$89$	21.0637	0.236671	0.118335	−	0.992974i	$-0.462244\pi$
$89$	21.0637	0.236671	0.118335	+	0.992974i	$0.462244\pi$
$90$	0	0
$91$	−20.3498	−0.223624
$92$	0	0
$93$	1.76386i	0.0189662i
$94$	0	0
$95$	− 35.7228i	− 0.376029i
$96$	0	0
$97$	160.594	1.65561	0.827806	−	0.561014i	$-0.189589\pi$
$97$	160.594	1.65561	0.827806	+	0.561014i	$0.189589\pi$
$98$	0	0
$99$	−17.3939	−0.175696
$100$	0	0
$101$	76.2681i	0.755130i	0.925983	+	0.377565i	$0.123239\pi$
$101$	76.2681i	0.755130i	−0.925983	+	0.377565i	$0.876761\pi$
$102$	0	0
$103$	− 182.763i	− 1.77440i	−0.461383	−	0.887201i	$-0.652647\pi$
$103$	− 182.763i	− 1.77440i	0.461383	−	0.887201i	$-0.347353\pi$
$104$	0	0
$105$	2.93524	0.0279547
$106$	0	0
$107$	−31.8533	−0.297694	−0.148847	−	0.988860i	$-0.547556\pi$
$107$	−31.8533	−0.297694	−0.148847	+	0.988860i	$0.547556\pi$
$108$	0	0
$109$	11.3289i	0.103935i	0.998649	+	0.0519676i	$0.0165493\pi$
$109$	11.3289i	0.103935i	−0.998649	+	0.0519676i	$0.983451\pi$
$110$	0	0
$111$	− 25.9717i	− 0.233979i
$112$	0	0
$113$	49.9587	0.442113	0.221056	−	0.975261i	$-0.429050\pi$
$113$	49.9587	0.442113	0.221056	+	0.975261i	$0.429050\pi$
$114$	0	0
$115$	34.2605	0.297917
$116$	0	0
$117$	49.1490i	0.420077i
$118$	0	0
$119$	− 6.22412i	− 0.0523035i
$120$	0	0
$121$	−87.3837	−0.722179
$122$	0	0
$123$	125.610	1.02122
$124$	0	0
$125$	− 65.6767i	− 0.525414i
$126$	0	0
$127$	208.236i	1.63965i	0.572614	+	0.819825i	$0.305929\pi$
$127$	208.236i	1.63965i	−0.572614	+	0.819825i	$0.694071\pi$
$128$	0	0
$129$	−58.0102	−0.449691
$130$	0	0
$131$	−220.549	−1.68358	−0.841791	−	0.539804i	$-0.818498\pi$
$131$	−220.549	−1.68358	−0.841791	+	0.539804i	$0.818498\pi$
$132$	0	0
$133$	32.5231i	0.244535i
$134$	0	0
$135$	− 7.08924i	− 0.0525129i
$136$	0	0
$137$	−15.2664	−0.111433	−0.0557167	−	0.998447i	$-0.517744\pi$
$137$	−15.2664	−0.111433	−0.0557167	+	0.998447i	$0.517744\pi$
$138$	0	0
$139$	86.7117	0.623825	0.311912	−	0.950111i	$-0.399030\pi$
$139$	86.7117	0.623825	0.311912	+	0.950111i	$0.399030\pi$
$140$	0	0
$141$	− 115.292i	− 0.817675i
$142$	0	0
$143$	− 94.9881i	− 0.664252i
$144$	0	0
$145$	44.7148	0.308378
$146$	0	0
$147$	82.1982	0.559171
$148$	0	0
$149$	146.849i	0.985561i	0.870154	+	0.492780i	$0.164019\pi$
$149$	146.849i	0.985561i	−0.870154	+	0.492780i	$0.835981\pi$
$150$	0	0
$151$	195.933i	1.29757i	0.760972	+	0.648785i	$0.224723\pi$
$151$	195.933i	1.29757i	−0.760972	+	0.648785i	$0.775277\pi$
$152$	0	0
$153$	−15.0326	−0.0982522
$154$	0	0
$155$	1.38938	0.00896374
$156$	0	0
$157$	4.65454i	0.0296468i	0.999890	+	0.0148234i	$0.00471860\pi$
$157$	4.65454i	0.0296468i	−0.999890	+	0.0148234i	$0.995281\pi$
$158$	0	0
$159$	94.6589i	0.595339i
$160$	0	0
$161$	−31.1918	−0.193738
$162$	0	0
$163$	59.5489	0.365331	0.182665	−	0.983175i	$-0.441528\pi$
$163$	59.5489	0.365331	0.182665	+	0.983175i	$0.441528\pi$
$164$	0	0
$165$	13.7010i	0.0830367i
$166$	0	0
$167$	− 209.012i	− 1.25157i	−0.779996	−	0.625785i	$-0.784779\pi$
$167$	− 209.012i	− 1.25157i	0.779996	−	0.625785i	$-0.215221\pi$
$168$	0	0
$169$	−99.4032	−0.588185
$170$	0	0
$171$	78.5504	0.459359
$172$	0	0
$173$	− 96.7635i	− 0.559326i	−0.960098	−	0.279663i	$-0.909777\pi$
$173$	− 96.7635i	− 0.559326i	0.960098	−	0.279663i	$-0.0902228\pi$
$174$	0	0
$175$	28.7411i	0.164235i
$176$	0	0
$177$	−35.6594	−0.201466
$178$	0	0
$179$	−49.5039	−0.276558	−0.138279	−	0.990393i	$-0.544157\pi$
$179$	−49.5039	−0.276558	−0.138279	+	0.990393i	$0.544157\pi$
$180$	0	0
$181$	− 141.417i	− 0.781310i	−0.920537	−	0.390655i	$-0.872248\pi$
$181$	− 141.417i	− 0.781310i	0.920537	−	0.390655i	$-0.127752\pi$
$182$	0	0
$183$	192.303i	1.05084i
$184$	0	0
$185$	−20.4578	−0.110582
$186$	0	0
$187$	29.0528	0.155362
$188$	0	0
$189$	6.45427i	0.0341496i
$190$	0	0
$191$	− 116.994i	− 0.612533i	−0.951946	−	0.306267i	$-0.900920\pi$
$191$	− 116.994i	− 0.612533i	0.951946	−	0.306267i	$-0.0990799\pi$
$192$	0	0
$193$	−90.7357	−0.470133	−0.235067	−	0.971979i	$-0.575531\pi$
$193$	−90.7357	−0.470133	−0.235067	+	0.971979i	$0.575531\pi$
$194$	0	0
$195$	38.7144	0.198535
