LMFDB - Embedded newform 4032.3.d.o.449.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,3,Mod(449,4032)]

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([0, 0, 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("4032.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4032.d (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:	$109.864042590$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 44x^{10} + 719x^{8} + 5356x^{6} + 17809x^{4} + 20000x^{2} + 144 $ x^12 + 44x^10 + 719x^8 + 5356x^6 + 17809x^4 + 20000*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.8
Root			$-0.0851273i$ of defining polynomial
Character	$\chi$	$=$	4032.449
Dual form			4032.3.d.o.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.88654i q^{5} +2.64575 q^{7} +O(q^{10})$	$q+2.88654i q^{5} +2.64575 q^{7} +18.0578i q^{11} +5.20437 q^{13} +9.56562i q^{17} -2.80500 q^{19} -0.616004i q^{23} +16.6679 q^{25} +34.2440i q^{29} +12.0673 q^{31} +7.63707i q^{35} +19.4073 q^{37} -38.4511i q^{41} +35.0366 q^{43} +27.6215i q^{47} +7.00000 q^{49} +52.7429i q^{53} -52.1245 q^{55} -30.7587i q^{59} +56.2655 q^{61} +15.0226i q^{65} -32.9285 q^{67} +69.4826i q^{71} -143.199 q^{73} +47.7764i q^{77} +10.9392 q^{79} -2.13994i q^{83} -27.6115 q^{85} +1.32827i q^{89} +13.7695 q^{91} -8.09675i q^{95} -10.2059 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 16 q^{13} + 64 q^{19} - 124 q^{25} - 160 q^{31} - 56 q^{37} - 64 q^{43} + 84 q^{49} + 160 q^{55} - 104 q^{61} - 64 q^{67} - 64 q^{73} - 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) $ 12 * q - 16 * q^13 + 64 * q^19 - 124 * q^25 - 160 * q^31 - 56 * q^37 - 64 * q^43 + 84 * q^49 + 160 * q^55 - 104 * q^61 - 64 * q^67 - 64 * q^73 - 32 * q^79 + 184 * q^85 - 96 * q^97

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
$5$	2.88654i	0.577308i	0.957433	+	0.288654i	$0.0932077\pi$
$5$	2.88654i	0.577308i	−0.957433	+	0.288654i	$0.906792\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.0578i	1.64162i	0.571203	+	0.820809i	$0.306477\pi$
$11$	18.0578i	1.64162i	−0.571203	+	0.820809i	$0.693523\pi$
$12$	0	0
$13$	5.20437	0.400336	0.200168	−	0.979762i	$-0.435851\pi$
$13$	5.20437	0.400336	0.200168	+	0.979762i	$0.435851\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.56562i	0.562683i	0.959608	+	0.281342i	$0.0907794\pi$
$17$	9.56562i	0.562683i	−0.959608	+	0.281342i	$0.909221\pi$
$18$	0	0
$19$	−2.80500	−0.147632	−0.0738159	−	0.997272i	$-0.523518\pi$
$19$	−2.80500	−0.147632	−0.0738159	+	0.997272i	$0.523518\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 0.616004i	− 0.0267828i	−0.999910	−	0.0133914i	$-0.995737\pi$
$23$	− 0.616004i	− 0.0267828i	0.999910	−	0.0133914i	$-0.00426274\pi$
$24$	0	0
$25$	16.6679	0.666716
$26$	0	0
$27$	0	0
$28$	0	0
$29$	34.2440i	1.18083i	0.807101	+	0.590413i	$0.201035\pi$
$29$	34.2440i	1.18083i	−0.807101	+	0.590413i	$0.798965\pi$
$30$	0	0
$31$	12.0673	0.389266	0.194633	−	0.980876i	$-0.437648\pi$
$31$	12.0673	0.389266	0.194633	+	0.980876i	$0.437648\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.63707i	0.218202i
$36$	0	0
$37$	19.4073	0.524521	0.262261	−	0.964997i	$-0.415532\pi$
$37$	19.4073	0.524521	0.262261	+	0.964997i	$0.415532\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 38.4511i	− 0.937833i	−0.883243	−	0.468916i	$-0.844645\pi$
$41$	− 38.4511i	− 0.937833i	0.883243	−	0.468916i	$-0.155355\pi$
$42$	0	0
$43$	35.0366	0.814805	0.407403	−	0.913249i	$-0.366435\pi$
$43$	35.0366	0.814805	0.407403	+	0.913249i	$0.366435\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	27.6215i	0.587691i	0.955853	+	0.293845i	$0.0949350\pi$
$47$	27.6215i	0.587691i	−0.955853	+	0.293845i	$0.905065\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	52.7429i	0.995149i	0.867421	+	0.497574i	$0.165776\pi$
$53$	52.7429i	0.995149i	−0.867421	+	0.497574i	$0.834224\pi$
$54$	0	0
$55$	−52.1245	−0.947719
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 30.7587i	− 0.521334i	−0.965429	−	0.260667i	$-0.916058\pi$
