LMFDB - Embedded newform 1728.3.e.v.1025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1025,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1728.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.e (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2441150464.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196 $ x^8 - 14x^6 + 77x^4 - 188*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$1.52833 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.1025
Dual form			1728.3.e.v.1025.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.37942i q^{5} +1.26611 q^{7} +O(q^{10})$	$q-7.37942i q^{5} +1.26611 q^{7} -5.82843i q^{11} +10.4853 q^{13} +18.3399i q^{17} +20.8722 q^{19} -20.4853i q^{23} -29.4558 q^{25} -11.1777i q^{29} +61.3503 q^{31} -9.34315i q^{35} +38.4264 q^{37} -33.0988i q^{41} +49.3410 q^{43} +21.5980i q^{47} -47.3970 q^{49} +77.5925i q^{53} -43.0104 q^{55} -25.7157i q^{59} +55.8823 q^{61} -77.3753i q^{65} -91.0853 q^{67} -114.770i q^{71} -120.338 q^{73} -7.37942i q^{77} -8.21107 q^{79} -150.338i q^{83} +135.338 q^{85} +118.505i q^{89} +13.2755 q^{91} -154.024i q^{95} -72.8823 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{13} - 32 q^{25} - 32 q^{37} + 96 q^{49} - 96 q^{61} - 216 q^{73} + 336 q^{85} - 40 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 - 32 * q^25 - 32 * q^37 + 96 * q^49 - 96 * q^61 - 216 * q^73 + 336 * q^85 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 7.37942i	− 1.47588i	−0.674864	−	0.737942i	$-0.735798\pi$
$5$	− 7.37942i	− 1.47588i	0.674864	−	0.737942i	$-0.264202\pi$
$6$	0	0
$7$	1.26611	0.180873	0.0904363	−	0.995902i	$-0.471174\pi$
$7$	1.26611	0.180873	0.0904363	+	0.995902i	$0.471174\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.82843i	− 0.529857i	−0.964268	−	0.264929i	$-0.914652\pi$
$11$	− 5.82843i	− 0.529857i	0.964268	−	0.264929i	$-0.0853484\pi$
$12$	0	0
$13$	10.4853	0.806560	0.403280	−	0.915077i	$-0.367870\pi$
$13$	10.4853	0.806560	0.403280	+	0.915077i	$0.367870\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.3399i	1.07882i	0.842043	+	0.539410i	$0.181353\pi$
$17$	18.3399i	1.07882i	−0.842043	+	0.539410i	$0.818647\pi$
$18$	0	0
$19$	20.8722	1.09853	0.549267	−	0.835647i	$-0.314907\pi$
$19$	20.8722	1.09853	0.549267	+	0.835647i	$0.314907\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 20.4853i	− 0.890664i	−0.895365	−	0.445332i	$-0.853086\pi$
$23$	− 20.4853i	− 0.890664i	0.895365	−	0.445332i	$-0.146914\pi$
$24$	0	0
$25$	−29.4558	−1.17823
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 11.1777i	− 0.385439i	−0.981254	−	0.192720i	$-0.938269\pi$
$29$	− 11.1777i	− 0.385439i	0.981254	−	0.192720i	$-0.0617308\pi$
$30$	0	0
$31$	61.3503	1.97904	0.989522	−	0.144384i	$-0.0461200\pi$
$31$	61.3503	1.97904	0.989522	+	0.144384i	$0.0461200\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 9.34315i	− 0.266947i
$36$	0	0
$37$	38.4264	1.03855	0.519276	−	0.854607i	$-0.326202\pi$
$37$	38.4264	1.03855	0.519276	+	0.854607i	$0.326202\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 33.0988i	− 0.807287i	−0.914916	−	0.403644i	$-0.867744\pi$
$41$	− 33.0988i	− 0.807287i	0.914916	−	0.403644i	$-0.132256\pi$
$42$	0	0
$43$	49.3410	1.14746	0.573732	−	0.819043i	$-0.305495\pi$
$43$	49.3410	1.14746	0.573732	+	0.819043i	$0.305495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	21.5980i	0.459531i	0.973246	+	0.229766i	$0.0737960\pi$
$47$	21.5980i	0.459531i	−0.973246	+	0.229766i	$0.926204\pi$
$48$	0	0
$49$	−47.3970	−0.967285
$50$	0	0
$51$	0	0
$52$	0	0
$53$	77.5925i	1.46401i	0.681299	+	0.732005i	$0.261415\pi$
$53$	77.5925i	1.46401i	−0.681299	+	0.732005i	$0.738585\pi$
$54$	0	0
$55$	−43.0104	−0.782008
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 25.7157i	− 0.435860i	−0.975964	−	0.217930i	$-0.930070\pi$
$59$	− 25.7157i	− 0.435860i	0.975964	−	0.217930i	$-0.0699304\pi$
$60$	0	0
$61$	55.8823	0.916102	0.458051	−	0.888926i	$-0.348548\pi$
$61$	55.8823	0.916102	0.458051	+	0.888926i	$0.348548\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 77.3753i	− 1.19039i
$66$	0	0
$67$	−91.0853	−1.35948	−0.679741	−	0.733452i	$-0.737908\pi$
$67$	−91.0853	−1.35948	−0.679741	+	0.733452i	$0.737908\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 114.770i	− 1.61647i	−0.588858	−	0.808236i	$-0.700422\pi$
$71$	− 114.770i	− 1.61647i	0.588858	−	0.808236i	$-0.299578\pi$
$72$	0	0
$73$	−120.338	−1.64847	−0.824234	−	0.566250i	$-0.808394\pi$
$73$	−120.338	−1.64847	−0.824234	+	0.566250i	$0.808394\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7.37942i	− 0.0958366i