$196$	0	0
$197$	− 380.039i	− 1.92913i	−0.263840	−	0.964566i	$-0.584989\pi$
$197$	− 380.039i	− 1.92913i	0.263840	−	0.964566i	$-0.415011\pi$
$198$	0	0
$199$	− 77.6563i	− 0.390233i	−0.980780	−	0.195116i	$-0.937492\pi$
$199$	− 77.6563i	− 0.390233i	0.980780	−	0.195116i	$-0.0625084\pi$
$200$	0	0
$201$	105.576	0.525251
$202$	0	0
$203$	−40.7098	−0.200541
$204$	0	0
$205$	− 98.9425i	− 0.482646i
$206$	0	0
$207$	75.3350i	0.363937i
$208$	0	0
$209$	−151.811	−0.726367
$210$	0	0
$211$	191.446	0.907325	0.453662	−	0.891174i	$-0.350117\pi$
$211$	191.446	0.907325	0.453662	+	0.891174i	$0.350117\pi$
$212$	0	0
$213$	− 139.357i	− 0.654256i
$214$	0	0
$215$	45.6943i	0.212531i
$216$	0	0
$217$	−1.26493	−0.00582919
$218$	0	0
$219$	−52.0525	−0.237683
$220$	0	0
$221$	− 82.0930i	− 0.371462i
$222$	0	0
$223$	168.451i	0.755387i	0.925931	+	0.377693i	$0.123283\pi$
$223$	168.451i	0.755387i	−0.925931	+	0.377693i	$0.876717\pi$
$224$	0	0
$225$	69.4158	0.308515
$226$	0	0
$227$	113.516	0.500071	0.250036	−	0.968237i	$-0.419558\pi$
$227$	113.516	0.500071	0.250036	+	0.968237i	$0.419558\pi$
$228$	0	0
$229$	− 117.618i	− 0.513615i	−0.966463	−	0.256808i	$-0.917329\pi$
$229$	− 117.618i	− 0.513615i	0.966463	−	0.256808i	$-0.0826707\pi$
$230$	0	0
$231$	− 12.4739i	− 0.0539994i
$232$	0	0
$233$	−277.085	−1.18921	−0.594604	−	0.804019i	$-0.702691\pi$
$233$	−277.085	−1.18921	−0.594604	+	0.804019i	$0.702691\pi$
$234$	0	0
$235$	−90.8150	−0.386447
$236$	0	0
$237$	140.199i	0.591558i
$238$	0	0
$239$	− 343.072i	− 1.43545i	−0.696327	−	0.717724i	$-0.745184\pi$
$239$	− 343.072i	− 1.43545i	0.696327	−	0.717724i	$-0.254816\pi$
$240$	0	0
$241$	−328.140	−1.36157	−0.680787	−	0.732481i	$-0.738362\pi$
$241$	−328.140	−1.36157	−0.680787	+	0.732481i	$0.738362\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	− 64.7470i	− 0.264273i
$246$	0	0
$247$	428.964i	1.73670i
$248$	0	0
$249$	−197.382	−0.792697
$250$	0	0
$251$	−452.914	−1.80444	−0.902219	−	0.431279i	$-0.858063\pi$
$251$	−452.914	−1.80444	−0.902219	+	0.431279i	$0.858063\pi$
$252$	0	0
$253$	− 145.596i	− 0.575480i
$254$	0	0
$255$	11.8411i	0.0464356i
$256$	0	0
$257$	−346.830	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$257$	−346.830	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$258$	0	0
$259$	18.6254	0.0719127
$260$	0	0
$261$	98.3229i	0.376716i
$262$	0	0
$263$	− 402.440i	− 1.53019i	−0.643917	−	0.765095i	$-0.722692\pi$
$263$	− 402.440i	− 1.53019i	0.643917	−	0.765095i	$-0.277308\pi$
$264$	0	0
$265$	74.5622	0.281367
$266$	0	0
$267$	36.4834	0.136642
$268$	0	0
$269$	321.562i	1.19540i	0.801721	+	0.597699i	$0.203918\pi$
$269$	321.562i	1.19540i	−0.801721	+	0.597699i	$0.796082\pi$
$270$	0	0
$271$	− 456.902i	− 1.68599i	−0.537924	−	0.842993i	$-0.680791\pi$
$271$	− 456.902i	− 1.68599i	0.537924	−	0.842993i	$-0.319209\pi$
$272$	0	0
$273$	−35.2468	−0.129109
$274$	0	0
$275$	−134.157	−0.487843
$276$	0	0
$277$	− 329.543i	− 1.18969i	−0.803842	−	0.594843i	$-0.797214\pi$
$277$	− 329.543i	− 1.18969i	0.803842	−	0.594843i	$-0.202786\pi$
$278$	0	0
$279$	3.05509i	0.0109501i
$280$	0	0
$281$	175.064	0.623005	0.311503	−	0.950245i	$-0.399168\pi$
$281$	175.064	0.623005	0.311503	+	0.950245i	$0.399168\pi$
$282$	0	0
$283$	150.298	0.531087	0.265544	−	0.964099i	$-0.414449\pi$
$283$	150.298	0.531087	0.265544	+	0.964099i	$0.414449\pi$
$284$	0	0
$285$	− 61.8736i	− 0.217101i
$286$	0	0
$287$	90.0804i	0.313869i
$288$	0	0
$289$	−263.891	−0.913119
$290$	0	0
$291$	278.158	0.955868
$292$	0	0
$293$	− 160.435i	− 0.547561i	−0.961792	−	0.273781i	$-0.911726\pi$
$293$	− 160.435i	− 0.547561i	0.961792	−	0.273781i	$-0.0882742\pi$
$294$	0	0
$295$	28.0887i	0.0952159i
$296$	0	0
$297$	−30.1271	−0.101438
$298$	0	0
$299$	−411.405	−1.37594
$300$	0	0
$301$	− 41.6015i	− 0.138211i
$302$	0	0
$303$	132.100i	0.435974i
$304$	0	0
$305$	151.476	0.496642
$306$	0	0
$307$	−168.120	−0.547621	−0.273811	−	0.961784i	$-0.588284\pi$
$307$	−168.120	−0.547621	−0.273811	+	0.961784i	$0.588284\pi$
$308$	0	0
$309$	− 316.555i	− 1.02445i
$310$	0	0