$59$	− 30.7587i	− 0.521334i	0.965429	−	0.260667i	$-0.0839425\pi$
$60$	0	0
$61$	56.2655	0.922385	0.461192	−	0.887300i	$-0.347422\pi$
$61$	56.2655	0.922385	0.461192	+	0.887300i	$0.347422\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	15.0226i	0.231117i
$66$	0	0
$67$	−32.9285	−0.491469	−0.245735	−	0.969337i	$-0.579029\pi$
$67$	−32.9285	−0.491469	−0.245735	+	0.969337i	$0.579029\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	69.4826i	0.978628i	0.872108	+	0.489314i	$0.162753\pi$
$71$	69.4826i	0.978628i	−0.872108	+	0.489314i	$0.837247\pi$
$72$	0	0
$73$	−143.199	−1.96163	−0.980814	−	0.194944i	$-0.937548\pi$
$73$	−143.199	−1.96163	−0.980814	+	0.194944i	$0.937548\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	47.7764i	0.620473i
$78$	0	0
$79$	10.9392	0.138471	0.0692357	−	0.997600i	$-0.477944\pi$
$79$	10.9392	0.138471	0.0692357	+	0.997600i	$0.477944\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.13994i	− 0.0257824i	−0.999917	−	0.0128912i	$-0.995896\pi$
$83$	− 2.13994i	− 0.0257824i	0.999917	−	0.0128912i	$-0.00410351\pi$
$84$	0	0
$85$	−27.6115	−0.324841
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.32827i	0.0149244i	0.999972	+	0.00746219i	$0.00237531\pi$
$89$	1.32827i	0.0149244i	−0.999972	+	0.00746219i	$0.997625\pi$
$90$	0	0
$91$	13.7695	0.151313
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 8.09675i	− 0.0852290i
$96$	0	0
$97$	−10.2059	−0.105215	−0.0526077	−	0.998615i	$-0.516753\pi$
$97$	−10.2059	−0.105215	−0.0526077	+	0.998615i	$0.516753\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 143.295i	− 1.41876i	−0.704827	−	0.709379i	$-0.748975\pi$
$101$	− 143.295i	− 1.41876i	0.704827	−	0.709379i	$-0.251025\pi$
$102$	0	0
$103$	97.3583	0.945226	0.472613	−	0.881270i	$-0.343311\pi$
$103$	97.3583	0.945226	0.472613	+	0.881270i	$0.343311\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	154.655i	1.44537i	0.691177	+	0.722686i	$0.257093\pi$
$107$	154.655i	1.44537i	−0.691177	+	0.722686i	$0.742907\pi$
$108$	0	0
$109$	78.8509	0.723402	0.361701	−	0.932294i	$-0.382196\pi$
$109$	78.8509	0.723402	0.361701	+	0.932294i	$0.382196\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 123.583i	− 1.09365i	−0.837246	−	0.546826i	$-0.815836\pi$
$113$	− 123.583i	− 1.09365i	0.837246	−	0.546826i	$-0.184164\pi$
$114$	0	0
$115$	1.77812	0.0154619
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.3082i	0.212674i
$120$	0	0
$121$	−205.084	−1.69491
$122$	0	0
$123$	0	0
$124$	0	0
$125$	120.276i	0.962208i
$126$	0	0
$127$	3.50114	0.0275680	0.0137840	−	0.999905i	$-0.495612\pi$
$127$	3.50114	0.0275680	0.0137840	+	0.999905i	$0.495612\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	130.464i	0.995906i	0.867204	+	0.497953i	$0.165915\pi$
$131$	130.464i	0.995906i	−0.867204	+	0.497953i	$0.834085\pi$
$132$	0	0
$133$	−7.42134	−0.0557995
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 196.102i	− 1.43140i	−0.698409	−	0.715699i	$-0.746108\pi$
$137$	− 196.102i	− 1.43140i	0.698409	−	0.715699i	$-0.253892\pi$
$138$	0	0
$139$	−242.473	−1.74441	−0.872204	−	0.489143i	$-0.837310\pi$
$139$	−242.473	−1.74441	−0.872204	+	0.489143i	$0.837310\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	93.9795i	0.657199i
$144$	0	0
$145$	−98.8465	−0.681700
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 37.1840i	− 0.249557i	−0.992185	−	0.124778i	$-0.960178\pi$
$149$	− 37.1840i	− 0.249557i	0.992185	−	0.124778i	$-0.0398220\pi$
$150$	0	0
$151$	−146.217	−0.968323	−0.484161	−	0.874979i	$-0.660875\pi$
$151$	−146.217	−0.968323	−0.484161	+	0.874979i	$0.660875\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	34.8326i	0.224727i
$156$	0	0
$157$	233.455	1.48698	0.743489	−	0.668749i	$-0.233170\pi$
$157$	233.455	1.48698	0.743489	+	0.668749i	$0.233170\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.62979i	− 0.0101229i
$162$	0	0
$163$	158.798	0.974219	0.487110	−	0.873341i	$-0.338051\pi$
