LMFDB - Embedded newform 4032.2.h.h.575.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(575,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4032.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.h (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.3
Root			$0.892524 - 1.09700i$ of defining polynomial
Character	$\chi$	$=$	4032.575
Dual form			4032.2.h.h.575.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56483i q^{5} -1.00000i q^{7} +O(q^{10})$	$q-2.56483i q^{5} -1.00000i q^{7} -1.15061 q^{11} +0.578337 q^{13} +5.39325i q^{17} +6.20555i q^{19} +7.62536 q^{23} -1.57834 q^{25} +1.41421i q^{29} +5.04888i q^{31} -2.56483 q^{35} -9.83276 q^{37} -6.21115i q^{41} +11.2544i q^{43} -11.0772 q^{47} -1.00000 q^{49} +4.53333i q^{53} +2.95112i q^{55} +4.83896 q^{59} -0.951124 q^{61} -1.48333i q^{65} -2.78389i q^{67} +3.68835 q^{71} +14.0383 q^{73} +1.15061i q^{77} +12.8816i q^{79} +8.77597 q^{83} +13.8328 q^{85} +5.68395i q^{89} -0.578337i q^{91} +15.9162 q^{95} -12.8816 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 12 q^{25} - 8 q^{37} - 12 q^{49} - 56 q^{61} + 56 q^{85}+O(q^{100}) $ 12 * q - 12 * q^25 - 8 * q^37 - 12 * q^49 - 56 * q^61 + 56 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$	− 2.56483i	− 1.14703i	−0.819197	−	0.573513i	$-0.805580\pi$
$5$	− 2.56483i	− 1.14703i	0.819197	−	0.573513i	$-0.194420\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.15061	−0.346923	−0.173461	−	0.984841i	$-0.555495\pi$
$11$	−1.15061	−0.346923	−0.173461	+	0.984841i	$0.555495\pi$
$12$	0	0
$13$	0.578337	0.160402	0.0802009	−	0.996779i	$-0.474444\pi$
$13$	0.578337	0.160402	0.0802009	+	0.996779i	$0.474444\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.39325i	1.30806i	0.756470	+	0.654028i	$0.226922\pi$
$17$	5.39325i	1.30806i	−0.756470	+	0.654028i	$0.773078\pi$
$18$	0	0
$19$	6.20555i	1.42365i	0.702356	+	0.711825i	$0.252131\pi$
$19$	6.20555i	1.42365i	−0.702356	+	0.711825i	$0.747869\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.62536	1.59000	0.794999	−	0.606611i	$-0.207471\pi$
$23$	7.62536	1.59000	0.794999	+	0.606611i	$0.207471\pi$
$24$	0	0
$25$	−1.57834	−0.315667
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.41421i	0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$	1.41421i	0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$	0	0
$31$	5.04888i	0.906805i	0.891306	+	0.453402i	$0.149790\pi$
$31$	5.04888i	0.906805i	−0.891306	+	0.453402i	$0.850210\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.56483	−0.433535
$36$	0	0
$37$	−9.83276	−1.61650	−0.808248	−	0.588842i	$-0.799584\pi$
$37$	−9.83276	−1.61650	−0.808248	+	0.588842i	$0.799584\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.21115i	− 0.970018i	−0.874509	−	0.485009i	$-0.838816\pi$
$41$	− 6.21115i	− 0.970018i	0.874509	−	0.485009i	$-0.161184\pi$
$42$	0	0
$43$	11.2544i	1.71628i	0.513413	+	0.858142i	$0.328381\pi$
$43$	11.2544i	1.71628i	−0.513413	+	0.858142i	$0.671619\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.0772	−1.61578	−0.807888	−	0.589336i	$-0.799389\pi$
$47$	−11.0772	−1.61578	−0.807888	+	0.589336i	$0.799389\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.53333i	0.622701i	0.950295	+	0.311351i	$0.100781\pi$
$53$	4.53333i	0.622701i	−0.950295	+	0.311351i	$0.899219\pi$
$54$	0	0
$55$	2.95112i	0.397929i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.83896	0.629979	0.314990	−	0.949095i	$-0.397999\pi$
$59$	4.83896	0.629979	0.314990	+	0.949095i	$0.397999\pi$
$60$	0	0
$61$	−0.951124	−0.121779	−0.0608895	−	0.998145i	$-0.519394\pi$
$61$	−0.951124	−0.121779	−0.0608895	+	0.998145i	$0.519394\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.48333i	− 0.183985i
$66$	0	0
$67$	− 2.78389i	− 0.340106i	−0.985435	−	0.170053i	$-0.945606\pi$
$67$	− 2.78389i	− 0.340106i	0.985435	−	0.170053i	$-0.0543939\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.68835	0.437726	0.218863	−	0.975756i	$-0.429765\pi$
$71$	3.68835	0.437726	0.218863	+	0.975756i	$0.429765\pi$
$72$	0	0
$73$	14.0383	1.64306	0.821530	−	0.570165i	$-0.193121\pi$
$73$	14.0383	1.64306	0.821530	+	0.570165i	$0.193121\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.15061i	0.131125i
$78$	0	0
$79$	12.8816i	1.44930i	0.689118	+	0.724649i	$0.257998\pi$
$79$	12.8816i	1.44930i	−0.689118	+	0.724649i	$0.742002\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.77597	0.963288	0.481644	−	0.876367i	$-0.340040\pi$
$83$	8.77597	0.963288	0.481644	+	0.876367i	$0.340040\pi$
$84$	0	0
$85$	13.8328	1.50037
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.68395i	0.602497i	0.953546	+	0.301249i	$0.0974034\pi$