$311$	470.376i	1.51246i	0.654305	+	0.756231i	$0.272961\pi$
$311$	470.376i	1.51246i	−0.654305	+	0.756231i	$0.727039\pi$
$312$	0	0
$313$	−19.4378	−0.0621016	−0.0310508	−	0.999518i	$-0.509885\pi$
$313$	−19.4378	−0.0621016	−0.0310508	+	0.999518i	$0.509885\pi$
$314$	0	0
$315$	5.08399	0.0161396
$316$	0	0
$317$	− 242.195i	− 0.764021i	−0.924158	−	0.382011i	$-0.875232\pi$
$317$	− 242.195i	− 0.764021i	0.924158	−	0.382011i	$-0.124768\pi$
$318$	0	0
$319$	− 190.024i	− 0.595686i
$320$	0	0
$321$	−55.1715	−0.171874
$322$	0	0
$323$	−131.202	−0.406197
$324$	0	0
$325$	379.080i	1.16640i
$326$	0	0
$327$	19.6223i	0.0600070i
$328$	0	0
$329$	82.6808	0.251309
$330$	0	0
$331$	440.951	1.33218	0.666090	−	0.745872i	$-0.267967\pi$
$331$	440.951	1.33218	0.666090	+	0.745872i	$0.267967\pi$
$332$	0	0
$333$	− 44.9843i	− 0.135088i
$334$	0	0
$335$	− 83.1612i	− 0.248242i
$336$	0	0
$337$	−250.841	−0.744335	−0.372167	−	0.928166i	$-0.621385\pi$
$337$	−250.841	−0.744335	−0.372167	+	0.928166i	$0.621385\pi$
$338$	0	0
$339$	86.5310	0.255254
$340$	0	0
$341$	− 5.90443i	− 0.0173150i
$342$	0	0
$343$	119.812i	0.349306i
$344$	0	0
$345$	59.3409	0.172003
$346$	0	0
$347$	16.1029	0.0464060	0.0232030	−	0.999731i	$-0.492614\pi$
$347$	16.1029	0.0464060	0.0232030	+	0.999731i	$0.492614\pi$
$348$	0	0
$349$	− 274.843i	− 0.787516i	−0.919214	−	0.393758i	$-0.871175\pi$
$349$	− 274.843i	− 0.787516i	0.919214	−	0.393758i	$-0.128825\pi$
$350$	0	0
$351$	85.1287i	0.242532i
$352$	0	0
$353$	165.428	0.468634	0.234317	−	0.972160i	$-0.424715\pi$
$353$	165.428	0.468634	0.234317	+	0.972160i	$0.424715\pi$
$354$	0	0
$355$	−109.770	−0.309212
$356$	0	0
$357$	− 10.7805i	− 0.0301974i
$358$	0	0
$359$	− 688.519i	− 1.91788i	−0.283607	−	0.958941i	$-0.591531\pi$
$359$	− 688.519i	− 1.91788i	0.283607	−	0.958941i	$-0.408469\pi$
$360$	0	0
$361$	324.574	0.899096
$362$	0	0
$363$	−151.353	−0.416950
$364$	0	0
$365$	41.0015i	0.112333i
$366$	0	0
$367$	− 102.170i	− 0.278393i	−0.990265	−	0.139196i	$-0.955548\pi$
$367$	− 102.170i	− 0.278393i	0.990265	−	0.139196i	$-0.0444519\pi$
$368$	0	0
$369$	217.564	0.589603
$370$	0	0
$371$	−67.8838	−0.182975
$372$	0	0
$373$	294.317i	0.789052i	0.918885	+	0.394526i	$0.129091\pi$
$373$	294.317i	0.789052i	−0.918885	+	0.394526i	$0.870909\pi$
$374$	0	0
$375$	− 113.755i	− 0.303348i
$376$	0	0
$377$	−536.942	−1.42425
$378$	0	0
$379$	81.1923	0.214228	0.107114	−	0.994247i	$-0.465839\pi$
$379$	81.1923	0.214228	0.107114	+	0.994247i	$0.465839\pi$
$380$	0	0
$381$	360.675i	0.946653i
$382$	0	0
$383$	198.838i	0.519160i	0.965722	+	0.259580i	$0.0835841\pi$
$383$	198.838i	0.519160i	−0.965722	+	0.259580i	$0.916416\pi$
$384$	0	0
$385$	−9.82559	−0.0255210
$386$	0	0
$387$	−100.477	−0.259629
$388$	0	0
$389$	− 368.767i	− 0.947987i	−0.880528	−	0.473993i	$-0.842812\pi$
$389$	− 368.767i	− 0.947987i	0.880528	−	0.473993i	$-0.157188\pi$
$390$	0	0
$391$	− 125.831i	− 0.321819i
$392$	0	0
$393$	−382.002	−0.972016
$394$	0	0
$395$	110.434	0.279580
$396$	0	0
$397$	− 114.315i	− 0.287947i	−0.989582	−	0.143973i	$-0.954012\pi$
$397$	− 114.315i	− 0.287947i	0.989582	−	0.143973i	$-0.0459880\pi$
$398$	0	0
$399$	56.3317i	0.141182i
$400$	0	0
$401$	39.9083	0.0995218	0.0497609	−	0.998761i	$-0.484154\pi$
$401$	39.9083	0.0995218	0.0497609	+	0.998761i	$0.484154\pi$
$402$	0	0
$403$	−16.6839	−0.0413992
$404$	0	0
$405$	− 12.2789i	− 0.0303183i
$406$	0	0
$407$	86.9391i	0.213610i
$408$	0	0
$409$	269.868	0.659825	0.329912	−	0.944012i	$-0.392981\pi$
$409$	269.868	0.659825	0.329912	+	0.944012i	$0.392981\pi$
$410$	0	0
$411$	−26.4421	−0.0643361
$412$	0	0
$413$	− 25.5728i	− 0.0619197i
$414$	0	0
$415$	155.476i	0.374641i
$416$	0	0
$417$	150.189	0.360165
$418$	0	0
$419$	20.3559	0.0485821	0.0242910	−	0.999705i	$-0.492267\pi$
$419$	20.3559	0.0485821	0.0242910	+	0.999705i	$0.492267\pi$
$420$	0	0
$421$	557.905i	1.32519i	0.748978	+	0.662595i	$0.230545\pi$
$421$	557.905i	1.32519i	−0.748978	+	0.662595i	$0.769455\pi$
$422$	0	0
$423$	− 199.692i	− 0.472085i
$424$	0	0
$425$	−115.944	−0.272810
$426$	0	0
$427$	−137.908	−0.322970
$428$	0	0