$163$	158.798	0.974219	0.487110	+	0.873341i	$0.338051\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 107.901i	− 0.646114i	−0.946380	−	0.323057i	$-0.895289\pi$
$167$	− 107.901i	− 0.646114i	0.946380	−	0.323057i	$-0.104711\pi$
$168$	0	0
$169$	−141.915	−0.839731
$170$	0	0
$171$	0	0
$172$	0	0
$173$	43.1821i	0.249608i	0.992181	+	0.124804i	$0.0398301\pi$
$173$	43.1821i	0.249608i	−0.992181	+	0.124804i	$0.960170\pi$
$174$	0	0
$175$	44.0991	0.251995
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 104.746i	− 0.585171i	−0.956239	−	0.292586i	$-0.905484\pi$
$179$	− 104.746i	− 0.585171i	0.956239	−	0.292586i	$-0.0945157\pi$
$180$	0	0
$181$	129.814	0.717204	0.358602	−	0.933491i	$-0.383254\pi$
$181$	129.814	0.717204	0.358602	+	0.933491i	$0.383254\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	56.0199i	0.302810i
$186$	0	0
$187$	−172.734	−0.923711
$188$	0	0
$189$	0	0
$190$	0	0
$191$	191.387i	1.00203i	0.865439	+	0.501014i	$0.167040\pi$
$191$	191.387i	1.00203i	−0.865439	+	0.501014i	$0.832960\pi$
$192$	0	0
$193$	−34.1271	−0.176825	−0.0884123	−	0.996084i	$-0.528179\pi$
$193$	−34.1271	−0.176825	−0.0884123	+	0.996084i	$0.528179\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	199.934i	1.01489i	0.861683	+	0.507447i	$0.169411\pi$
$197$	199.934i	1.01489i	−0.861683	+	0.507447i	$0.830589\pi$
$198$	0	0
$199$	−361.477	−1.81647	−0.908235	−	0.418461i	$-0.862570\pi$
$199$	−361.477	−1.81647	−0.908235	+	0.418461i	$0.862570\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	90.6010i	0.446310i
$204$	0	0
$205$	110.991	0.541418
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 50.6522i	− 0.242355i
$210$	0	0
$211$	26.2138	0.124236	0.0621179	−	0.998069i	$-0.480215\pi$
$211$	26.2138	0.124236	0.0621179	+	0.998069i	$0.480215\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	101.135i	0.470394i
$216$	0	0
$217$	31.9270	0.147129
$218$	0	0
$219$	0	0
$220$	0	0
$221$	49.7830i	0.225263i
$222$	0	0
$223$	−270.198	−1.21165	−0.605825	−	0.795598i	$-0.707157\pi$
$223$	−270.198	−1.21165	−0.605825	+	0.795598i	$0.707157\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	167.173i	0.736444i	0.929738	+	0.368222i	$0.120033\pi$
$227$	167.173i	0.736444i	−0.929738	+	0.368222i	$0.879967\pi$
$228$	0	0
$229$	270.416	1.18086	0.590428	−	0.807091i	$-0.298959\pi$
$229$	270.416	1.18086	0.590428	+	0.807091i	$0.298959\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 211.255i	− 0.906675i	−0.891339	−	0.453338i	$-0.850233\pi$
$233$	− 211.255i	− 0.906675i	0.891339	−	0.453338i	$-0.149767\pi$
$234$	0	0
$235$	−79.7304	−0.339278
$236$	0	0
$237$	0	0
$238$	0	0
$239$	452.287i	1.89241i	0.323563	+	0.946206i	$0.395119\pi$
$239$	452.287i	1.89241i	−0.323563	+	0.946206i	$0.604881\pi$
$240$	0	0
$241$	100.625	0.417530	0.208765	−	0.977966i	$-0.433056\pi$
$241$	100.625	0.417530	0.208765	+	0.977966i	$0.433056\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.2058i	0.0824726i
$246$	0	0
$247$	−14.5983	−0.0591023
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.251829i	0.00100330i	1.00000		0.000501652i	$0.000159681\pi$
$251$	0.251829i	0.00100330i	−1.00000		0.000501652i	$0.999840\pi$
$252$	0	0
$253$	11.1237	0.0439671
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 108.299i	− 0.421397i	−0.977551	−	0.210699i	$-0.932426\pi$
$257$	− 108.299i	− 0.421397i	0.977551	−	0.210699i	$-0.0675739\pi$
$258$	0	0
$259$	51.3468	0.198250
$260$	0	0
$261$	0	0
$262$	0	0
$263$	37.2101i	0.141483i	0.997495	+	0.0707417i	$0.0225366\pi$
$263$	37.2101i	0.141483i	−0.997495	+	0.0707417i	$0.977463\pi$
$264$	0	0
$265$	−152.244	−0.574507
$266$	0	0
$267$	0	0
$268$	0	0
$269$	462.280i	1.71851i	0.511546	+	0.859256i	$0.329073\pi$
$269$	462.280i	1.71851i	−0.511546	+	0.859256i	$0.670927\pi$
$270$	0	0
$271$	126.306	0.466073	0.233037	−	0.972468i	$-0.425134\pi$
$271$	126.306	0.466073	0.233037	+	0.972468i	$0.425134\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	300.985i	1.09449i
$276$	0	0
$277$	38.9274	0.140532	0.0702661	−	0.997528i	$-0.477615\pi$