$89$	5.68395i	0.602497i	−0.953546	+	0.301249i	$0.902597\pi$
$90$	0	0
$91$	− 0.578337i	− 0.0606262i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.9162	1.63296
$96$	0	0
$97$	−12.8816	−1.30793	−0.653966	−	0.756524i	$-0.726896\pi$
$97$	−12.8816	−1.30793	−0.653966	+	0.756524i	$0.726896\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.75747i	0.373882i	0.982371	+	0.186941i	$0.0598574\pi$
$101$	3.75747i	0.373882i	−0.982371	+	0.186941i	$0.940143\pi$
$102$	0	0
$103$	1.79445i	0.176812i	0.996085	+	0.0884062i	$0.0281774\pi$
$103$	1.79445i	0.176812i	−0.996085	+	0.0884062i	$0.971823\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.1631	0.982503	0.491252	−	0.871018i	$-0.336540\pi$
$107$	10.1631	0.982503	0.491252	+	0.871018i	$0.336540\pi$
$108$	0	0
$109$	15.2544	1.46111	0.730555	−	0.682854i	$-0.239262\pi$
$109$	15.2544	1.46111	0.730555	+	0.682854i	$0.239262\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.06053i	− 0.476055i	−0.971258	−	0.238027i	$-0.923499\pi$
$113$	− 5.06053i	− 0.476055i	0.971258	−	0.238027i	$-0.0765008\pi$
$114$	0	0
$115$	− 19.5577i	− 1.82377i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.39325	0.494399
$120$	0	0
$121$	−9.67609	−0.879644
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8.77597i	− 0.784947i
$126$	0	0
$127$	− 0.470539i	− 0.0417536i	−0.999782	−	0.0208768i	$-0.993354\pi$
$127$	− 0.470539i	− 0.0417536i	0.999782	−	0.0208768i	$-0.00664577\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.9162	1.39060	0.695301	−	0.718719i	$-0.255271\pi$
$131$	15.9162	1.39060	0.695301	+	0.718719i	$0.255271\pi$
$132$	0	0
$133$	6.20555	0.538089
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.55440i	− 0.730852i	−0.930840	−	0.365426i	$-0.880923\pi$
$137$	− 8.55440i	− 0.730852i	0.930840	−	0.365426i	$-0.119077\pi$
$138$	0	0
$139$	16.0000i	1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$	16.0000i	1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.665442	−0.0556471
$144$	0	0
$145$	3.62721	0.301224
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 16.8032i	− 1.37657i	−0.725441	−	0.688285i	$-0.758364\pi$
$149$	− 16.8032i	− 1.37657i	0.725441	−	0.688285i	$-0.241636\pi$
$150$	0	0
$151$	− 2.09775i	− 0.170713i	−0.996350	−	0.0853563i	$-0.972797\pi$
$151$	− 2.09775i	− 0.170713i	0.996350	−	0.0853563i	$-0.0272029\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.9495	1.04013
$156$	0	0
$157$	−17.3622	−1.38566	−0.692828	−	0.721103i	$-0.743636\pi$
$157$	−17.3622	−1.38566	−0.692828	+	0.721103i	$0.743636\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 7.62536i	− 0.600963i
$162$	0	0
$163$	− 0.470539i	− 0.0368554i	−0.999830	−	0.0184277i	$-0.994134\pi$
$163$	− 0.470539i	− 0.0368554i	0.999830	−	0.0184277i	$-0.00586606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.7551	1.60608	0.803040	−	0.595925i	$-0.203215\pi$
$167$	20.7551	1.60608	0.803040	+	0.595925i	$0.203215\pi$
$168$	0	0
$169$	−12.6655	−0.974271
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.1692i	1.07727i	0.842540	+	0.538633i	$0.181059\pi$
$173$	14.1692i	1.07727i	−0.842540	+	0.538633i	$0.818941\pi$
$174$	0	0
$175$	1.57834i	0.119311i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.15061	−0.0860009	−0.0430004	−	0.999075i	$-0.513692\pi$
$179$	−1.15061	−0.0860009	−0.0430004	+	0.999075i	$0.513692\pi$
$180$	0	0
$181$	−3.83276	−0.284887	−0.142444	−	0.989803i	$-0.545496\pi$
$181$	−3.83276	−0.284887	−0.142444	+	0.989803i	$0.545496\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	25.2193i	1.85416i
$186$	0	0
$187$	− 6.20555i	− 0.453795i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.92659	−0.718263	−0.359131	−	0.933287i	$-0.616927\pi$
$191$	−9.92659	−0.718263	−0.359131	+	0.933287i	$0.616927\pi$
$192$	0	0
$193$	12.6761	0.912445	0.456222	−	0.889866i	$-0.349202\pi$
$193$	12.6761	0.912445	0.456222	+	0.889866i	$0.349202\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.70491i	0.121469i	0.998154	+	0.0607347i	$0.0193444\pi$
$197$	1.70491i	0.121469i	−0.998154	+	0.0607347i	$0.980656\pi$
$198$	0	0
$199$	14.8433i	1.05222i	0.850418	+	0.526108i	$0.176349\pi$
$199$	14.8433i	1.05222i	−0.850418	+	0.526108i	$0.823651\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.41421	0.0992583
$204$	0	0
$205$	−15.9305	−1.11264
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7.14019i	− 0.493897i
$210$	0	0
$211$	− 1.15667i	− 0.0796287i	−0.999207	−	0.0398144i	$-0.987323\pi$
$211$	− 1.15667i	− 0.0796287i	0.999207	−	0.0398144i	$-0.0126767\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	28.8657	1.96862