$429$	− 164.524i	− 0.383506i
$430$	0	0
$431$	376.569i	0.873710i	0.899532	+	0.436855i	$0.143908\pi$
$431$	376.569i	0.873710i	−0.899532	+	0.436855i	$0.856092\pi$
$432$	0	0
$433$	602.876	1.39232	0.696162	−	0.717885i	$-0.254890\pi$
$433$	602.876	1.39232	0.696162	+	0.717885i	$0.254890\pi$
$434$	0	0
$435$	77.4483	0.178042
$436$	0	0
$437$	657.510i	1.50460i
$438$	0	0
$439$	381.087i	0.868080i	0.900894	+	0.434040i	$0.142912\pi$
$439$	381.087i	0.868080i	−0.900894	+	0.434040i	$0.857088\pi$
$440$	0	0
$441$	142.371	0.322838
$442$	0	0
$443$	599.838	1.35404	0.677018	−	0.735966i	$-0.263272\pi$
$443$	599.838	1.35404	0.677018	+	0.735966i	$0.263272\pi$
$444$	0	0
$445$	− 28.7377i	− 0.0645791i
$446$	0	0
$447$	254.349i	0.569014i
$448$	0	0
$449$	814.240	1.81345	0.906726	−	0.421720i	$-0.138573\pi$
$449$	814.240	1.81345	0.906726	+	0.421720i	$0.138573\pi$
$450$	0	0
$451$	−420.475	−0.932317
$452$	0	0
$453$	339.366i	0.749152i
$454$	0	0
$455$	27.7637i	0.0610191i
$456$	0	0
$457$	−111.281	−0.243502	−0.121751	−	0.992561i	$-0.538851\pi$
$457$	−111.281	−0.243502	−0.121751	+	0.992561i	$0.538851\pi$
$458$	0	0
$459$	−26.0372	−0.0567259
$460$	0	0
$461$	507.833i	1.10159i	0.834641	+	0.550795i	$0.185675\pi$
$461$	507.833i	1.10159i	−0.834641	+	0.550795i	$0.814325\pi$
$462$	0	0
$463$	397.302i	0.858103i	0.903280	+	0.429052i	$0.141152\pi$
$463$	397.302i	0.858103i	−0.903280	+	0.429052i	$0.858848\pi$
$464$	0	0
$465$	2.40648	0.00517522
$466$	0	0
$467$	830.195	1.77772	0.888860	−	0.458179i	$-0.151498\pi$
$467$	830.195	1.77772	0.888860	+	0.458179i	$0.151498\pi$
$468$	0	0
$469$	75.7126i	0.161434i
$470$	0	0
$471$	8.06190i	0.0171166i
$472$	0	0
$473$	194.186	0.410542
$474$	0	0
$475$	605.849	1.27547
$476$	0	0
$477$	163.954i	0.343719i
$478$	0	0
$479$	146.251i	0.305325i	0.988278	+	0.152662i	$0.0487847\pi$
$479$	146.251i	0.305325i	−0.988278	+	0.152662i	$0.951215\pi$
$480$	0	0
$481$	245.660	0.510727
$482$	0	0
$483$	−54.0258	−0.111855
$484$	0	0
$485$	− 219.103i	− 0.451759i
$486$	0	0
$487$	− 177.070i	− 0.363593i	−0.983336	−	0.181797i	$-0.941809\pi$
$487$	− 177.070i	− 0.363593i	0.983336	−	0.181797i	$-0.0581913\pi$
$488$	0	0
$489$	103.142	0.210924
$490$	0	0
$491$	94.9463	0.193373	0.0966866	−	0.995315i	$-0.469176\pi$
$491$	94.9463	0.193373	0.0966866	+	0.995315i	$0.469176\pi$
$492$	0	0
$493$	− 164.227i	− 0.333118i
$494$	0	0
$495$	23.7309i	0.0479412i
$496$	0	0
$497$	99.9384	0.201083
$498$	0	0
$499$	−744.720	−1.49243	−0.746213	−	0.665707i	$-0.768130\pi$
$499$	−744.720	−1.49243	−0.746213	+	0.665707i	$0.768130\pi$
$500$	0	0
$501$	− 362.020i	− 0.722594i
$502$	0	0
$503$	− 578.757i	− 1.15061i	−0.817939	−	0.575305i	$-0.804883\pi$
$503$	− 578.757i	− 1.15061i	0.817939	−	0.575305i	$-0.195117\pi$
$504$	0	0
$505$	104.055	0.206049
$506$	0	0
$507$	−172.171	−0.339589
$508$	0	0
$509$	− 323.101i	− 0.634777i	−0.948296	−	0.317388i	$-0.897194\pi$
$509$	− 323.101i	− 0.634777i	0.948296	−	0.317388i	$-0.102806\pi$
$510$	0	0
$511$	− 37.3290i	− 0.0730509i
$512$	0	0
$513$	136.053	0.265211
$514$	0	0
$515$	−249.349	−0.484172
$516$	0	0
$517$	385.935i	0.746490i
$518$	0	0
$519$	− 167.599i	− 0.322927i
$520$	0	0
$521$	582.929	1.11887	0.559433	−	0.828875i	$-0.311019\pi$
$521$	582.929	1.11887	0.559433	+	0.828875i	$0.311019\pi$
$522$	0	0
$523$	227.111	0.434247	0.217124	−	0.976144i	$-0.430332\pi$
$523$	227.111	0.434247	0.217124	+	0.976144i	$0.430332\pi$
$524$	0	0
$525$	49.7810i	0.0948209i
$526$	0	0
$527$	− 5.10288i	− 0.00968288i
$528$	0	0
$529$	−101.596	−0.192053
$530$	0	0
$531$	−61.7639	−0.116316
$532$	0	0
$533$	1188.12i	2.22911i
$534$	0	0
$535$	43.4582i	0.0812303i
$536$	0	0
$537$	−85.7433	−0.159671
$538$	0	0
$539$	−275.154	−0.510491
$540$	0	0
$541$	551.391i	1.01921i	0.860409	+	0.509603i	$0.170208\pi$
$541$	551.391i	1.01921i	−0.860409	+	0.509603i	$0.829792\pi$
$542$	0	0
$543$	− 244.942i	− 0.451090i
$544$	0	0
$545$	15.4564	0.0283603
$546$	0	0
$547$	745.659	1.36318	0.681590	−	0.731735i	$-0.261289\pi$
$547$	745.659	1.36318	0.681590	+	0.731735i	$0.261289\pi$
$548$	0	0