$277$	38.9274	0.140532	0.0702661	+	0.997528i	$0.477615\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	39.8614i	0.141855i	0.997481	+	0.0709277i	$0.0225960\pi$
$281$	39.8614i	0.141855i	−0.997481	+	0.0709277i	$0.977404\pi$
$282$	0	0
$283$	−285.104	−1.00743	−0.503717	−	0.863868i	$-0.668035\pi$
$283$	−285.104	−1.00743	−0.503717	+	0.863868i	$0.668035\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 101.732i	− 0.354468i
$288$	0	0
$289$	197.499	0.683388
$290$	0	0
$291$	0	0
$292$	0	0
$293$	71.5890i	0.244331i	0.992510	+	0.122165i	$0.0389839\pi$
$293$	71.5890i	0.244331i	−0.992510	+	0.122165i	$0.961016\pi$
$294$	0	0
$295$	88.7862	0.300970
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.20591i	− 0.0107221i
$300$	0	0
$301$	92.6982	0.307967
$302$	0	0
$303$	0	0
$304$	0	0
$305$	162.413i	0.532500i
$306$	0	0
$307$	−531.353	−1.73079	−0.865396	−	0.501088i	$-0.832933\pi$
$307$	−531.353	−1.73079	−0.865396	+	0.501088i	$0.832933\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 96.5666i	− 0.310504i	−0.987875	−	0.155252i	$-0.950381\pi$
$311$	− 96.5666i	− 0.310504i	0.987875	−	0.155252i	$-0.0496189\pi$
$312$	0	0
$313$	260.857	0.833407	0.416704	−	0.909042i	$-0.363185\pi$
$313$	260.857	0.833407	0.416704	+	0.909042i	$0.363185\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 58.4994i	− 0.184541i	−0.995734	−	0.0922704i	$-0.970588\pi$
$317$	− 58.4994i	− 0.184541i	0.995734	−	0.0922704i	$-0.0294124\pi$
$318$	0	0
$319$	−618.370	−1.93847
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 26.8316i	− 0.0830699i
$324$	0	0
$325$	86.7459	0.266910
$326$	0	0
$327$	0	0
$328$	0	0
$329$	73.0795i	0.222126i
$330$	0	0
$331$	−250.683	−0.757351	−0.378675	−	0.925530i	$-0.623620\pi$
$331$	−250.683	−0.757351	−0.378675	+	0.925530i	$0.623620\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 95.0493i	− 0.283729i
$336$	0	0
$337$	−350.919	−1.04130	−0.520651	−	0.853769i	$-0.674311\pi$
$337$	−350.919	−1.04130	−0.520651	+	0.853769i	$0.674311\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	217.908i	0.639027i
$342$	0	0
$343$	18.5203	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	464.715i	1.33924i	0.742705	+	0.669618i	$0.233542\pi$
$347$	464.715i	1.33924i	−0.742705	+	0.669618i	$0.766458\pi$
$348$	0	0
$349$	−305.481	−0.875304	−0.437652	−	0.899144i	$-0.644190\pi$
$349$	−305.481	−0.875304	−0.437652	+	0.899144i	$0.644190\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 325.130i	− 0.921047i	−0.887647	−	0.460524i	$-0.847662\pi$
$353$	− 325.130i	− 0.921047i	0.887647	−	0.460524i	$-0.152338\pi$
$354$	0	0
$355$	−200.564	−0.564969
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 97.2891i	− 0.271000i	−0.990777	−	0.135500i	$-0.956736\pi$
$359$	− 97.2891i	− 0.271000i	0.990777	−	0.135500i	$-0.0432641\pi$
$360$	0	0
$361$	−353.132	−0.978205
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 413.349i	− 1.13246i
$366$	0	0
$367$	5.51516	0.0150277	0.00751385	−	0.999972i	$-0.497608\pi$
$367$	5.51516	0.0150277	0.00751385	+	0.999972i	$0.497608\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	139.545i	0.376131i
$372$	0	0
$373$	−142.221	−0.381290	−0.190645	−	0.981659i	$-0.561058\pi$
$373$	−142.221	−0.381290	−0.190645	+	0.981659i	$0.561058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	178.218i	0.472728i
$378$	0	0
$379$	−10.3654	−0.0273494	−0.0136747	−	0.999906i	$-0.504353\pi$
$379$	−10.3654	−0.0273494	−0.0136747	+	0.999906i	$0.504353\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	663.874i	1.73335i	0.498870	+	0.866677i	$0.333749\pi$
$383$	663.874i	1.73335i	−0.498870	+	0.866677i	$0.666251\pi$
$384$	0	0
$385$	−137.909	−0.358204
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 175.840i	− 0.452030i	−0.974124	−	0.226015i	$-0.927430\pi$
$389$	− 175.840i	− 0.452030i	0.974124	−	0.226015i	$-0.0725699\pi$
$390$	0	0
$391$	5.89246	0.0150702
$392$	0	0
$393$	0	0
$394$	0	0
$395$	31.5765i	0.0799406i
$396$	0	0
$397$	−295.433	−0.744163	−0.372081	−	0.928200i	$-0.621356\pi$