$216$	0	0
$217$	5.04888	0.342740
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.11912i	0.209815i
$222$	0	0
$223$	− 23.6655i	− 1.58476i	−0.610027	−	0.792380i	$-0.708842\pi$
$223$	− 23.6655i	− 1.58476i	0.610027	−	0.792380i	$-0.291158\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.44142	0.626649	0.313324	−	0.949646i	$-0.398557\pi$
$227$	9.44142	0.626649	0.313324	+	0.949646i	$0.398557\pi$
$228$	0	0
$229$	0.578337	0.0382176	0.0191088	−	0.999817i	$-0.493917\pi$
$229$	0.578337	0.0382176	0.0191088	+	0.999817i	$0.493917\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.305630i	0.0200225i	0.999950	+	0.0100112i	$0.00318673\pi$
$233$	0.305630i	0.0200225i	−0.999950	+	0.0100112i	$0.996813\pi$
$234$	0	0
$235$	28.4111i	1.85334i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.92659	0.642098	0.321049	−	0.947063i	$-0.395965\pi$
$239$	9.92659	0.642098	0.321049	+	0.947063i	$0.395965\pi$
$240$	0	0
$241$	9.29274	0.598598	0.299299	−	0.954159i	$-0.403247\pi$
$241$	9.29274	0.598598	0.299299	+	0.954159i	$0.403247\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.56483i	0.163861i
$246$	0	0
$247$	3.58890i	0.228356i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.93701	−0.248502	−0.124251	−	0.992251i	$-0.539653\pi$
$251$	−3.93701	−0.248502	−0.124251	+	0.992251i	$0.539653\pi$
$252$	0	0
$253$	−8.77384	−0.551607
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.8891i	0.991133i	0.868570	+	0.495566i	$0.165040\pi$
$257$	15.8891i	0.991133i	−0.868570	+	0.495566i	$0.834960\pi$
$258$	0	0
$259$	9.83276i	0.610978i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.38712	0.0855336	0.0427668	−	0.999085i	$-0.486383\pi$
$263$	1.38712	0.0855336	0.0427668	+	0.999085i	$0.486383\pi$
$264$	0	0
$265$	11.6272	0.714254
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.60869i	0.0980837i	0.998797	+	0.0490419i	$0.0156168\pi$
$269$	1.60869i	0.0980837i	−0.998797	+	0.0490419i	$0.984383\pi$
$270$	0	0
$271$	8.30330i	0.504390i	0.967676	+	0.252195i	$0.0811524\pi$
$271$	8.30330i	0.504390i	−0.967676	+	0.252195i	$0.918848\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.81606	0.109512
$276$	0	0
$277$	3.83276	0.230288	0.115144	−	0.993349i	$-0.463267\pi$
$277$	3.83276	0.230288	0.115144	+	0.993349i	$0.463267\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 28.9348i	− 1.72610i	−0.505115	−	0.863052i	$-0.668550\pi$
$281$	− 28.9348i	− 1.72610i	0.505115	−	0.863052i	$-0.331450\pi$
$282$	0	0
$283$	31.0278i	1.84441i	0.386703	+	0.922204i	$0.373614\pi$
$283$	31.0278i	1.84441i	−0.386703	+	0.922204i	$0.626386\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.21115	−0.366632
$288$	0	0
$289$	−12.0872	−0.711011
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.01850i	0.293184i	0.989197	+	0.146592i	$0.0468305\pi$
$293$	5.01850i	0.293184i	−0.989197	+	0.146592i	$0.953170\pi$
$294$	0	0
$295$	− 12.4111i	− 0.722602i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.41003	0.255039
$300$	0	0
$301$	11.2544	0.648694
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.43947i	0.139684i
$306$	0	0
$307$	− 15.3622i	− 0.876768i	−0.898788	−	0.438384i	$-0.855551\pi$
$307$	− 15.3622i	− 0.876768i	0.898788	−	0.438384i	$-0.144449\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.83896	0.274392	0.137196	−	0.990544i	$-0.456191\pi$
$311$	4.83896	0.274392	0.137196	+	0.990544i	$0.456191\pi$
$312$	0	0
$313$	19.7633	1.11709	0.558543	−	0.829475i	$-0.311360\pi$
$313$	19.7633	1.11709	0.558543	+	0.829475i	$0.311360\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.96248i	0.334886i	0.985882	+	0.167443i	$0.0535511\pi$
$317$	5.96248i	0.334886i	−0.985882	+	0.167443i	$0.946449\pi$
$318$	0	0
$319$	− 1.62721i	− 0.0911064i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−33.4681	−1.86222
$324$	0	0
$325$	−0.912811	−0.0506336
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.0772i	0.610706i
$330$	0	0
$331$	− 1.15667i	− 0.0635766i	−0.999495	−	0.0317883i	$-0.989880\pi$
$331$	− 1.15667i	− 0.0635766i	0.999495	−	0.0317883i	$-0.0101202\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.14019	−0.390110
$336$	0	0
$337$	3.25443	0.177280	0.0886399	−	0.996064i	$-0.471748\pi$
$337$	3.25443	0.177280	0.0886399	+	0.996064i	$0.471748\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 5.80930i	− 0.314591i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.4013	0.880470	0.440235	−	0.897883i	$-0.354895\pi$
$347$	16.4013	0.880470	0.440235	+	0.897883i	$0.354895\pi$
$348$	0	0
$349$	8.20555	0.439233	0.219617	−	0.975586i	$-0.429519\pi$