$549$	333.078i	0.606700i
$550$	0	0
$551$	858.144i	1.55743i
$552$	0	0
$553$	−100.543	−0.181813
$554$	0	0
$555$	−35.4339	−0.0638448
$556$	0	0
$557$	755.207i	1.35585i	0.735132	+	0.677924i	$0.237120\pi$
$557$	755.207i	1.35585i	−0.735132	+	0.677924i	$0.762880\pi$
$558$	0	0
$559$	− 548.703i	− 0.981580i
$560$	0	0
$561$	50.3209	0.0896985
$562$	0	0
$563$	−699.309	−1.24211	−0.621056	−	0.783766i	$-0.713296\pi$
$563$	−699.309	−1.24211	−0.621056	+	0.783766i	$0.713296\pi$
$564$	0	0
$565$	− 68.1600i	− 0.120637i
$566$	0	0
$567$	11.1791i	0.0197163i
$568$	0	0
$569$	−35.8709	−0.0630419	−0.0315210	−	0.999503i	$-0.510035\pi$
$569$	−35.8709	−0.0630419	−0.0315210	+	0.999503i	$0.510035\pi$
$570$	0	0
$571$	−828.429	−1.45084	−0.725420	−	0.688307i	$-0.758354\pi$
$571$	−828.429	−1.45084	−0.725420	+	0.688307i	$0.758354\pi$
$572$	0	0
$573$	− 202.639i	− 0.353646i
$574$	0	0
$575$	581.049i	1.01052i
$576$	0	0
$577$	−471.333	−0.816867	−0.408434	−	0.912788i	$-0.633925\pi$
$577$	−471.333	−0.816867	−0.408434	+	0.912788i	$0.633925\pi$
$578$	0	0
$579$	−157.159	−0.271432
$580$	0	0
$581$	− 141.550i	− 0.243632i
$582$	0	0
$583$	− 316.866i	− 0.543510i
$584$	0	0
$585$	67.0553	0.114624
$586$	0	0
$587$	645.149	1.09906	0.549531	−	0.835473i	$-0.314807\pi$
$587$	645.149	1.09906	0.549531	+	0.835473i	$0.314807\pi$
$588$	0	0
$589$	26.6643i	0.0452704i
$590$	0	0
$591$	− 658.247i	− 1.11379i
$592$	0	0
$593$	203.619	0.343370	0.171685	−	0.985152i	$-0.445079\pi$
$593$	203.619	0.343370	0.171685	+	0.985152i	$0.445079\pi$
$594$	0	0
$595$	−8.49172	−0.0142718
$596$	0	0
$597$	− 134.505i	− 0.225301i
$598$	0	0
$599$	603.605i	1.00769i	0.863795	+	0.503844i	$0.168081\pi$
$599$	603.605i	1.00769i	−0.863795	+	0.503844i	$0.831919\pi$
$600$	0	0
$601$	626.271	1.04205	0.521024	−	0.853542i	$-0.325550\pi$
$601$	626.271	1.04205	0.521024	+	0.853542i	$0.325550\pi$
$602$	0	0
$603$	182.862	0.303254
$604$	0	0
$605$	119.220i	0.197057i
$606$	0	0
$607$	− 421.012i	− 0.693595i	−0.937940	−	0.346797i	$-0.887269\pi$
$607$	− 421.012i	− 0.693595i	0.937940	−	0.346797i	$-0.112731\pi$
$608$	0	0
$609$	−70.5114	−0.115782
$610$	0	0
$611$	1090.52	1.78481
$612$	0	0
$613$	− 12.9743i	− 0.0211652i	−0.999944	−	0.0105826i	$-0.996631\pi$
$613$	− 12.9743i	− 0.0211652i	0.999944	−	0.0105826i	$-0.00336861\pi$
$614$	0	0
$615$	− 171.373i	− 0.278656i
$616$	0	0
$617$	423.164	0.685842	0.342921	−	0.939364i	$-0.388584\pi$
$617$	423.164	0.685842	0.342921	+	0.939364i	$0.388584\pi$
$618$	0	0
$619$	−625.820	−1.01102	−0.505509	−	0.862821i	$-0.668695\pi$
$619$	−625.820	−1.01102	−0.505509	+	0.862821i	$0.668695\pi$
$620$	0	0
$621$	130.484i	0.210119i
$622$	0	0
$623$	26.1637i	0.0419963i
$624$	0	0
$625$	488.861	0.782178
$626$	0	0
$627$	−262.944	−0.419368
$628$	0	0
$629$	75.1367i	0.119454i
$630$	0	0
$631$	− 690.848i	− 1.09485i	−0.836856	−	0.547423i	$-0.815609\pi$
$631$	− 690.848i	− 1.09485i	0.836856	−	0.547423i	$-0.184391\pi$
$632$	0	0
$633$	331.593	0.523844
$634$	0	0
$635$	284.101	0.447403
$636$	0	0
$637$	777.491i	1.22055i
$638$	0	0
$639$	− 241.373i	− 0.377735i
$640$	0	0
$641$	−369.160	−0.575913	−0.287957	−	0.957643i	$-0.592976\pi$
$641$	−369.160	−0.575913	−0.287957	+	0.957643i	$0.592976\pi$
$642$	0	0
$643$	−666.030	−1.03582	−0.517909	−	0.855436i	$-0.673289\pi$
$643$	−666.030	−1.03582	−0.517909	+	0.855436i	$0.673289\pi$
$644$	0	0
$645$	79.1448i	0.122705i
$646$	0	0
$647$	651.886i	1.00755i	0.863835	+	0.503776i	$0.168056\pi$
$647$	651.886i	1.00755i	−0.863835	+	0.503776i	$0.831944\pi$
$648$	0	0
$649$	119.368	0.183926
$650$	0	0
$651$	−2.19093	−0.00336549
$652$	0	0
$653$	− 903.324i	− 1.38334i	−0.722212	−	0.691672i	$-0.756874\pi$
$653$	− 903.324i	− 1.38334i	0.722212	−	0.691672i	$-0.243126\pi$
$654$	0	0
$655$	300.901i	0.459391i
$656$	0	0
$657$	−90.1576	−0.137226
$658$	0	0
$659$	643.621	0.976664	0.488332	−	0.872658i	$-0.337606\pi$
$659$	643.621	0.976664	0.488332	+	0.872658i	$0.337606\pi$
$660$	0	0
$661$	− 860.187i	− 1.30134i	−0.759360	−	0.650671i	$-0.774488\pi$
$661$	− 860.187i	− 1.30134i	0.759360	−	0.650671i	$-0.225512\pi$