$397$	−295.433	−0.744163	−0.372081	+	0.928200i	$0.621356\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 442.722i	− 1.10405i	−0.833829	−	0.552023i	$-0.813856\pi$
$401$	− 442.722i	− 1.10405i	0.833829	−	0.552023i	$-0.186144\pi$
$402$	0	0
$403$	62.8025	0.155837
$404$	0	0
$405$	0	0
$406$	0	0
$407$	350.453i	0.861063i
$408$	0	0
$409$	701.640	1.71550	0.857751	−	0.514066i	$-0.171861\pi$
$409$	701.640	1.71550	0.857751	+	0.514066i	$0.171861\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 81.3799i	− 0.197046i
$414$	0	0
$415$	6.17701	0.0148844
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 79.8175i	− 0.190495i	−0.995454	−	0.0952477i	$-0.969636\pi$
$419$	− 79.8175i	− 0.190495i	0.995454	−	0.0952477i	$-0.0303643\pi$
$420$	0	0
$421$	34.2457	0.0813436	0.0406718	−	0.999173i	$-0.487050\pi$
$421$	34.2457	0.0813436	0.0406718	+	0.999173i	$0.487050\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	159.439i	0.375150i
$426$	0	0
$427$	148.864	0.348629
$428$	0	0
$429$	0	0
$430$	0	0
$431$	93.0878i	0.215981i	0.994152	+	0.107991i	$0.0344416\pi$
$431$	93.0878i	0.215981i	−0.994152	+	0.107991i	$0.965558\pi$
$432$	0	0
$433$	−752.553	−1.73800	−0.868998	−	0.494815i	$-0.835236\pi$
$433$	−752.553	−1.73800	−0.868998	+	0.494815i	$0.835236\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.72789i	0.00395399i
$438$	0	0
$439$	264.987	0.603616	0.301808	−	0.953369i	$-0.402410\pi$
$439$	264.987	0.603616	0.301808	+	0.953369i	$0.402410\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	238.731i	0.538896i	0.963015	+	0.269448i	$0.0868412\pi$
$443$	238.731i	0.538896i	−0.963015	+	0.269448i	$0.913159\pi$
$444$	0	0
$445$	−3.83411	−0.00861597
$446$	0	0
$447$	0	0
$448$	0	0
$449$	444.777i	0.990595i	0.868723	+	0.495297i	$0.164941\pi$
$449$	444.777i	0.990595i	−0.868723	+	0.495297i	$0.835059\pi$
$450$	0	0
$451$	694.343	1.53956
$452$	0	0
$453$	0	0
$454$	0	0
$455$	39.7461i	0.0873541i
$456$	0	0
$457$	−219.152	−0.479546	−0.239773	−	0.970829i	$-0.577073\pi$
$457$	−219.152	−0.479546	−0.239773	+	0.970829i	$0.577073\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 552.018i	− 1.19744i	−0.800960	−	0.598718i	$-0.795677\pi$
$461$	− 552.018i	− 1.19744i	0.800960	−	0.598718i	$-0.204323\pi$
$462$	0	0
$463$	−329.958	−0.712652	−0.356326	−	0.934362i	$-0.615971\pi$
$463$	−329.958	−0.712652	−0.356326	+	0.934362i	$0.615971\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 60.7577i	− 0.130102i	−0.997882	−	0.0650511i	$-0.979279\pi$
$467$	− 60.7577i	− 0.130102i	0.997882	−	0.0650511i	$-0.0207210\pi$
$468$	0	0
$469$	−87.1205	−0.185758
$470$	0	0
$471$	0	0
$472$	0	0
$473$	632.684i	1.33760i
$474$	0	0
$475$	−46.7535	−0.0984284
$476$	0	0
$477$	0	0
$478$	0	0
$479$	560.435i	1.17001i	0.811030	+	0.585005i	$0.198907\pi$
$479$	560.435i	1.17001i	−0.811030	+	0.585005i	$0.801093\pi$
$480$	0	0
$481$	101.003	0.209985
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 29.4597i	− 0.0607417i
$486$	0	0
$487$	31.4608	0.0646013	0.0323006	−	0.999478i	$-0.489717\pi$
$487$	31.4608	0.0646013	0.0323006	+	0.999478i	$0.489717\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	226.582i	0.461471i	0.973017	+	0.230735i	$0.0741132\pi$
$491$	226.582i	0.461471i	−0.973017	+	0.230735i	$0.925887\pi$
$492$	0	0
$493$	−327.564	−0.664431
$494$	0	0
$495$	0	0
$496$	0	0
$497$	183.834i	0.369886i
$498$	0	0
$499$	566.072	1.13441	0.567206	−	0.823576i	$-0.308024\pi$
$499$	566.072	1.13441	0.567206	+	0.823576i	$0.308024\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	507.115i	1.00818i	0.863651	+	0.504090i	$0.168172\pi$
$503$	507.115i	1.00818i	−0.863651	+	0.504090i	$0.831828\pi$
$504$	0	0
$505$	413.625	0.819060
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 361.354i	− 0.709929i	−0.934880	−	0.354964i	$-0.884493\pi$
$509$	− 361.354i	− 0.709929i	0.934880	−	0.354964i	$-0.115507\pi$
$510$	0	0
$511$	−378.869	−0.741426
$512$	0	0
$513$	0	0
$514$	0	0
$515$	281.029i	0.545687i
$516$	0	0
$517$	−498.783	−0.964763
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 401.146i	− 0.769955i	−0.922926	−	0.384977i	$-0.874209\pi$