$349$	8.20555	0.439233	0.219617	+	0.975586i	$0.429519\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.9557i	1.32826i	0.747617	+	0.664130i	$0.231198\pi$
$353$	24.9557i	1.32826i	−0.747617	+	0.664130i	$0.768802\pi$
$354$	0	0
$355$	− 9.45998i	− 0.502083i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.38712	0.0732095	0.0366048	−	0.999330i	$-0.488346\pi$
$359$	1.38712	0.0732095	0.0366048	+	0.999330i	$0.488346\pi$
$360$	0	0
$361$	−19.5089	−1.02678
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 36.0058i	− 1.88463i
$366$	0	0
$367$	10.9200i	0.570017i	0.958525	+	0.285008i	$0.0919964\pi$
$367$	10.9200i	0.570017i	−0.958525	+	0.285008i	$0.908004\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.53333	0.235359
$372$	0	0
$373$	−29.6655	−1.53602	−0.768011	−	0.640436i	$-0.778754\pi$
$373$	−29.6655	−1.53602	−0.768011	+	0.640436i	$0.778754\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.817892i	0.0421236i
$378$	0	0
$379$	− 7.66553i	− 0.393752i	−0.980428	−	0.196876i	$-0.936920\pi$
$379$	− 7.66553i	− 0.393752i	0.980428	−	0.196876i	$-0.0630796\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.8407	0.553933	0.276967	−	0.960880i	$-0.410671\pi$
$383$	10.8407	0.553933	0.276967	+	0.960880i	$0.410671\pi$
$384$	0	0
$385$	2.95112	0.150403
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 22.6125i	− 1.14650i	−0.819381	−	0.573249i	$-0.805683\pi$
$389$	− 22.6125i	− 1.14650i	0.819381	−	0.573249i	$-0.194317\pi$
$390$	0	0
$391$	41.1255i	2.07981i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	33.0392	1.66238
$396$	0	0
$397$	21.5577	1.08195	0.540976	−	0.841038i	$-0.318055\pi$
$397$	21.5577	1.08195	0.540976	+	0.841038i	$0.318055\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.26176i	− 0.0630095i	−0.999504	−	0.0315047i	$-0.989970\pi$
$401$	− 1.26176i	− 0.0630095i	0.999504	−	0.0315047i	$-0.0100299\pi$
$402$	0	0
$403$	2.91995i	0.145453i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.3137	0.560800
$408$	0	0
$409$	−34.4494	−1.70341	−0.851707	−	0.524018i	$-0.824432\pi$
$409$	−34.4494	−1.70341	−0.851707	+	0.524018i	$0.824432\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4.83896i	− 0.238110i
$414$	0	0
$415$	− 22.5089i	− 1.10492i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.5816	−0.810064	−0.405032	−	0.914302i	$-0.632740\pi$
$419$	−16.5816	−0.810064	−0.405032	+	0.914302i	$0.632740\pi$
$420$	0	0
$421$	8.36274	0.407575	0.203788	−	0.979015i	$-0.434675\pi$
$421$	8.36274	0.407575	0.203788	+	0.979015i	$0.434675\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 8.51237i	− 0.412911i
$426$	0	0
$427$	0.951124i	0.0460281i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.1565	−1.78976	−0.894882	−	0.446303i	$-0.852740\pi$
$431$	−37.1565	−1.78976	−0.894882	+	0.446303i	$0.852740\pi$
$432$	0	0
$433$	−7.56777	−0.363684	−0.181842	−	0.983328i	$-0.558206\pi$
$433$	−7.56777	−0.363684	−0.181842	+	0.983328i	$0.558206\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	47.3196i	2.26360i
$438$	0	0
$439$	16.8222i	0.802880i	0.915885	+	0.401440i	$0.131490\pi$
$439$	16.8222i	0.802880i	−0.915885	+	0.401440i	$0.868510\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.4643	−0.592198	−0.296099	−	0.955157i	$-0.595686\pi$
$443$	−12.4643	−0.592198	−0.296099	+	0.955157i	$0.595686\pi$
$444$	0	0
$445$	14.5783	0.691079
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 7.44582i	− 0.351390i	−0.984445	−	0.175695i	$-0.943783\pi$
$449$	− 7.44582i	− 0.351390i	0.984445	−	0.175695i	$-0.0562172\pi$
$450$	0	0
$451$	7.14663i	0.336522i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.48333	−0.0695398
$456$	0	0
$457$	−5.68665	−0.266010	−0.133005	−	0.991115i	$-0.542463\pi$
$457$	−5.68665	−0.266010	−0.133005	+	0.991115i	$0.542463\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 0.790801i	− 0.0368313i	−0.999830	−	0.0184156i	$-0.994138\pi$
$461$	− 0.790801i	− 0.0368313i	0.999830	−	0.0184156i	$-0.00586221\pi$
$462$	0	0
$463$	25.2927i	1.17545i	0.809060	+	0.587727i	$0.199977\pi$
$463$	25.2927i	1.17545i	−0.809060	+	0.587727i	$0.800023\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−40.8448	−1.89007	−0.945036	−	0.326966i	$-0.893974\pi$
$467$	−40.8448	−1.89007	−0.945036	+	0.326966i	$0.893974\pi$
$468$	0	0
$469$	−2.78389	−0.128548
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 12.9495i	− 0.595418i
$474$	0	0
$475$	− 9.79445i	− 0.449400i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.2507	−0.696823	−0.348412	−	0.937342i	$-0.613279\pi$
$479$	−15.2507	−0.696823	−0.348412	+	0.937342i	$0.613279\pi$
$480$	0	0