$662$	0	0
$663$	− 142.189i	− 0.214463i
$664$	0	0
$665$	44.3722	0.0667250
$666$	0	0
$667$	−823.017	−1.23391
$668$	0	0
$669$	291.766i	0.436123i
$670$	0	0
$671$	− 643.725i	− 0.959351i
$672$	0	0
$673$	866.535	1.28757	0.643785	−	0.765206i	$-0.277363\pi$
$673$	866.535	1.28757	0.643785	+	0.765206i	$0.277363\pi$
$674$	0	0
$675$	120.232	0.178121
$676$	0	0
$677$	− 1307.26i	− 1.93095i	−0.260489	−	0.965477i	$-0.583884\pi$
$677$	− 1307.26i	− 1.93095i	0.260489	−	0.965477i	$-0.416116\pi$
$678$	0	0
$679$	199.478i	0.293783i
$680$	0	0
$681$	196.616	0.288716
$682$	0	0
$683$	−783.569	−1.14725	−0.573623	−	0.819120i	$-0.694462\pi$
$683$	−783.569	−1.14725	−0.573623	+	0.819120i	$0.694462\pi$
$684$	0	0
$685$	20.8283i	0.0304063i
$686$	0	0
$687$	− 203.720i	− 0.296536i
$688$	0	0
$689$	−895.354	−1.29950
$690$	0	0
$691$	1014.95	1.46882	0.734408	−	0.678708i	$-0.237460\pi$
$691$	1014.95	1.46882	0.734408	+	0.678708i	$0.237460\pi$
$692$	0	0
$693$	− 21.6054i	− 0.0311766i
$694$	0	0
$695$	− 118.303i	− 0.170220i
$696$	0	0
$697$	−363.394	−0.521368
$698$	0	0
$699$	−479.926	−0.686590
$700$	0	0
$701$	957.527i	1.36595i	0.730444	+	0.682973i	$0.239313\pi$
$701$	957.527i	1.36595i	−0.730444	+	0.682973i	$0.760687\pi$
$702$	0	0
$703$	− 392.615i	− 0.558485i
$704$	0	0
$705$	−157.296	−0.223115
$706$	0	0
$707$	−94.7346	−0.133995
$708$	0	0
$709$	− 65.7503i	− 0.0927366i	−0.998924	−	0.0463683i	$-0.985235\pi$
$709$	− 65.7503i	− 0.0927366i	0.998924	−	0.0463683i	$-0.0147648\pi$
$710$	0	0
$711$	242.832i	0.341536i
$712$	0	0
$713$	−25.5728	−0.0358665
$714$	0	0
$715$	−129.595	−0.181251
$716$	0	0
$717$	− 594.219i	− 0.828757i
$718$	0	0
$719$	573.085i	0.797058i	0.917156	+	0.398529i	$0.130479\pi$
$719$	573.085i	0.797058i	−0.917156	+	0.398529i	$0.869521\pi$
$720$	0	0
$721$	227.015	0.314861
$722$	0	0
$723$	−568.354	−0.786106
$724$	0	0
$725$	758.352i	1.04600i
$726$	0	0
$727$	− 249.632i	− 0.343373i	−0.985152	−	0.171686i	$-0.945078\pi$
$727$	− 249.632i	− 0.343373i	0.985152	−	0.171686i	$-0.0549216\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	167.825	0.229582
$732$	0	0
$733$	662.187i	0.903393i	0.892172	+	0.451696i	$0.149181\pi$
$733$	662.187i	0.903393i	−0.892172	+	0.451696i	$0.850819\pi$
$734$	0	0
$735$	− 112.145i	− 0.152578i
$736$	0	0
$737$	−353.409	−0.479524
$738$	0	0
$739$	98.7372	0.133609	0.0668046	−	0.997766i	$-0.478720\pi$
$739$	98.7372	0.133609	0.0668046	+	0.997766i	$0.478720\pi$
$740$	0	0
$741$	742.988i	1.00268i
$742$	0	0
$743$	906.520i	1.22008i	0.792370	+	0.610041i	$0.208847\pi$
$743$	906.520i	1.22008i	−0.792370	+	0.610041i	$0.791153\pi$
$744$	0	0
$745$	200.349	0.268925
$746$	0	0
$747$	−341.875	−0.457664
$748$	0	0
$749$	− 39.5657i	− 0.0528248i
$750$	0	0
$751$	286.284i	0.381204i	0.981667	+	0.190602i	$0.0610440\pi$
$751$	286.284i	0.381204i	−0.981667	+	0.190602i	$0.938956\pi$
$752$	0	0
$753$	−784.470	−1.04179
$754$	0	0
$755$	267.316	0.354062
$756$	0	0
$757$	− 1162.42i	− 1.53556i	−0.640712	−	0.767781i	$-0.721361\pi$
$757$	− 1162.42i	− 1.53556i	0.640712	−	0.767781i	$-0.278639\pi$
$758$	0	0
$759$	− 252.180i	− 0.332253i
$760$	0	0
$761$	994.905	1.30737	0.653683	−	0.756769i	$-0.273223\pi$
$761$	994.905	1.30737	0.653683	+	0.756769i	$0.273223\pi$
$762$	0	0
$763$	−14.0720	−0.0184429
$764$	0	0
$765$	20.5093i	0.0268096i
$766$	0	0
$767$	− 337.293i	− 0.439756i
$768$	0	0
$769$	−614.473	−0.799055	−0.399527	−	0.916721i	$-0.630826\pi$
$769$	−614.473	−0.799055	−0.399527	+	0.916721i	$0.630826\pi$
$770$	0	0
$771$	−600.727	−0.779153
$772$	0	0
$773$	− 927.633i	− 1.20004i	−0.799984	−	0.600021i	$-0.795159\pi$
$773$	− 927.633i	− 1.20004i	0.799984	−	0.600021i	$-0.204841\pi$
$774$	0	0
$775$	23.5635i	0.0304045i
$776$	0	0
$777$	32.2601	0.0415188
$778$	0	0
$779$	1898.86	2.43756
$780$	0	0
$781$	466.490i	0.597298i
$782$	0	0
$783$	170.300i	0.217497i
$784$	0	0
$785$	6.35031	0.00808957
$786$	0	0
$787$	781.920	0.993545	0.496772	−	0.867881i	$-0.334518\pi$
$787$	781.920	0.993545	0.496772	+	0.867881i	$0.334518\pi$
$788$	0	0
$789$	− 697.047i	− 0.883456i