$78$	0	0
$79$	−8.21107	−0.103938	−0.0519688	−	0.998649i	$-0.516550\pi$
$79$	−8.21107	−0.103938	−0.0519688	+	0.998649i	$0.516550\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 150.338i	− 1.81130i	−0.424024	−	0.905651i	$-0.639383\pi$
$83$	− 150.338i	− 1.81130i	0.424024	−	0.905651i	$-0.360617\pi$
$84$	0	0
$85$	135.338	1.59221
$86$	0	0
$87$	0	0
$88$	0	0
$89$	118.505i	1.33152i	0.746166	+	0.665759i	$0.231892\pi$
$89$	118.505i	1.33152i	−0.746166	+	0.665759i	$0.768108\pi$
$90$	0	0
$91$	13.2755	0.145885
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 154.024i	− 1.62131i
$96$	0	0
$97$	−72.8823	−0.751363	−0.375682	−	0.926749i	$-0.622591\pi$
$97$	−72.8823	−0.751363	−0.375682	+	0.926749i	$0.622591\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 113.838i	− 1.12711i	−0.826079	−	0.563554i	$-0.809434\pi$
$101$	− 113.838i	− 1.12711i	0.826079	−	0.563554i	$-0.190566\pi$
$102$	0	0
$103$	−158.766	−1.54142	−0.770709	−	0.637187i	$-0.780098\pi$
$103$	−158.766	−1.54142	−0.770709	+	0.637187i	$0.780098\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 77.7452i	− 0.726590i	−0.931674	−	0.363295i	$-0.881652\pi$
$107$	− 77.7452i	− 0.726590i	0.931674	−	0.363295i	$-0.118348\pi$
$108$	0	0
$109$	15.3970	0.141257	0.0706283	−	0.997503i	$-0.477500\pi$
$109$	15.3970	0.141257	0.0706283	+	0.997503i	$0.477500\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	76.9408i	0.680892i	0.940264	+	0.340446i	$0.110578\pi$
$113$	76.9408i	0.680892i	−0.940264	+	0.340446i	$0.889422\pi$
$114$	0	0
$115$	−151.170	−1.31452
$116$	0	0
$117$	0	0
$118$	0	0
$119$	23.2203i	0.195129i
$120$	0	0
$121$	87.0294	0.719252
$122$	0	0
$123$	0	0
$124$	0	0
$125$	32.8815i	0.263052i
$126$	0	0
$127$	72.0936	0.567666	0.283833	−	0.958874i	$-0.408394\pi$
$127$	72.0936	0.567666	0.283833	+	0.958874i	$0.408394\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	84.4214i	0.644438i	0.946665	+	0.322219i	$0.104429\pi$
$131$	84.4214i	0.644438i	−0.946665	+	0.322219i	$0.895571\pi$
$132$	0	0
$133$	26.4264	0.198695
$134$	0	0
$135$	0	0
$136$	0	0
$137$	29.0832i	0.212286i	0.994351	+	0.106143i	$0.0338502\pi$
$137$	29.0832i	0.212286i	−0.994351	+	0.106143i	$0.966150\pi$
$138$	0	0
$139$	130.297	0.937391	0.468695	−	0.883360i	$-0.344724\pi$
$139$	130.297	0.937391	0.468695	+	0.883360i	$0.344724\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 61.1127i	− 0.427362i
$144$	0	0
$145$	−82.4853	−0.568864
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 92.3514i	− 0.619808i	−0.950768	−	0.309904i	$-0.899703\pi$
$149$	− 92.3514i	− 0.619808i	0.950768	−	0.309904i	$-0.100297\pi$
$150$	0	0
$151$	−44.9282	−0.297538	−0.148769	−	0.988872i	$-0.547531\pi$
$151$	−44.9282	−0.297538	−0.148769	+	0.988872i	$0.547531\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 452.730i	− 2.92084i
$156$	0	0
$157$	131.029	0.834582	0.417291	−	0.908773i	$-0.362980\pi$
$157$	131.029	0.834582	0.417291	+	0.908773i	$0.362980\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 25.9366i	− 0.161097i
$162$	0	0
$163$	−258.677	−1.58697	−0.793487	−	0.608587i	$-0.791737\pi$
$163$	−258.677	−1.58697	−0.793487	+	0.608587i	$0.791737\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 304.108i	− 1.82100i	−0.413505	−	0.910502i	$-0.635696\pi$
$167$	− 304.108i	− 1.82100i	0.413505	−	0.910502i	$-0.364304\pi$
$168$	0	0
$169$	−59.0589	−0.349461
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 0.651690i	− 0.00376699i	−0.999998	−	0.00188350i	$-0.999400\pi$
$173$	− 0.651690i	− 0.00376699i	0.999998	−	0.00188350i	$-0.000599536\pi$
$174$	0	0
$175$	−37.2943	−0.213110
$176$	0	0
$177$	0	0
$178$	0	0
$179$	126.765i	0.708182i	0.935211	+	0.354091i	$0.115210\pi$
$179$	126.765i	0.708182i	−0.935211	+	0.354091i	$0.884790\pi$
$180$	0	0
$181$	−304.735	−1.68362	−0.841810	−	0.539775i	$-0.818509\pi$
$181$	−304.735	−1.68362	−0.841810	+	0.539775i	$0.818509\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 283.565i	− 1.53278i
$186$	0	0
$187$	106.893	0.571620
$188$	0	0
$189$	0	0
$190$	0	0
$191$	134.132i	0.702262i	0.936326	+	0.351131i	$0.114203\pi$
$191$	134.132i	0.702262i	−0.936326	+	0.351131i	$0.885797\pi$
$192$	0	0
$193$	182.397	0.945062	0.472531	−	0.881314i	$-0.343340\pi$
$193$	182.397	0.945062	0.472531	+	0.881314i	$0.343340\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	154.533i	0.784433i	0.919873	+	0.392217i	$0.128292\pi$
$197$	154.533i	0.784433i	−0.919873	+	0.392217i	$0.871708\pi$
$198$	0	0
$199$	36.7171	0.184508	0.0922541	−	0.995735i	$-0.470593\pi$
$199$	36.7171	0.184508	0.0922541	+	0.995735i	$0.470593\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 14.1522i	− 0.0697155i