$481$	−5.68665	−0.259289
$482$	0	0
$483$	0	0
$484$	0	0
$485$	33.0392i	1.50023i
$486$	0	0
$487$	24.6066i	1.11503i	0.830166	+	0.557516i	$0.188245\pi$
$487$	24.6066i	1.11503i	−0.830166	+	0.557516i	$0.811755\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.3472	1.41468	0.707339	−	0.706875i	$-0.249896\pi$
$491$	31.3472	1.41468	0.707339	+	0.706875i	$0.249896\pi$
$492$	0	0
$493$	−7.62721	−0.343512
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.68835i	− 0.165445i
$498$	0	0
$499$	11.2544i	0.503817i	0.967751	+	0.251909i	$0.0810583\pi$
$499$	11.2544i	0.503817i	−0.967751	+	0.251909i	$0.918942\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.6588	−1.23325	−0.616623	−	0.787259i	$-0.711500\pi$
$503$	−27.6588	−1.23325	−0.616623	+	0.787259i	$0.711500\pi$
$504$	0	0
$505$	9.63726	0.428852
$506$	0	0
$507$	0	0
$508$	0	0
$509$	29.7947i	1.32063i	0.750990	+	0.660313i	$0.229577\pi$
$509$	29.7947i	1.32063i	−0.750990	+	0.660313i	$0.770423\pi$
$510$	0	0
$511$	− 14.0383i	− 0.621018i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.60245	0.202808
$516$	0	0
$517$	12.7456	0.560550
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.84520i	0.0808397i	0.999183	+	0.0404199i	$0.0128696\pi$
$521$	1.84520i	0.0808397i	−0.999183	+	0.0404199i	$0.987130\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−27.2299	−1.18615
$528$	0	0
$529$	35.1461	1.52809
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.59214i	− 0.155593i
$534$	0	0
$535$	− 26.0666i	− 1.12696i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.15061	0.0495604
$540$	0	0
$541$	26.3416	1.13251	0.566257	−	0.824229i	$-0.308391\pi$
$541$	26.3416	1.13251	0.566257	+	0.824229i	$0.308391\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 39.1250i	− 1.67593i
$546$	0	0
$547$	− 0.805013i	− 0.0344199i	−0.999852	−	0.0172099i	$-0.994522\pi$
$547$	− 0.805013i	− 0.0344199i	0.999852	−	0.0172099i	$-0.00547836\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.77597	−0.373869
$552$	0	0
$553$	12.8816	0.547783
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.8909i	− 0.630948i	−0.948934	−	0.315474i	$-0.897836\pi$
$557$	− 14.8909i	− 0.630948i	0.948934	−	0.315474i	$-0.102164\pi$
$558$	0	0
$559$	6.50885i	0.275295i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−43.3825	−1.82836	−0.914178	−	0.405313i	$-0.867163\pi$
$563$	−43.3825	−1.82836	−0.914178	+	0.405313i	$0.867163\pi$
$564$	0	0
$565$	−12.9794	−0.546047
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.8786i	0.917201i	0.888643	+	0.458600i	$0.151649\pi$
$569$	21.8786i	0.917201i	−0.888643	+	0.458600i	$0.848351\pi$
$570$	0	0
$571$	− 37.7038i	− 1.57786i	−0.614485	−	0.788928i	$-0.710636\pi$
$571$	− 37.7038i	− 1.57786i	0.614485	−	0.788928i	$-0.289364\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0354	−0.501910
$576$	0	0
$577$	−26.4111	−1.09951	−0.549754	−	0.835326i	$-0.685279\pi$
$577$	−26.4111	−1.09951	−0.549754	+	0.835326i	$0.685279\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.77597i	− 0.364089i
$582$	0	0
$583$	− 5.21611i	− 0.216029i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.1527	0.666692	0.333346	−	0.942805i	$-0.391822\pi$
$587$	16.1527	0.666692	0.333346	+	0.942805i	$0.391822\pi$
$588$	0	0
$589$	−31.3311	−1.29097
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.44765i	0.264773i	0.991198	+	0.132387i	$0.0422641\pi$
$593$	6.44765i	0.264773i	−0.991198	+	0.132387i	$0.957736\pi$
$594$	0	0
$595$	− 13.8328i	− 0.567088i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.9391	0.773829	0.386915	−	0.922116i	$-0.373541\pi$
$599$	18.9391	0.773829	0.386915	+	0.922116i	$0.373541\pi$
$600$	0	0
$601$	18.4111	0.751004	0.375502	−	0.926821i	$-0.377470\pi$
$601$	18.4111	0.751004	0.375502	+	0.926821i	$0.377470\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.8175i	1.00897i
$606$	0	0
$607$	− 24.6066i	− 0.998751i	−0.866386	−	0.499376i	$-0.833563\pi$
$607$	− 24.6066i	− 0.998751i	0.866386	−	0.499376i	$-0.166437\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.40636	−0.259173
$612$	0	0
$613$	−8.09775	−0.327065	−0.163533	−	0.986538i	$-0.552289\pi$
$613$	−8.09775	−0.327065	−0.163533	+	0.986538i	$0.552289\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.14459i	− 0.207113i	−0.994624	−	0.103557i	$-0.966978\pi$
$617$	− 5.14459i	− 0.207113i	0.994624	−	0.103557i	$-0.0330223\pi$
$618$	0	0
$619$	− 36.2922i	− 1.45871i	−0.684137	−	0.729354i	$-0.739821\pi$
$619$	− 36.2922i	− 1.45871i	0.684137	−	0.729354i	$-0.260179\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.68395	0.227722
$624$	0	0