$521$	− 401.146i	− 0.769955i	0.922926	−	0.384977i	$-0.125791\pi$
$522$	0	0
$523$	−960.181	−1.83591	−0.917955	−	0.396685i	$-0.870160\pi$
$523$	−960.181	−1.83591	−0.917955	+	0.396685i	$0.870160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	115.431i	0.219034i
$528$	0	0
$529$	528.621	0.999283
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 200.114i	− 0.375449i
$534$	0	0
$535$	−446.417	−0.834425
$536$	0	0
$537$	0	0
$538$	0	0
$539$	126.405i	0.234517i
$540$	0	0
$541$	−867.386	−1.60330	−0.801650	−	0.597793i	$-0.796044\pi$
$541$	−867.386	−1.60330	−0.801650	+	0.597793i	$0.796044\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	227.606i	0.417626i
$546$	0	0
$547$	−865.665	−1.58257	−0.791284	−	0.611449i	$-0.790587\pi$
$547$	−865.665	−1.58257	−0.791284	+	0.611449i	$0.790587\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 96.0544i	− 0.174327i
$552$	0	0
$553$	28.9425	0.0523372
$554$	0	0
$555$	0	0
$556$	0	0
$557$	338.801i	0.608260i	0.952631	+	0.304130i	$0.0983657\pi$
$557$	338.801i	0.608260i	−0.952631	+	0.304130i	$0.901634\pi$
$558$	0	0
$559$	182.344	0.326196
$560$	0	0
$561$	0	0
$562$	0	0
$563$	466.907i	0.829320i	0.909977	+	0.414660i	$0.136099\pi$
$563$	466.907i	0.829320i	−0.909977	+	0.414660i	$0.863901\pi$
$564$	0	0
$565$	356.726	0.631374
$566$	0	0
$567$	0	0
$568$	0	0
$569$	319.398i	0.561332i	0.959806	+	0.280666i	$0.0905552\pi$
$569$	319.398i	0.561332i	−0.959806	+	0.280666i	$0.909445\pi$
$570$	0	0
$571$	488.166	0.854931	0.427465	−	0.904032i	$-0.359407\pi$
$571$	488.166	0.854931	0.427465	+	0.904032i	$0.359407\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 10.2675i	− 0.0178565i
$576$	0	0
$577$	534.405	0.926178	0.463089	−	0.886312i	$-0.346741\pi$
$577$	534.405	0.926178	0.463089	+	0.886312i	$0.346741\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.66174i	− 0.00974482i
$582$	0	0
$583$	−952.420	−1.63365
$584$	0	0
$585$	0	0
$586$	0	0
$587$	611.025i	1.04093i	0.853884	+	0.520464i	$0.174241\pi$
$587$	611.025i	1.04093i	−0.853884	+	0.520464i	$0.825759\pi$
$588$	0	0
$589$	−33.8487	−0.0574681
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 240.712i	− 0.405922i	−0.979187	−	0.202961i	$-0.934944\pi$
$593$	− 240.712i	− 0.405922i	0.979187	−	0.202961i	$-0.0650564\pi$
$594$	0	0
$595$	−73.0532	−0.122779
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 390.995i	− 0.652747i	−0.945241	−	0.326373i	$-0.894173\pi$
$599$	− 390.995i	− 0.652747i	0.945241	−	0.326373i	$-0.105827\pi$
$600$	0	0
$601$	972.568	1.61825	0.809125	−	0.587637i	$-0.199942\pi$
$601$	972.568	1.61825	0.809125	+	0.587637i	$0.199942\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 591.983i	− 0.978485i
$606$	0	0
$607$	901.264	1.48478	0.742392	−	0.669965i	$-0.233691\pi$
$607$	901.264	1.48478	0.742392	+	0.669965i	$0.233691\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	143.752i	0.235274i
$612$	0	0
$613$	106.581	0.173868	0.0869338	−	0.996214i	$-0.472293\pi$
$613$	106.581	0.173868	0.0869338	+	0.996214i	$0.472293\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	323.494i	0.524301i	0.965027	+	0.262151i	$0.0844318\pi$
$617$	323.494i	0.524301i	−0.965027	+	0.262151i	$0.915568\pi$
$618$	0	0
$619$	−490.488	−0.792388	−0.396194	−	0.918167i	$-0.629669\pi$
$619$	−490.488	−0.792388	−0.396194	+	0.918167i	$0.629669\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.51427i	0.00564089i
$624$	0	0
$625$	69.5158	0.111225
$626$	0	0
$627$	0	0
$628$	0	0
$629$	185.643i	0.295139i
$630$	0	0
$631$	977.120	1.54853	0.774263	−	0.632864i	$-0.218121\pi$
$631$	977.120	1.54853	0.774263	+	0.632864i	$0.218121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.1062i	0.0159153i
$636$	0	0
$637$	36.4306	0.0571909
$638$	0	0
$639$	0	0
$640$	0	0
$641$	860.677i	1.34271i	0.741136	+	0.671355i	$0.234287\pi$
$641$	860.677i	1.34271i	−0.741136	+	0.671355i	$0.765713\pi$
$642$	0	0
$643$	544.835	0.847333	0.423666	−	0.905818i	$-0.360743\pi$