$204$	0	0
$205$	−244.250	−1.19146
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 121.652i	− 0.582066i
$210$	0	0
$211$	359.891	1.70564	0.852822	−	0.522201i	$-0.174889\pi$
$211$	359.891	1.70564	0.852822	+	0.522201i	$0.174889\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 364.108i	− 1.69352i
$216$	0	0
$217$	77.6762	0.357955
$218$	0	0
$219$	0	0
$220$	0	0
$221$	192.299i	0.870133i
$222$	0	0
$223$	273.870	1.22812	0.614059	−	0.789260i	$-0.289536\pi$
$223$	273.870	1.22812	0.614059	+	0.789260i	$0.289536\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 234.500i	− 1.03304i	−0.856276	−	0.516519i	$-0.827228\pi$
$227$	− 234.500i	− 1.03304i	0.856276	−	0.516519i	$-0.172772\pi$
$228$	0	0
$229$	−25.6468	−0.111995	−0.0559973	−	0.998431i	$-0.517834\pi$
$229$	−25.6468	−0.111995	−0.0559973	+	0.998431i	$0.517834\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.30338i	− 0.00559390i	−0.999996	−	0.00279695i	$-0.999110\pi$
$233$	− 1.30338i	− 0.00559390i	0.999996	−	0.00279695i	$-0.000890298\pi$
$234$	0	0
$235$	159.381	0.678215
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 75.1615i	− 0.314483i	−0.987560	−	0.157242i	$-0.949740\pi$
$239$	− 75.1615i	− 0.314483i	0.987560	−	0.157242i	$-0.0502601\pi$
$240$	0	0
$241$	252.558	1.04796	0.523980	−	0.851730i	$-0.324447\pi$
$241$	252.558	1.04796	0.523980	+	0.851730i	$0.324447\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	349.762i	1.42760i
$246$	0	0
$247$	218.850	0.886034
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 246.000i	− 0.980080i	−0.871700	−	0.490040i	$-0.836982\pi$
$251$	− 246.000i	− 0.980080i	0.871700	−	0.490040i	$-0.163018\pi$
$252$	0	0
$253$	−119.397	−0.471925
$254$	0	0
$255$	0	0
$256$	0	0
$257$	359.097i	1.39726i	0.715482	+	0.698632i	$0.246207\pi$
$257$	359.097i	1.39726i	−0.715482	+	0.698632i	$0.753793\pi$
$258$	0	0
$259$	48.6520	0.187846
$260$	0	0
$261$	0	0
$262$	0	0
$263$	215.897i	0.820899i	0.911883	+	0.410450i	$0.134628\pi$
$263$	215.897i	0.820899i	−0.911883	+	0.410450i	$0.865372\pi$
$264$	0	0
$265$	572.588	2.16071
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 488.885i	− 1.81742i	−0.417432	−	0.908708i	$-0.637070\pi$
$269$	− 488.885i	− 1.81742i	0.417432	−	0.908708i	$-0.362930\pi$
$270$	0	0
$271$	161.261	0.595059	0.297530	−	0.954713i	$-0.403837\pi$
$271$	161.261	0.595059	0.297530	+	0.954713i	$0.403837\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	171.681i	0.624295i
$276$	0	0
$277$	−248.617	−0.897535	−0.448768	−	0.893648i	$-0.648137\pi$
$277$	−248.617	−0.897535	−0.448768	+	0.893648i	$0.648137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	329.579i	1.17288i	0.809993	+	0.586439i	$0.199471\pi$
$281$	329.579i	1.17288i	−0.809993	+	0.586439i	$0.800529\pi$
$282$	0	0
$283$	−262.438	−0.927343	−0.463671	−	0.886007i	$-0.653468\pi$
$283$	−262.438	−0.927343	−0.463671	+	0.886007i	$0.653468\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 41.9066i	− 0.146016i
$288$	0	0
$289$	−47.3532	−0.163852
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 335.872i	− 1.14632i	−0.819443	−	0.573161i	$-0.805717\pi$
$293$	− 335.872i	− 1.14632i	0.819443	−	0.573161i	$-0.194283\pi$
$294$	0	0
$295$	−189.767	−0.643279
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 214.794i	− 0.718374i
$300$	0	0
$301$	62.4710	0.207545
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 412.379i	− 1.35206i
$306$	0	0
$307$	580.045	1.88940	0.944698	−	0.327941i	$-0.106355\pi$
$307$	580.045	1.88940	0.944698	+	0.327941i	$0.106355\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 452.132i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$311$	− 452.132i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$312$	0	0
$313$	−126.632	−0.404577	−0.202288	−	0.979326i	$-0.564838\pi$
$313$	−126.632	−0.404577	−0.202288	+	0.979326i	$0.564838\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 237.767i	− 0.750055i	−0.927014	−	0.375028i	$-0.877633\pi$
$317$	− 237.767i	− 0.750055i	0.927014	−	0.375028i	$-0.122367\pi$
$318$	0	0
$319$	−65.1487	−0.204228
$320$	0	0
$321$	0	0
$322$	0	0
$323$	382.794i	1.18512i
$324$	0	0
$325$	−308.853	−0.950316
$326$	0	0
$327$	0	0
$328$	0	0
$329$	27.3454i	0.0831167i
$330$	0	0
$331$	−222.611	−0.672542	−0.336271	−	0.941765i	$-0.609166\pi$
$331$	−222.611	−0.672542	−0.336271	+	0.941765i	$0.609166\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	672.156i	2.00644i
$336$	0	0
$337$	396.794	1.17743	0.588715	−	0.808341i	$-0.299634\pi$