$625$	−30.4005	−1.21602
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 53.0306i	− 2.11447i
$630$	0	0
$631$	19.7250i	0.785238i	0.919701	+	0.392619i	$0.128431\pi$
$631$	19.7250i	0.785238i	−0.919701	+	0.392619i	$0.871569\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.20685	−0.0478924
$636$	0	0
$637$	−0.578337	−0.0229145
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 20.6860i	− 0.817048i	−0.912748	−	0.408524i	$-0.866044\pi$
$641$	− 20.6860i	− 0.817048i	0.912748	−	0.408524i	$-0.133956\pi$
$642$	0	0
$643$	15.0278i	0.592637i	0.955089	+	0.296318i	$0.0957589\pi$
$643$	15.0278i	0.592637i	−0.955089	+	0.296318i	$0.904241\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.3909	0.880277	0.440139	−	0.897930i	$-0.354929\pi$
$647$	22.3909	0.880277	0.440139	+	0.897930i	$0.354929\pi$
$648$	0	0
$649$	−5.56777	−0.218554
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 31.0978i	− 1.21695i	−0.793573	−	0.608475i	$-0.791782\pi$
$653$	− 31.0978i	− 1.21695i	0.793573	−	0.608475i	$-0.208218\pi$
$654$	0	0
$655$	− 40.8222i	− 1.59506i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.1631	−0.395898	−0.197949	−	0.980212i	$-0.563428\pi$
$659$	−10.1631	−0.395898	−0.197949	+	0.980212i	$0.563428\pi$
$660$	0	0
$661$	15.1255	0.588314	0.294157	−	0.955757i	$-0.404961\pi$
$661$	15.1255	0.588314	0.294157	+	0.955757i	$0.404961\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 15.9162i	− 0.617202i
$666$	0	0
$667$	10.7839i	0.417554i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.09438	0.0422479
$672$	0	0
$673$	36.8716	1.42130	0.710648	−	0.703548i	$-0.248402\pi$
$673$	36.8716	1.42130	0.710648	+	0.703548i	$0.248402\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10.3705i	− 0.398569i	−0.979942	−	0.199285i	$-0.936138\pi$
$677$	− 10.3705i	− 0.398569i	0.979942	−	0.199285i	$-0.0638618\pi$
$678$	0	0
$679$	12.8816i	0.494352i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.19275	0.351751	0.175875	−	0.984412i	$-0.443724\pi$
$683$	9.19275	0.351751	0.175875	+	0.984412i	$0.443724\pi$
$684$	0	0
$685$	−21.9406	−0.838306
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.62179i	0.0998824i
$690$	0	0
$691$	− 22.8433i	− 0.869001i	−0.900672	−	0.434501i	$-0.856925\pi$
$691$	− 22.8433i	− 0.869001i	0.900672	−	0.434501i	$-0.143075\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	41.0372	1.55663
$696$	0	0
$697$	33.4983	1.26884
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 19.1044i	− 0.721563i	−0.932650	−	0.360782i	$-0.882510\pi$
$701$	− 19.1044i	− 0.721563i	0.932650	−	0.360782i	$-0.117490\pi$
$702$	0	0
$703$	− 61.0177i	− 2.30133i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.75747	0.141314
$708$	0	0
$709$	9.56777	0.359325	0.179663	−	0.983728i	$-0.442499\pi$
$709$	9.56777	0.359325	0.179663	+	0.983728i	$0.442499\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	38.4995i	1.44182i
$714$	0	0
$715$	1.70674i	0.0638286i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30.5014	−1.13751	−0.568756	−	0.822506i	$-0.692575\pi$
$719$	−30.5014	−1.13751	−0.568756	+	0.822506i	$0.692575\pi$
$720$	0	0
$721$	1.79445	0.0668288
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.23211i	− 0.0828983i
$726$	0	0
$727$	− 32.5189i	− 1.20606i	−0.797719	−	0.603030i	$-0.793960\pi$
$727$	− 32.5189i	− 1.20606i	0.797719	−	0.603030i	$-0.206040\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−60.6980	−2.24500
$732$	0	0
$733$	−19.0106	−0.702171	−0.351086	−	0.936343i	$-0.614187\pi$
$733$	−19.0106	−0.702171	−0.351086	+	0.936343i	$0.614187\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.20318i	0.117991i
$738$	0	0
$739$	− 19.7250i	− 0.725595i	−0.931868	−	0.362797i	$-0.881822\pi$
$739$	− 19.7250i	− 0.725595i	0.931868	−	0.362797i	$-0.118178\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.8849	−1.24312	−0.621558	−	0.783368i	$-0.713500\pi$
$743$	−33.8849	−1.24312	−0.621558	+	0.783368i	$0.713500\pi$
$744$	0	0
$745$	−43.0972	−1.57896
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10.1631i	− 0.371351i
$750$	0	0
$751$	− 30.5089i	− 1.11328i	−0.830753	−	0.556642i	$-0.812090\pi$
$751$	− 30.5089i	− 1.11328i	0.830753	−	0.556642i	$-0.187910\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.38037	−0.195812
$756$	0	0
$757$	48.0766	1.74737	0.873687	−	0.486488i	$-0.161722\pi$
$757$	48.0766	1.74737	0.873687	+	0.486488i	$0.161722\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.4055i	0.485950i	0.970033	+	0.242975i	$0.0781232\pi$
$761$	13.4055i	0.485950i	−0.970033	+	0.242975i	$0.921877\pi$
$762$	0	0