$790$	0	0
$791$	62.0550i	0.0784513i
$792$	0	0
$793$	−1818.94	−2.29375
$794$	0	0
$795$	129.146	0.162447
$796$	0	0
$797$	− 1160.22i	− 1.45574i	−0.685718	−	0.727868i	$-0.740511\pi$
$797$	− 1160.22i	− 1.45574i	0.685718	−	0.727868i	$-0.259489\pi$
$798$	0	0
$799$	333.543i	0.417450i
$800$	0	0
$801$	63.1910	0.0788902
$802$	0	0
$803$	174.243	0.216991
$804$	0	0
$805$	42.5558i	0.0528644i
$806$	0	0
$807$	556.962i	0.690163i
$808$	0	0
$809$	−1512.26	−1.86930	−0.934651	−	0.355568i	$-0.884288\pi$
$809$	−1512.26	−1.86930	−0.934651	+	0.355568i	$0.884288\pi$
$810$	0	0
$811$	−1586.92	−1.95674	−0.978371	−	0.206858i	$-0.933676\pi$
$811$	−1586.92	−1.95674	−0.978371	+	0.206858i	$0.933676\pi$
$812$	0	0
$813$	− 791.378i	− 0.973405i
$814$	0	0
$815$	− 81.2441i	− 0.0996860i
$816$	0	0
$817$	−876.942	−1.07337
$818$	0	0
$819$	−61.0493	−0.0745412
$820$	0	0
$821$	1118.96i	1.36292i	0.731857	+	0.681459i	$0.238654\pi$
$821$	1118.96i	1.36292i	−0.731857	+	0.681459i	$0.761346\pi$
$822$	0	0
$823$	− 1628.26i	− 1.97844i	−0.146438	−	0.989220i	$-0.546781\pi$
$823$	− 1628.26i	− 1.97844i	0.146438	−	0.989220i	$-0.453219\pi$
$824$	0	0
$825$	−232.366	−0.281656
$826$	0	0
$827$	421.552	0.509736	0.254868	−	0.966976i	$-0.417968\pi$
$827$	421.552	0.509736	0.254868	+	0.966976i	$0.417968\pi$
$828$	0	0
$829$	− 475.263i	− 0.573297i	−0.958036	−	0.286649i	$-0.907459\pi$
$829$	− 475.263i	− 0.573297i	0.958036	−	0.286649i	$-0.0925412\pi$
$830$	0	0
$831$	− 570.786i	− 0.686866i
$832$	0	0
$833$	−237.801	−0.285475
$834$	0	0
$835$	−285.161	−0.341510
$836$	0	0
$837$	5.29157i	0.00632207i
$838$	0	0
$839$	653.590i	0.779010i	0.921024	+	0.389505i	$0.127354\pi$
$839$	653.590i	0.779010i	−0.921024	+	0.389505i	$0.872646\pi$
$840$	0	0
$841$	−233.154	−0.277234
$842$	0	0
$843$	303.221	0.359692
$844$	0	0
$845$	135.618i	0.160495i
$846$	0	0
$847$	− 108.541i	− 0.128148i
$848$	0	0
$849$	260.323	0.306623
$850$	0	0
$851$	376.544	0.442472
$852$	0	0
$853$	− 140.493i	− 0.164705i	−0.996603	−	0.0823523i	$-0.973757\pi$
$853$	− 140.493i	− 0.164705i	0.996603	−	0.0823523i	$-0.0262433\pi$
$854$	0	0
$855$	− 107.168i	− 0.125343i
$856$	0	0
$857$	562.796	0.656704	0.328352	−	0.944555i	$-0.393507\pi$
$857$	562.796	0.656704	0.328352	+	0.944555i	$0.393507\pi$
$858$	0	0
$859$	228.316	0.265792	0.132896	−	0.991130i	$-0.457572\pi$
$859$	228.316	0.265792	0.132896	+	0.991130i	$0.457572\pi$
$860$	0	0
$861$	156.024i	0.181212i
$862$	0	0
$863$	892.187i	1.03382i	0.856040	+	0.516910i	$0.172918\pi$
$863$	892.187i	1.03382i	−0.856040	+	0.516910i	$0.827082\pi$
$864$	0	0
$865$	−132.017	−0.152621
$866$	0	0
$867$	−457.073	−0.527189
$868$	0	0
$869$	− 469.310i	− 0.540058i
$870$	0	0
$871$	998.611i	1.14651i
$872$	0	0
$873$	481.783	0.551871
$874$	0	0
$875$	81.5787	0.0932328
$876$	0	0
$877$	− 1406.66i	− 1.60395i	−0.597359	−	0.801974i	$-0.703783\pi$
$877$	− 1406.66i	− 1.60395i	0.597359	−	0.801974i	$-0.296217\pi$
$878$	0	0
$879$	− 277.882i	− 0.316135i
$880$	0	0
$881$	943.043	1.07042	0.535212	−	0.844718i	$-0.320232\pi$
$881$	943.043	1.07042	0.535212	+	0.844718i	$0.320232\pi$
$882$	0	0
$883$	−1146.63	−1.29856	−0.649280	−	0.760549i	$-0.724930\pi$
$883$	−1146.63	−1.29856	−0.649280	+	0.760549i	$0.724930\pi$
$884$	0	0
$885$	48.6511i	0.0549729i
$886$	0	0
$887$	894.171i	1.00808i	0.863679	+	0.504042i	$0.168154\pi$
$887$	894.171i	1.00808i	−0.863679	+	0.504042i	$0.831846\pi$
$888$	0	0
$889$	−258.655	−0.290950
$890$	0	0
$891$	−52.1816	−0.0585652
$892$	0	0
$893$	− 1742.88i	− 1.95171i
$894$	0	0
$895$	67.5395i	0.0754631i
$896$	0	0
$897$	−712.574	−0.794397
$898$	0	0
$899$	−33.3761	−0.0371258
$900$	0	0
$901$	− 273.850i	− 0.303940i
$902$	0	0
$903$	− 72.0559i	− 0.0797961i
$904$	0	0
$905$	−192.939	−0.213192
$906$	0	0
$907$	1358.60	1.49790	0.748950	−	0.662626i	$-0.230558\pi$
$907$	1358.60	1.49790	0.748950	+	0.662626i	$0.230558\pi$
$908$	0	0
$909$	228.804i	0.251710i
$910$	0	0
$911$	804.510i	0.883106i	0.897235	+	0.441553i	$0.145572\pi$
$911$	804.510i	0.883106i	−0.897235	+	0.441553i	$0.854428\pi$
$912$	0	0
$913$	660.726	0.723686