$337$	396.794	1.17743	0.588715	+	0.808341i	$0.299634\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 357.576i	− 1.04861i
$342$	0	0
$343$	−122.049	−0.355828
$344$	0	0
$345$	0	0
$346$	0	0
$347$	218.387i	0.629357i	0.949198	+	0.314678i	$0.101897\pi$
$347$	218.387i	0.629357i	−0.949198	+	0.314678i	$0.898103\pi$
$348$	0	0
$349$	−499.161	−1.43026	−0.715131	−	0.698990i	$-0.753633\pi$
$349$	−499.161	−1.43026	−0.715131	+	0.698990i	$0.753633\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	423.991i	1.20111i	0.799585	+	0.600554i	$0.205053\pi$
$353$	423.991i	1.20111i	−0.799585	+	0.600554i	$0.794947\pi$
$354$	0	0
$355$	−846.933	−2.38573
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.2254i	0.0897643i	0.998992	+	0.0448822i	$0.0142912\pi$
$359$	32.2254i	0.0897643i	−0.998992	+	0.0448822i	$0.985709\pi$
$360$	0	0
$361$	74.6468	0.206778
$362$	0	0
$363$	0	0
$364$	0	0
$365$	888.025i	2.43295i
$366$	0	0
$367$	−449.710	−1.22537	−0.612684	−	0.790328i	$-0.709910\pi$
$367$	−449.710	−1.22537	−0.612684	+	0.790328i	$0.709910\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	98.2405i	0.264799i
$372$	0	0
$373$	386.368	1.03584	0.517919	−	0.855430i	$-0.326707\pi$
$373$	386.368	1.03584	0.517919	+	0.855430i	$0.326707\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 117.202i	− 0.310880i
$378$	0	0
$379$	−508.603	−1.34196	−0.670980	−	0.741475i	$-0.734126\pi$
$379$	−508.603	−1.34196	−0.670980	+	0.741475i	$0.734126\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	308.142i	0.804549i	0.915519	+	0.402274i	$0.131780\pi$
$383$	308.142i	0.804549i	−0.915519	+	0.402274i	$0.868220\pi$
$384$	0	0
$385$	−54.4558	−0.141444
$386$	0	0
$387$	0	0
$388$	0	0
$389$	356.707i	0.916985i	0.888698	+	0.458492i	$0.151610\pi$
$389$	356.707i	0.916985i	−0.888698	+	0.458492i	$0.848390\pi$
$390$	0	0
$391$	375.699	0.960866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	60.5929i	0.153400i
$396$	0	0
$397$	−524.191	−1.32038	−0.660190	−	0.751099i	$-0.729524\pi$
$397$	−524.191	−1.32038	−0.660190	+	0.751099i	$0.729524\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	236.141i	0.588881i	0.955670	+	0.294441i	$0.0951333\pi$
$401$	236.141i	0.588881i	−0.955670	+	0.294441i	$0.904867\pi$
$402$	0	0
$403$	643.276	1.59622
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 223.966i	− 0.550284i
$408$	0	0
$409$	261.235	0.638718	0.319359	−	0.947634i	$-0.396532\pi$
$409$	261.235	0.638718	0.319359	+	0.947634i	$0.396532\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 32.5589i	− 0.0788351i
$414$	0	0
$415$	−1109.41	−2.67327
$416$	0	0
$417$	0	0
$418$	0	0
$419$	392.745i	0.937339i	0.883374	+	0.468670i	$0.155267\pi$
$419$	392.745i	0.937339i	−0.883374	+	0.468670i	$0.844733\pi$
$420$	0	0
$421$	266.794	0.633715	0.316857	−	0.948473i	$-0.397372\pi$
$421$	266.794	0.633715	0.316857	+	0.948473i	$0.397372\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 540.218i	− 1.27110i
$426$	0	0
$427$	70.7530	0.165698
$428$	0	0
$429$	0	0
$430$	0	0
$431$	601.029i	1.39450i	0.716828	+	0.697250i	$0.245593\pi$
$431$	601.029i	1.39450i	−0.716828	+	0.697250i	$0.754407\pi$
$432$	0	0
$433$	−605.500	−1.39838	−0.699191	−	0.714935i	$-0.746456\pi$
$433$	−605.500	−1.39838	−0.699191	+	0.714935i	$0.746456\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 427.572i	− 0.978425i
$438$	0	0
$439$	333.992	0.760801	0.380401	−	0.924822i	$-0.375786\pi$
$439$	333.992	0.760801	0.380401	+	0.924822i	$0.375786\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 203.823i	− 0.460098i	−0.973179	−	0.230049i	$-0.926111\pi$
$443$	− 203.823i	− 0.460098i	0.973179	−	0.230049i	$-0.0738886\pi$
$444$	0	0
$445$	874.500	1.96517
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 733.492i	− 1.63361i	−0.576912	−	0.816806i	$-0.695742\pi$
$449$	− 733.492i	− 1.63361i	0.576912	−	0.816806i	$-0.304258\pi$
$450$	0	0
$451$	−192.914	−0.427747
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 97.9655i	− 0.215309i
$456$	0	0
$457$	−514.765	−1.12640	−0.563200	−	0.826321i	$-0.690430\pi$
$457$	−514.765	−1.12640	−0.563200	+	0.826321i	$0.690430\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	232.778i	0.504941i	0.967605	+	0.252470i	$0.0812430\pi$
$461$	232.778i	0.504941i	−0.967605	+	0.252470i	$0.918757\pi$
$462$	0	0
$463$	−72.7826	−0.157198	−0.0785989	−	0.996906i	$-0.525045\pi$
$463$	−72.7826	−0.157198	−0.0785989	+	0.996906i	$0.525045\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	304.643i	0.652340i	0.945311	+	0.326170i	$0.105758\pi$
$467$	304.643i	0.652340i	−0.945311	+	0.326170i	$0.894242\pi$
$468$	0	0
$469$	−115.324	−0.245893
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 287.580i	− 0.607992i