$763$	− 15.2544i	− 0.552247i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.79855	0.101050
$768$	0	0
$769$	13.4700	0.485741	0.242871	−	0.970059i	$-0.421911\pi$
$769$	13.4700	0.485741	0.242871	+	0.970059i	$0.421911\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.5479i	1.02680i	0.858151	+	0.513398i	$0.171613\pi$
$773$	28.5479i	1.02680i	−0.858151	+	0.513398i	$0.828387\pi$
$774$	0	0
$775$	− 7.96883i	− 0.286249i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	38.5436	1.38097
$780$	0	0
$781$	−4.24386	−0.151857
$782$	0	0
$783$	0	0
$784$	0	0
$785$	44.5311i	1.58938i
$786$	0	0
$787$	25.7633i	0.918362i	0.888343	+	0.459181i	$0.151857\pi$
$787$	25.7633i	0.918362i	−0.888343	+	0.459181i	$0.848143\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.06053	−0.179932
$792$	0	0
$793$	−0.550070	−0.0195336
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.3741i	0.827954i	0.910287	+	0.413977i	$0.135861\pi$
$797$	23.3741i	0.827954i	−0.910287	+	0.413977i	$0.864139\pi$
$798$	0	0
$799$	− 59.7422i	− 2.11353i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.1527	−0.570015
$804$	0	0
$805$	−19.5577	−0.689319
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.6019i	0.724326i	0.932115	+	0.362163i	$0.117962\pi$
$809$	20.6019i	0.724326i	−0.932115	+	0.362163i	$0.882038\pi$
$810$	0	0
$811$	2.64782i	0.0929776i	0.998919	+	0.0464888i	$0.0148032\pi$
$811$	2.64782i	0.0929776i	−0.998919	+	0.0464888i	$0.985197\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.20685	−0.0422741
$816$	0	0
$817$	−69.8399	−2.44339
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.5201i	1.51886i	0.650589	+	0.759430i	$0.274522\pi$
$821$	43.5201i	1.51886i	−0.650589	+	0.759430i	$0.725478\pi$
$822$	0	0
$823$	10.5683i	0.368387i	0.982890	+	0.184194i	$0.0589674\pi$
$823$	10.5683i	0.368387i	−0.982890	+	0.184194i	$0.941033\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.92486	−0.136481	−0.0682403	−	0.997669i	$-0.521738\pi$
$827$	−3.92486	−0.136481	−0.0682403	+	0.997669i	$0.521738\pi$
$828$	0	0
$829$	−15.2061	−0.528129	−0.264064	−	0.964505i	$-0.585063\pi$
$829$	−15.2061	−0.528129	−0.264064	+	0.964505i	$0.585063\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.39325i	− 0.186865i
$834$	0	0
$835$	− 53.2333i	− 1.84221i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.2841	0.424093	0.212046	−	0.977260i	$-0.431987\pi$
$839$	12.2841	0.424093	0.212046	+	0.977260i	$0.431987\pi$
$840$	0	0
$841$	27.0000	0.931034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	32.4849i	1.11751i
$846$	0	0
$847$	9.67609i	0.332474i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−74.9784	−2.57022
$852$	0	0
$853$	−16.2056	−0.554867	−0.277434	−	0.960745i	$-0.589484\pi$
$853$	−16.2056	−0.554867	−0.277434	+	0.960745i	$0.589484\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.9873i	0.546117i	0.961997	+	0.273059i	$0.0880353\pi$
$857$	15.9873i	0.546117i	−0.961997	+	0.273059i	$0.911965\pi$
$858$	0	0
$859$	31.9688i	1.09076i	0.838188	+	0.545381i	$0.183615\pi$
$859$	31.9688i	1.09076i	−0.838188	+	0.545381i	$0.816385\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.9847	1.32705	0.663527	−	0.748153i	$-0.269059\pi$
$863$	38.9847	1.32705	0.663527	+	0.748153i	$0.269059\pi$
$864$	0	0
$865$	36.3416	1.23565
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 14.8218i	− 0.502795i
$870$	0	0
$871$	− 1.61003i	− 0.0545536i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.77597	−0.296682
$876$	0	0
$877$	6.07663	0.205193	0.102597	−	0.994723i	$-0.467285\pi$
$877$	6.07663	0.205193	0.102597	+	0.994723i	$0.467285\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.9575i	0.537621i	0.963193	+	0.268810i	$0.0866305\pi$
$881$	15.9575i	0.537621i	−0.963193	+	0.268810i	$0.913370\pi$
$882$	0	0
$883$	− 43.2544i	− 1.45563i	−0.685775	−	0.727814i	$-0.740537\pi$
$883$	− 43.2544i	− 1.45563i	0.685775	−	0.727814i	$-0.259463\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.1811	−1.55061	−0.775305	−	0.631587i	$-0.782404\pi$
$887$	−46.1811	−1.55061	−0.775305	+	0.631587i	$0.782404\pi$
$888$	0	0
$889$	−0.470539	−0.0157814
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 68.7401i	− 2.30030i
$894$	0	0
$895$	2.95112i	0.0986452i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.14019	−0.238139
$900$	0	0
$901$	−24.4494	−0.814528
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.83037i	0.326773i
$906$	0	0
$907$	− 8.94108i	− 0.296884i	−0.988921	−	0.148442i	$-0.952574\pi$
$907$	− 8.94108i	− 0.296884i	0.988921	−	0.148442i	$-0.0474258\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.8219	−1.25310	−0.626548	−	0.779383i	$-0.715533\pi$