$914$	0	0
$915$	262.364	0.286736
$916$	0	0
$917$	− 273.950i	− 0.298745i
$918$	0	0
$919$	− 1704.73i	− 1.85498i	−0.373849	−	0.927490i	$-0.621962\pi$
$919$	− 1704.73i	− 1.85498i	0.373849	−	0.927490i	$-0.378038\pi$
$920$	0	0
$921$	−291.192	−0.316169
$922$	0	0
$923$	1318.14	1.42810
$924$	0	0
$925$	− 346.958i	− 0.375090i
$926$	0	0
$927$	− 548.290i	− 0.591467i
$928$	0	0
$929$	−1351.05	−1.45431	−0.727154	−	0.686475i	$-0.759157\pi$
$929$	−1351.05	−1.45431	−0.727154	+	0.686475i	$0.759157\pi$
$930$	0	0
$931$	1242.59	1.33469
$932$	0	0
$933$	814.715i	0.873220i
$934$	0	0
$935$	− 39.6374i	− 0.0423930i
$936$	0	0
$937$	−672.646	−0.717872	−0.358936	−	0.933362i	$-0.616860\pi$
$937$	−672.646	−0.717872	−0.358936	+	0.933362i	$0.616860\pi$
$938$	0	0
$939$	−33.6672	−0.0358544
$940$	0	0
$941$	528.671i	0.561818i	0.959734	+	0.280909i	$0.0906359\pi$
$941$	528.671i	0.561818i	−0.959734	+	0.280909i	$0.909364\pi$
$942$	0	0
$943$	1821.13i	1.93121i
$944$	0	0
$945$	8.80573	0.00931823
$946$	0	0
$947$	−661.066	−0.698063	−0.349032	−	0.937111i	$-0.613489\pi$
$947$	−661.066	−0.698063	−0.349032	+	0.937111i	$0.613489\pi$
$948$	0	0
$949$	− 492.351i	− 0.518811i
$950$	0	0
$951$	− 419.494i	− 0.441108i
$952$	0	0
$953$	−1545.41	−1.62163	−0.810815	−	0.585303i	$-0.800976\pi$
$953$	−1545.41	−1.62163	−0.810815	+	0.585303i	$0.800976\pi$
$954$	0	0
$955$	−159.618	−0.167139
$956$	0	0
$957$	− 329.131i	− 0.343920i
$958$	0	0
$959$	− 18.9627i	− 0.0197735i
$960$	0	0
$961$	959.963	0.998921
$962$	0	0
$963$	−95.5598	−0.0992313
$964$	0	0
$965$	123.793i	0.128283i
$966$	0	0
$967$	161.279i	0.166782i	0.996517	+	0.0833912i	$0.0265751\pi$
$967$	161.279i	0.166782i	−0.996517	+	0.0833912i	$0.973425\pi$
$968$	0	0
$969$	−227.248	−0.234518
$970$	0	0
$971$	4.20412	0.00432969	0.00216484	−	0.999998i	$-0.499311\pi$
$971$	4.20412	0.00432969	0.00216484	+	0.999998i	$0.499311\pi$
$972$	0	0
$973$	107.707i	0.110696i
$974$	0	0
$975$	656.586i	0.673422i
$976$	0	0
$977$	−1348.19	−1.37993	−0.689967	−	0.723841i	$-0.742375\pi$
$977$	−1348.19	−1.37993	−0.689967	+	0.723841i	$0.742375\pi$
$978$	0	0
$979$	−122.126	−0.124746
$980$	0	0
$981$	33.9868i	0.0346451i
$982$	0	0
$983$	− 984.262i	− 1.00128i	−0.865655	−	0.500642i	$-0.833097\pi$
$983$	− 984.262i	− 1.00128i	0.865655	−	0.500642i	$-0.166903\pi$
$984$	0	0
$985$	−518.497	−0.526393
$986$	0	0
$987$	143.207	0.145094
$988$	0	0
$989$	− 841.045i	− 0.850399i
$990$	0	0
$991$	− 1013.87i	− 1.02308i	−0.859259	−	0.511541i	$-0.829075\pi$
$991$	− 1013.87i	− 1.02308i	0.859259	−	0.511541i	$-0.170925\pi$
$992$	0	0
$993$	763.750	0.769134
$994$	0	0
$995$	−105.948	−0.106481
$996$	0	0
$997$	− 311.310i	− 0.312246i	−0.987738	−	0.156123i	$-0.950100\pi$
$997$	− 311.310i	− 0.312246i	0.987738	−	0.156123i	$-0.0498997\pi$
$998$	0	0
$999$	− 77.9151i	− 0.0779931i

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -1.36433i q^{5} +1.24213i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} -1.36433i q^{5} +1.24213i q^{7} +3.00000 q^{9} -5.79796 q^{11} +16.3830i q^{13} -2.36308i q^{15} -5.01086 q^{17} +26.1835 q^{19} +2.15142i q^{21} +25.1117i q^{23} +23.1386 q^{25} +5.19615 q^{27} +32.7743i q^{29} +1.01836i q^{31} -10.0424 q^{33} +1.69466 q^{35} -14.9948i q^{37} +28.3762i q^{39} +72.5212 q^{41} -33.4922 q^{43} -4.09298i q^{45} -66.5640i q^{47} +47.4571 q^{49} -8.67906 q^{51} +54.6513i q^{53} +7.91030i q^{55} +45.3511 q^{57} -20.5880 q^{59} +111.026i q^{61} +3.72638i q^{63} +22.3518 q^{65} +60.9540 q^{67} +43.4947i q^{69} -80.4576i q^{71} -30.0525 q^{73} +40.0773 q^{75} -7.20179i q^{77} +80.9441i q^{79} +9.00000 q^{81} -113.958 q^{83} +6.83644i q^{85} +56.7667i q^{87} +21.0637 q^{89} -20.3498 q^{91} +1.76386i q^{93} -35.7228i q^{95} +160.594 q^{97} -17.3939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 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

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Modular forms

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Database

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