$474$	0	0
$475$	−614.807	−1.29433
$476$	0	0
$477$	0	0
$478$	0	0
$479$	310.118i	0.647427i	0.946155	+	0.323714i	$0.104931\pi$
$479$	310.118i	0.647427i	−0.946155	+	0.323714i	$0.895069\pi$
$480$	0	0
$481$	402.912	0.837654
$482$	0	0
$483$	0	0
$484$	0	0
$485$	537.829i	1.10893i
$486$	0	0
$487$	744.415	1.52857	0.764287	−	0.644877i	$-0.223091\pi$
$487$	744.415	1.52857	0.764287	+	0.644877i	$0.223091\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 480.754i	− 0.979133i	−0.871966	−	0.489567i	$-0.837155\pi$
$491$	− 480.754i	− 0.979133i	0.871966	−	0.489567i	$-0.162845\pi$
$492$	0	0
$493$	204.999	0.415820
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 145.311i	− 0.292376i
$498$	0	0
$499$	694.534	1.39185	0.695926	−	0.718113i	$-0.254994\pi$
$499$	694.534	1.39185	0.695926	+	0.718113i	$0.254994\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 266.309i	− 0.529441i	−0.964325	−	0.264720i	$-0.914720\pi$
$503$	− 266.309i	− 0.529441i	0.964325	−	0.264720i	$-0.0852796\pi$
$504$	0	0
$505$	−840.058	−1.66348
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 205.538i	− 0.403807i	−0.979405	−	0.201903i	$-0.935287\pi$
$509$	− 205.538i	− 0.403807i	0.979405	−	0.201903i	$-0.0647127\pi$
$510$	0	0
$511$	−152.361	−0.298163
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1171.60i	2.27496i
$516$	0	0
$517$	125.882	0.243486
$518$	0	0
$519$	0	0
$520$	0	0
$521$	940.550i	1.80528i	0.430398	+	0.902639i	$0.358373\pi$
$521$	940.550i	1.80528i	−0.430398	+	0.902639i	$0.641627\pi$
$522$	0	0
$523$	94.3064	0.180318	0.0901591	−	0.995927i	$-0.471262\pi$
$523$	94.3064	0.180318	0.0901591	+	0.995927i	$0.471262\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1125.16i	2.13503i
$528$	0	0
$529$	109.353	0.206717
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 347.050i	− 0.651126i
$534$	0	0
$535$	−573.714	−1.07236
$536$	0	0
$537$	0	0
$538$	0	0
$539$	276.250i	0.512523i
$540$	0	0
$541$	35.7645	0.0661081	0.0330541	−	0.999454i	$-0.489477\pi$
$541$	35.7645	0.0661081	0.0330541	+	0.999454i	$0.489477\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 113.621i	− 0.208478i
$546$	0	0
$547$	−443.305	−0.810430	−0.405215	−	0.914221i	$-0.632803\pi$
$547$	−443.305	−0.810430	−0.405215	+	0.914221i	$0.632803\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 233.304i	− 0.423419i
$552$	0	0
$553$	−10.3961	−0.0187995
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 438.098i	− 0.786531i	−0.919425	−	0.393266i	$-0.871345\pi$
$557$	− 438.098i	− 0.786531i	0.919425	−	0.393266i	$-0.128655\pi$
$558$	0	0
$559$	517.354	0.925499
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1065.07i	1.89178i	0.324485	+	0.945891i	$0.394809\pi$
$563$	1065.07i	1.89178i	−0.324485	+	0.945891i	$0.605191\pi$
$564$	0	0
$565$	567.779	1.00492
$566$	0	0
$567$	0	0
$568$	0	0
$569$	939.141i	1.65051i	0.564759	+	0.825256i	$0.308969\pi$
$569$	939.141i	1.65051i	−0.564759	+	0.825256i	$0.691031\pi$
$570$	0	0
$571$	294.128	0.515110	0.257555	−	0.966264i	$-0.417083\pi$
$571$	294.128	0.515110	0.257555	+	0.966264i	$0.417083\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	603.411i	1.04941i
$576$	0	0
$577$	−546.500	−0.947140	−0.473570	−	0.880756i	$-0.657035\pi$
$577$	−546.500	−0.947140	−0.473570	+	0.880756i	$0.657035\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 190.344i	− 0.327615i
$582$	0	0
$583$	452.242	0.775716
$584$	0	0
$585$	0	0
$586$	0	0
$587$	43.7939i	0.0746064i	0.999304	+	0.0373032i	$0.0118767\pi$
$587$	43.7939i	0.0746064i	−0.999304	+	0.0373032i	$0.988123\pi$
$588$	0	0
$589$	1280.51	2.17405
$590$	0	0
$591$	0	0
$592$	0	0
$593$	823.454i	1.38862i	0.719674	+	0.694312i	$0.244291\pi$
$593$	823.454i	1.38862i	−0.719674	+	0.694312i	$0.755709\pi$
$594$	0	0
$595$	171.353	0.287988
$596$	0	0
$597$	0	0
$598$	0	0
$599$	826.244i	1.37937i	0.724109	+	0.689686i	$0.242251\pi$
$599$	826.244i	1.37937i	−0.724109	+	0.689686i	$0.757749\pi$
$600$	0	0
$601$	−378.632	−0.630004	−0.315002	−	0.949091i	$-0.602005\pi$
$601$	−378.632	−0.630004	−0.315002	+	0.949091i	$0.602005\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 642.227i	− 1.06153i
$606$	0	0
$607$	−139.737	−0.230210	−0.115105	−	0.993353i	$-0.536720\pi$
$607$	−139.737	−0.230210	−0.115105	+	0.993353i	$0.536720\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	226.461i	0.370640i
$612$	0	0
$613$	508.587	0.829669	0.414834	−	0.909897i	$-0.363839\pi$
$613$	508.587	0.829669	0.414834	+	0.909897i	$0.363839\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 429.415i	− 0.695973i	−0.937500	−	0.347986i	$-0.886866\pi$