$911$	−37.8219	−1.25310	−0.626548	+	0.779383i	$0.715533\pi$
$912$	0	0
$913$	−10.0978	−0.334187
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 15.9162i	− 0.525598i
$918$	0	0
$919$	38.7244i	1.27740i	0.769455	+	0.638701i	$0.220528\pi$
$919$	38.7244i	1.27740i	−0.769455	+	0.638701i	$0.779472\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.13311	0.0702121
$924$	0	0
$925$	15.5194	0.510275
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 11.9222i	− 0.391154i	−0.980688	−	0.195577i	$-0.937342\pi$
$929$	− 11.9222i	− 0.391154i	0.980688	−	0.195577i	$-0.0626580\pi$
$930$	0	0
$931$	− 6.20555i	− 0.203379i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15.9162	−0.520514
$936$	0	0
$937$	28.1744	0.920417	0.460208	−	0.887811i	$-0.347775\pi$
$937$	28.1744	0.920417	0.460208	+	0.887811i	$0.347775\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.9054i	1.75727i	0.477496	+	0.878634i	$0.341544\pi$
$941$	53.9054i	1.75727i	−0.477496	+	0.878634i	$0.658456\pi$
$942$	0	0
$943$	− 47.3622i	− 1.54233i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.7275	−1.19348	−0.596742	−	0.802433i	$-0.703538\pi$
$947$	−36.7275	−1.19348	−0.596742	+	0.802433i	$0.703538\pi$
$948$	0	0
$949$	8.11888	0.263550
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 58.9090i	− 1.90825i	−0.299408	−	0.954125i	$-0.596789\pi$
$953$	− 58.9090i	− 1.90825i	0.299408	−	0.954125i	$-0.403211\pi$
$954$	0	0
$955$	25.4600i	0.823865i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.55440	−0.276236
$960$	0	0
$961$	5.50885	0.177705
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 32.5120i	− 1.04660i
$966$	0	0
$967$	43.6061i	1.40228i	0.713025	+	0.701139i	$0.247325\pi$
$967$	43.6061i	1.40228i	−0.713025	+	0.701139i	$0.752675\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	59.7960	1.91895	0.959473	−	0.281801i	$-0.0909317\pi$
$971$	59.7960	1.91895	0.959473	+	0.281801i	$0.0909317\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 41.5794i	− 1.33024i	−0.746736	−	0.665121i	$-0.768380\pi$
$977$	− 41.5794i	− 1.33024i	0.746736	−	0.665121i	$-0.231620\pi$
$978$	0	0
$979$	− 6.54002i	− 0.209020i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.428934	0.0136809	0.00684043	−	0.999977i	$-0.497823\pi$
$983$	0.428934	0.0136809	0.00684043	+	0.999977i	$0.497823\pi$
$984$	0	0
$985$	4.37279	0.139329
$986$	0	0
$987$	0	0
$988$	0	0
$989$	85.8190i	2.72889i
$990$	0	0
$991$	31.5466i	1.00211i	0.865415	+	0.501056i	$0.167055\pi$
$991$	31.5466i	1.00211i	−0.865415	+	0.501056i	$0.832945\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	38.0706	1.20692
$996$	0	0
$997$	−34.7144	−1.09942	−0.549708	−	0.835357i	$-0.685261\pi$
$997$	−34.7144	−1.09942	−0.549708	+	0.835357i	$0.685261\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.h.h.575.3		12
3.2	odd	2	inner	4032.2.h.h.575.9		12
4.3	odd	2	inner	4032.2.h.h.575.4		12
8.3	odd	2		252.2.e.a.71.6	yes	12
8.5	even	2		252.2.e.a.71.8	yes	12
12.11	even	2	inner	4032.2.h.h.575.10		12
24.5	odd	2		252.2.e.a.71.5	✓	12
24.11	even	2		252.2.e.a.71.7	yes	12
56.13	odd	2		1764.2.e.g.1079.8		12
56.27	even	2		1764.2.e.g.1079.6		12
168.83	odd	2		1764.2.e.g.1079.7		12
168.125	even	2		1764.2.e.g.1079.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.e.a.71.5	✓	12	24.5	odd	2
252.2.e.a.71.6	yes	12	8.3	odd	2
252.2.e.a.71.7	yes	12	24.11	even	2
252.2.e.a.71.8	yes	12	8.5	even	2
1764.2.e.g.1079.5		12	168.125	even	2
1764.2.e.g.1079.6		12	56.27	even	2
1764.2.e.g.1079.7		12	168.83	odd	2
1764.2.e.g.1079.8		12	56.13	odd	2
4032.2.h.h.575.3		12	1.1	even	1	trivial
4032.2.h.h.575.4		12	4.3	odd	2	inner
4032.2.h.h.575.9		12	3.2	odd	2	inner
4032.2.h.h.575.10		12	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56483i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-2.56483i q^{5} -1.00000i q^{7} -1.15061 q^{11} +0.578337 q^{13} +5.39325i q^{17} +6.20555i q^{19} +7.62536 q^{23} -1.57834 q^{25} +1.41421i q^{29} +5.04888i q^{31} -2.56483 q^{35} -9.83276 q^{37} -6.21115i q^{41} +11.2544i q^{43} -11.0772 q^{47} -1.00000 q^{49} +4.53333i q^{53} +2.95112i q^{55} +4.83896 q^{59} -0.951124 q^{61} -1.48333i q^{65} -2.78389i q^{67} +3.68835 q^{71} +14.0383 q^{73} +1.15061i q^{77} +12.8816i q^{79} +8.77597 q^{83} +13.8328 q^{85} +5.68395i q^{89} -0.578337i q^{91} +15.9162 q^{95} -12.8816 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 12 q^{25} - 8 q^{37} - 12 q^{49} - 56 q^{61} + 56 q^{85}+O(q^{100}) \) 12 * q - 12 * q^25 - 8 * q^37 - 12 * q^49 - 56 * q^61 + 56 * q^85

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