$643$	544.835	0.847333	0.423666	+	0.905818i	$0.360743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 858.937i	− 1.32757i	−0.747924	−	0.663784i	$-0.768949\pi$
$647$	− 858.937i	− 1.32757i	0.747924	−	0.663784i	$-0.231051\pi$
$648$	0	0
$649$	555.434	0.855831
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 936.003i	− 1.43339i	−0.697387	−	0.716694i	$-0.745654\pi$
$653$	− 936.003i	− 1.43339i	0.697387	−	0.716694i	$-0.254346\pi$
$654$	0	0
$655$	−376.589	−0.574944
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 815.909i	− 1.23810i	−0.785351	−	0.619051i	$-0.787517\pi$
$659$	− 815.909i	− 1.23810i	0.785351	−	0.619051i	$-0.212483\pi$
$660$	0	0
$661$	270.032	0.408520	0.204260	−	0.978917i	$-0.434521\pi$
$661$	270.032	0.408520	0.204260	+	0.978917i	$0.434521\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 21.4220i	− 0.0322135i
$666$	0	0
$667$	21.0944	0.0316258
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1016.03i	1.51420i
$672$	0	0
$673$	337.898	0.502078	0.251039	−	0.967977i	$-0.419228\pi$
$673$	337.898	0.502078	0.251039	+	0.967977i	$0.419228\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 135.693i	− 0.200433i	−0.994966	−	0.100216i	$-0.968046\pi$
$677$	− 135.693i	− 0.200433i	0.994966	−	0.100216i	$-0.0319535\pi$
$678$	0	0
$679$	−27.0023	−0.0397677
$680$	0	0
$681$	0	0
$682$	0	0
$683$	640.105i	0.937196i	0.883411	+	0.468598i	$0.155241\pi$
$683$	640.105i	0.937196i	−0.883411	+	0.468598i	$0.844759\pi$
$684$	0	0
$685$	566.055	0.826358
$686$	0	0
$687$	0	0
$688$	0	0
$689$	274.494i	0.398394i
$690$	0	0
$691$	−664.633	−0.961843	−0.480921	−	0.876764i	$-0.659698\pi$
$691$	−664.633	−0.961843	−0.480921	+	0.876764i	$0.659698\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 699.907i	− 1.00706i
$696$	0	0
$697$	367.809	0.527703
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1096.66i	1.56443i	0.623012	+	0.782213i	$0.285909\pi$
$701$	1096.66i	1.56443i	−0.623012	+	0.782213i	$0.714091\pi$
$702$	0	0
$703$	−54.4375	−0.0774359
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 379.122i	− 0.536240i
$708$	0	0
$709$	−208.277	−0.293762	−0.146881	−	0.989154i	$-0.546923\pi$
$709$	−208.277	−0.293762	−0.146881	+	0.989154i	$0.546923\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 7.43348i	− 0.0104256i
$714$	0	0
$715$	−271.276	−0.379406
$716$	0	0
$717$	0	0
$718$	0	0
$719$	983.038i	1.36723i	0.729843	+	0.683615i	$0.239593\pi$
$719$	983.038i	1.36723i	−0.729843	+	0.683615i	$0.760407\pi$
$720$	0	0
$721$	257.586	0.357262
$722$	0	0
$723$	0	0
$724$	0	0
$725$	570.774i	0.787275i
$726$	0	0
$727$	992.970	1.36585	0.682923	−	0.730490i	$-0.260708\pi$
$727$	992.970	1.36585	0.682923	+	0.730490i	$0.260708\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	335.147i	0.458477i
$732$	0	0
$733$	−684.369	−0.933655	−0.466828	−	0.884348i	$-0.654603\pi$
$733$	−684.369	−0.933655	−0.466828	+	0.884348i	$0.654603\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 594.615i	− 0.806805i
$738$	0	0
$739$	624.694	0.845323	0.422662	−	0.906288i	$-0.361096\pi$
$739$	624.694	0.845323	0.422662	+	0.906288i	$0.361096\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 79.5633i	− 0.107084i	−0.998566	−	0.0535419i	$-0.982949\pi$
$743$	− 79.5633i	− 0.107084i	0.998566	−	0.0535419i	$-0.0170511\pi$
$744$	0	0
$745$	107.333	0.144071
$746$	0	0
$747$	0	0
$748$	0	0
$749$	409.178i	0.546299i
$750$	0	0
$751$	−887.755	−1.18210	−0.591048	−	0.806636i	$-0.701286\pi$
$751$	−887.755	−1.18210	−0.591048	+	0.806636i	$0.701286\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 422.060i	− 0.559020i
$756$	0	0
$757$	−1292.59	−1.70752	−0.853762	−	0.520664i	$-0.825684\pi$
$757$	−1292.59	−1.70752	−0.853762	+	0.520664i	$0.825684\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	304.376i	0.399968i	0.979799	+	0.199984i	$0.0640891\pi$
$761$	304.376i	0.399968i	−0.979799	+	0.199984i	$0.935911\pi$
$762$	0	0
$763$	208.620	0.273420
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 160.080i	− 0.208709i
$768$	0	0
$769$	824.614	1.07232	0.536160	−	0.844116i	$-0.319874\pi$