$617$	− 429.415i	− 0.695973i	0.937500	−	0.347986i	$-0.113134\pi$
$618$	0	0
$619$	720.471	1.16393	0.581964	−	0.813215i	$-0.302285\pi$
$619$	720.471	1.16393	0.581964	+	0.813215i	$0.302285\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	150.040i	0.240835i
$624$	0	0
$625$	−493.749	−0.789999
$626$	0	0
$627$	0	0
$628$	0	0
$629$	704.738i	1.12041i
$630$	0	0
$631$	479.482	0.759877	0.379938	−	0.925012i	$-0.375945\pi$
$631$	479.482	0.759877	0.379938	+	0.925012i	$0.375945\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 532.009i	− 0.837810i
$636$	0	0
$637$	−496.971	−0.780174
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 562.679i	− 0.877815i	−0.898532	−	0.438907i	$-0.855366\pi$
$641$	− 562.679i	− 0.877815i	0.898532	−	0.438907i	$-0.144634\pi$
$642$	0	0
$643$	785.471	1.22157	0.610786	−	0.791796i	$-0.290854\pi$
$643$	785.471	1.22157	0.610786	+	0.791796i	$0.290854\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	571.529i	0.883352i	0.897175	+	0.441676i	$0.145616\pi$
$647$	571.529i	0.883352i	−0.897175	+	0.441676i	$0.854384\pi$
$648$	0	0
$649$	−149.882	−0.230943
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 325.346i	− 0.498233i	−0.968474	−	0.249117i	$-0.919860\pi$
$653$	− 325.346i	− 0.498233i	0.968474	−	0.249117i	$-0.0801402\pi$
$654$	0	0
$655$	622.981	0.951116
$656$	0	0
$657$	0	0
$658$	0	0
$659$	902.294i	1.36919i	0.728926	+	0.684593i	$0.240020\pi$
$659$	902.294i	1.36919i	−0.728926	+	0.684593i	$0.759980\pi$
$660$	0	0
$661$	59.6325	0.0902155	0.0451078	−	0.998982i	$-0.485637\pi$
$661$	59.6325	0.0902155	0.0451078	+	0.998982i	$0.485637\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 195.012i	− 0.293250i
$666$	0	0
$667$	−228.979	−0.343297
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 325.706i	− 0.485403i
$672$	0	0
$673$	−1005.17	−1.49357	−0.746787	−	0.665064i	$-0.768404\pi$
$673$	−1005.17	−1.49357	−0.746787	+	0.665064i	$0.768404\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	175.697i	0.259523i	0.991545	+	0.129762i	$0.0414212\pi$
$677$	175.697i	0.259523i	−0.991545	+	0.129762i	$0.958579\pi$
$678$	0	0
$679$	−92.2768	−0.135901
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 768.823i	− 1.12566i	−0.826574	−	0.562828i	$-0.809713\pi$
$683$	− 768.823i	− 1.12566i	0.826574	−	0.562828i	$-0.190287\pi$
$684$	0	0
$685$	214.617	0.313310
$686$	0	0
$687$	0	0
$688$	0	0
$689$	813.579i	1.18081i
$690$	0	0
$691$	590.714	0.854868	0.427434	−	0.904047i	$-0.359418\pi$
$691$	590.714	0.854868	0.427434	+	0.904047i	$0.359418\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 961.519i	− 1.38348i
$696$	0	0
$697$	607.029	0.870917
$698$	0	0
$699$	0	0
$700$	0	0
$701$	967.139i	1.37966i	0.723974	+	0.689828i	$0.242314\pi$
$701$	967.139i	1.37966i	−0.723974	+	0.689828i	$0.757686\pi$
$702$	0	0
$703$	802.042	1.14088
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 144.131i	− 0.203863i
$708$	0	0
$709$	−147.647	−0.208246	−0.104123	−	0.994564i	$-0.533204\pi$
$709$	−147.647	−0.208246	−0.104123	+	0.994564i	$0.533204\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1256.78i	− 1.76266i
$714$	0	0
$715$	−450.976	−0.630736
$716$	0	0
$717$	0	0
$718$	0	0
$719$	627.098i	0.872181i	0.899903	+	0.436091i	$0.143637\pi$
$719$	627.098i	0.872181i	−0.899903	+	0.436091i	$0.856363\pi$
$720$	0	0
$721$	−201.015	−0.278800
$722$	0	0
$723$	0	0
$724$	0	0
$725$	329.250i	0.454138i
$726$	0	0
$727$	−150.518	−0.207040	−0.103520	−	0.994627i	$-0.533011\pi$
$727$	−150.518	−0.207040	−0.103520	+	0.994627i	$0.533011\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	904.910i	1.23791i
$732$	0	0
$733$	430.073	0.586730	0.293365	−	0.956000i	$-0.405225\pi$
$733$	430.073	0.586730	0.293365	+	0.956000i	$0.405225\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	530.884i	0.720331i
$738$	0	0
$739$	−347.230	−0.469865	−0.234932	−	0.972012i	$-0.575487\pi$
$739$	−347.230	−0.469865	−0.234932	+	0.972012i	$0.575487\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 48.7797i	− 0.0656523i	−0.999461	−	0.0328261i	$-0.989549\pi$
$743$	− 48.7797i	− 0.0656523i	0.999461	−	0.0328261i	$-0.0104508\pi$
$744$	0	0
$745$	−681.500	−0.914765
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 98.4338i	− 0.131420i
$750$	0	0
$751$	−323.248	−0.430424	−0.215212	−	0.976567i	$-0.569044\pi$
$751$	−323.248	−0.430424	−0.215212	+	0.976567i	$0.569044\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	331.544i	0.439131i
$756$	0	0
$757$	−452.912	−0.598298	−0.299149	−	0.954206i	$-0.596703\pi$
$757$	−452.912	−0.598298	−0.299149	+	0.954206i	$0.596703\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 824.757i	− 1.08378i	−0.840449	−	0.541890i	$-0.817709\pi$
$761$	− 824.757i	− 1.08378i	0.840449	−	0.541890i	$-0.182291\pi$
$762$	0	0
$763$	19.4942	0.0255495
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 269.637i	− 0.351547i
$768$	0	0
$769$	248.955	0.323739	0.161870	−	0.986812i	$-0.448248\pi$
$769$	248.955	0.323739	0.161870	+	0.986812i	$0.448248\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 351.066i	− 0.454160i	−0.973876	−	0.227080i	$-0.927082\pi$
$773$	− 351.066i	− 0.454160i	0.973876	−	0.227080i	$-0.0729179\pi$
$774$	0	0
$775$	−1807.13	−2.33178
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 690.843i	− 0.886833i
$780$	0	0
$781$	−668.926	−0.856499
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 966.921i	− 1.23175i
$786$	0	0
$787$	−342.705	−0.435458	−0.217729	−	0.976009i	$-0.569865\pi$
$787$	−342.705	−0.435458	−0.217729	+	0.976009i	$0.569865\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	97.4154i	0.123155i
$792$	0	0
$793$	585.941	0.738892
$794$	0	0
$795$	0	0
$796$	0	0
$797$	485.087i	0.608641i	0.952570	+	0.304320i	$0.0984293\pi$
$797$	485.087i	0.608641i	−0.952570	+	0.304320i	$0.901571\pi$
$798$	0	0
$799$	−396.106	−0.495752
$800$	0	0
$801$	0	0
$802$	0	0
$803$	701.382i	0.873452i
$804$	0	0
$805$	−191.397	−0.237760
$806$	0	0
$807$	0	0
$808$	0	0
$809$	179.278i	0.221605i	0.993842	+	0.110802i	$0.0353421\pi$
$809$	179.278i	0.221605i	−0.993842	+	0.110802i	$0.964658\pi$
$810$	0	0
$811$	475.144	0.585874	0.292937	−	0.956132i	$-0.405367\pi$
$811$	475.144	0.585874	0.292937	+	0.956132i	$0.405367\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1908.89i	2.34219i
$816$	0	0
$817$	1029.85	1.26053
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 598.056i	− 0.728448i	−0.931311	−	0.364224i	$-0.881334\pi$
$821$	− 598.056i	− 0.728448i	0.931311	−	0.364224i	$-0.118666\pi$
$822$	0	0
$823$	433.288	0.526474	0.263237	−	0.964731i	$-0.415210\pi$
$823$	433.288	0.526474	0.263237	+	0.964731i	$0.415210\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	62.2153i	0.0752301i	0.999292	+	0.0376151i	$0.0119761\pi$
$827$	62.2153i	0.0752301i	−0.999292	+	0.0376151i	$0.988024\pi$
$828$	0	0
$829$	−3.70563	−0.00447000	−0.00223500	−	0.999998i	$-0.500711\pi$
$829$	−3.70563	−0.00447000	−0.00223500	+	0.999998i	$0.500711\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 869.257i	− 1.04353i
$834$	0	0
$835$	−2244.14	−2.68759
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1382.59i	− 1.64790i	−0.566660	−	0.823952i	$-0.691765\pi$
$839$	− 1382.59i	− 1.64790i	0.566660	−	0.823952i	$-0.308235\pi$
$840$	0	0
$841$	716.058	0.851436
$842$	0	0
$843$	0	0
$844$	0	0
$845$	435.820i	0.515764i
$846$	0	0
$847$	110.189	0.130093
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 787.176i	− 0.925001i
$852$	0	0
$853$	1273.53	1.49300	0.746500	−	0.665385i	$-0.231733\pi$
$853$	1273.53	1.49300	0.746500	+	0.665385i	$0.231733\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1009.57i	1.17802i	0.808125	+	0.589011i	$0.200483\pi$
$857$	1009.57i	1.17802i	−0.808125	+	0.589011i	$0.799517\pi$
$858$	0	0
$859$	−719.633	−0.837757	−0.418878	−	0.908042i	$-0.637577\pi$
$859$	−719.633	−0.837757	−0.418878	+	0.908042i	$0.637577\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 429.608i	− 0.497808i	−0.968528	−	0.248904i	$-0.919930\pi$
$863$	− 429.608i	− 0.497808i	0.968528	−	0.248904i	$-0.0800703\pi$
$864$	0	0
$865$	−4.80909	−0.00555964
$866$	0	0
$867$	0	0
$868$	0	0
$869$	47.8576i	0.0550721i
$870$	0	0
$871$	−955.055	−1.09650
$872$	0	0
$873$	0	0
$874$	0	0
$875$	41.6316i	0.0475790i
$876$	0	0
$877$	164.030	0.187036	0.0935178	−	0.995618i	$-0.470189\pi$
$877$	164.030	0.187036	0.0935178	+	0.995618i	$0.470189\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 333.265i	− 0.378281i	−0.981950	−	0.189140i	$-0.939430\pi$
$881$	− 333.265i	− 0.378281i	0.981950	−	0.189140i	$-0.0605701\pi$
$882$	0	0
$883$	−289.138	−0.327450	−0.163725	−	0.986506i	$-0.552351\pi$
$883$	−289.138	−0.327450	−0.163725	+	0.986506i	$0.552351\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 483.338i	− 0.544913i	−0.962168	−	0.272457i	$-0.912164\pi$
$887$	− 483.338i	− 0.544913i	0.962168	−	0.272457i	$-0.0878361\pi$
$888$	0	0
$889$	91.2784	0.102675
$890$	0	0
$891$	0	0
$892$	0	0
$893$	450.796i	0.504811i
$894$	0	0
$895$	935.449	1.04519
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 685.759i	− 0.762802i
$900$	0	0
$901$	−1423.04	−1.57940
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2248.77i	2.48483i
$906$	0	0
$907$	769.663	0.848581	0.424290	−	0.905526i	$-0.360524\pi$
$907$	769.663	0.848581	0.424290	+	0.905526i	$0.360524\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	332.059i	0.364499i	0.983252	+	0.182250i	$0.0583379\pi$
$911$	332.059i	0.364499i	−0.983252	+	0.182250i	$0.941662\pi$
$912$	0	0
$913$	−876.235	−0.959731
$914$	0	0
$915$	0	0
$916$	0	0
$917$	106.887i	0.116561i
$918$	0	0
$919$	−1339.15	−1.45718	−0.728591	−	0.684949i	$-0.759825\pi$
$919$	−1339.15	−1.45718	−0.728591	+	0.684949i	$0.759825\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1203.39i	− 1.30378i
$924$	0	0
$925$	−1131.88	−1.22366
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1194.16i	1.28543i	0.766106	+	0.642714i	$0.222192\pi$
$929$	1194.16i	1.28543i	−0.766106	+	0.642714i	$0.777808\pi$
$930$	0	0
$931$	−989.277	−1.06260
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 788.808i	− 0.843645i
$936$	0	0
$937$	631.705	0.674178	0.337089	−	0.941473i	$-0.390558\pi$
$937$	631.705	0.674178	0.337089	+	0.941473i	$0.390558\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1616.20i	− 1.71753i	−0.512367	−	0.858766i	$-0.671231\pi$
$941$	− 1616.20i	− 1.71753i	0.512367	−	0.858766i	$-0.328769\pi$
$942$	0	0
$943$	−678.038	−0.719022
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1625.50i	1.71647i	0.513256	+	0.858235i	$0.328439\pi$
$947$	1625.50i	1.71647i	−0.513256	+	0.858235i	$0.671561\pi$
$948$	0	0
$949$	−1261.78	−1.32959
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 143.033i	− 0.150087i	−0.997180	−	0.0750435i	$-0.976090\pi$
$953$	− 143.033i	− 0.150087i	0.997180	−	0.0750435i	$-0.0239096\pi$
$954$	0	0
$955$	989.817	1.03646
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.8225i	0.0383968i
$960$	0	0
$961$	2802.87	2.91661
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1345.98i	− 1.39480i
$966$	0	0
$967$	132.178	0.136689	0.0683443	−	0.997662i	$-0.478228\pi$
$967$	132.178	0.136689	0.0683443	+	0.997662i	$0.478228\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1171.53i	− 1.20652i	−0.797543	−	0.603262i	$-0.793867\pi$
$971$	− 1171.53i	− 1.20652i	0.797543	−	0.603262i	$-0.206133\pi$
$972$	0	0
$973$	164.971	0.169548
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 797.623i	− 0.816400i	−0.912893	−	0.408200i	$-0.866157\pi$
$977$	− 797.623i	− 0.816400i	0.912893	−	0.408200i	$-0.133843\pi$
$978$	0	0
$979$	690.699	0.705515
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 960.676i	− 0.977290i	−0.872483	−	0.488645i	$-0.837491\pi$
$983$	− 960.676i	− 0.977290i	0.872483	−	0.488645i	$-0.162509\pi$
$984$	0	0
$985$	1140.37	1.15773
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1010.76i	− 1.02201i
$990$	0	0
$991$	474.492	0.478802	0.239401	−	0.970921i	$-0.423049\pi$
$991$	474.492	0.478802	0.239401	+	0.970921i	$0.423049\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 270.951i	− 0.272313i
$996$	0	0
$997$	−180.219	−0.180762	−0.0903809	−	0.995907i	$-0.528808\pi$
$997$	−180.219	−0.180762	−0.0903809	+	0.995907i	$0.528808\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.e.v.1025.2		8
3.2	odd	2	inner	1728.3.e.v.1025.8		8
4.3	odd	2	inner	1728.3.e.v.1025.1		8
8.3	odd	2		864.3.e.e.161.7	yes	8
8.5	even	2		864.3.e.e.161.8	yes	8
12.11	even	2	inner	1728.3.e.v.1025.7		8
24.5	odd	2		864.3.e.e.161.2	yes	8
24.11	even	2		864.3.e.e.161.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.e.e.161.1	✓	8	24.11	even	2
864.3.e.e.161.2	yes	8	24.5	odd	2
864.3.e.e.161.7	yes	8	8.3	odd	2
864.3.e.e.161.8	yes	8	8.5	even	2
1728.3.e.v.1025.1		8	4.3	odd	2	inner
1728.3.e.v.1025.2		8	1.1	even	1	trivial
1728.3.e.v.1025.7		8	12.11	even	2	inner
1728.3.e.v.1025.8		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.37942i q^{5} +1.26611 q^{7} +O(q^{10})\)	\(q-7.37942i q^{5} +1.26611 q^{7} -5.82843i q^{11} +10.4853 q^{13} +18.3399i q^{17} +20.8722 q^{19} -20.4853i q^{23} -29.4558 q^{25} -11.1777i q^{29} +61.3503 q^{31} -9.34315i q^{35} +38.4264 q^{37} -33.0988i q^{41} +49.3410 q^{43} +21.5980i q^{47} -47.3970 q^{49} +77.5925i q^{53} -43.0104 q^{55} -25.7157i q^{59} +55.8823 q^{61} -77.3753i q^{65} -91.0853 q^{67} -114.770i q^{71} -120.338 q^{73} -7.37942i q^{77} -8.21107 q^{79} -150.338i q^{83} +135.338 q^{85} +118.505i q^{89} +13.2755 q^{91} -154.024i q^{95} -72.8823 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{13} - 32 q^{25} - 32 q^{37} + 96 q^{49} - 96 q^{61} - 216 q^{73} + 336 q^{85} - 40 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 - 32 * q^25 - 32 * q^37 + 96 * q^49 - 96 * q^61 - 216 * q^73 + 336 * q^85 - 40 * q^97

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