$769$	824.614	1.07232	0.536160	+	0.844116i	$0.319874\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1388.95i	− 1.79683i	−0.439143	−	0.898417i	$-0.644718\pi$
$773$	− 1388.95i	− 1.79683i	0.439143	−	0.898417i	$-0.355282\pi$
$774$	0	0
$775$	201.136	0.259530
$776$	0	0
$777$	0	0
$778$	0	0
$779$	107.856i	0.138454i
$780$	0	0
$781$	−1254.70	−1.60653
$782$	0	0
$783$	0	0
$784$	0	0
$785$	673.878i	0.858444i
$786$	0	0
$787$	1105.44	1.40463	0.702313	−	0.711868i	$-0.252151\pi$
$787$	1105.44	1.40463	0.702313	+	0.711868i	$0.252151\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 326.969i	− 0.413362i
$792$	0	0
$793$	292.826	0.369264
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.5855i	− 0.0333570i	−0.999861	−	0.0166785i	$-0.994691\pi$
$797$	− 26.5855i	− 0.0333570i	0.999861	−	0.0166785i	$-0.00530917\pi$
$798$	0	0
$799$	−264.216	−0.330684
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2585.86i	− 3.22025i
$804$	0	0
$805$	4.70446	0.00584405
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 904.336i	− 1.11784i	−0.829220	−	0.558922i	$-0.811215\pi$
$809$	− 904.336i	− 1.11784i	0.829220	−	0.558922i	$-0.188785\pi$
$810$	0	0
$811$	715.294	0.881990	0.440995	−	0.897510i	$-0.354626\pi$
$811$	715.294	0.881990	0.440995	+	0.897510i	$0.354626\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	458.376i	0.562424i
$816$	0	0
$817$	−98.2778	−0.120291
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 702.656i	− 0.855853i	−0.903813	−	0.427927i	$-0.859244\pi$
$821$	− 702.656i	− 0.855853i	0.903813	−	0.427927i	$-0.140756\pi$
$822$	0	0
$823$	707.567	0.859741	0.429870	−	0.902891i	$-0.358559\pi$
$823$	707.567	0.859741	0.429870	+	0.902891i	$0.358559\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	230.216i	0.278375i	0.990266	+	0.139187i	$0.0444490\pi$
$827$	230.216i	0.278375i	−0.990266	+	0.139187i	$0.955551\pi$
$828$	0	0
$829$	1106.47	1.33470	0.667352	−	0.744743i	$-0.267428\pi$
$829$	1106.47	1.33470	0.667352	+	0.744743i	$0.267428\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	66.9593i	0.0803833i
$834$	0	0
$835$	311.461	0.373007
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.6182i	0.0126558i	0.999980	+	0.00632788i	$0.00201424\pi$
$839$	10.6182i	0.0126558i	−0.999980	+	0.00632788i	$0.997986\pi$
$840$	0	0
$841$	−331.649	−0.394350
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 409.642i	− 0.484783i
$846$	0	0
$847$	−542.601	−0.640616
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 11.9550i	− 0.0140481i
$852$	0	0
$853$	−1069.06	−1.25330	−0.626649	−	0.779301i	$-0.715574\pi$
$853$	−1069.06	−1.25330	−0.626649	+	0.779301i	$0.715574\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	160.927i	0.187780i	0.995583	+	0.0938898i	$0.0299301\pi$
$857$	160.927i	0.187780i	−0.995583	+	0.0938898i	$0.970070\pi$
$858$	0	0
$859$	−1415.71	−1.64810	−0.824048	−	0.566521i	$-0.808289\pi$
$859$	−1415.71	−1.64810	−0.824048	+	0.566521i	$0.808289\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 592.839i	− 0.686952i	−0.939161	−	0.343476i	$-0.888396\pi$
$863$	− 592.839i	− 0.686952i	0.939161	−	0.343476i	$-0.111604\pi$
$864$	0	0
$865$	−124.647	−0.144100
$866$	0	0
$867$	0	0
$868$	0	0
$869$	197.538i	0.227317i
$870$	0	0
$871$	−171.372	−0.196753
$872$	0	0
$873$	0	0
$874$	0	0
$875$	318.220i	0.363680i
$876$	0	0
$877$	807.074	0.920267	0.460134	−	0.887850i	$-0.347802\pi$
$877$	807.074	0.920267	0.460134	+	0.887850i	$0.347802\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 827.182i	− 0.938913i	−0.882956	−	0.469456i	$-0.844450\pi$
$881$	− 827.182i	− 0.938913i	0.882956	−	0.469456i	$-0.155550\pi$
$882$	0	0
$883$	−885.408	−1.00273	−0.501363	−	0.865237i	$-0.667168\pi$
$883$	−885.408	−1.00273	−0.501363	+	0.865237i	$0.667168\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1012.05i	1.14098i	0.821303	+	0.570492i	$0.193247\pi$
$887$	1012.05i	1.14098i	−0.821303	+	0.570492i	$0.806753\pi$
$888$	0	0
$889$	9.26315	0.0104197
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 77.4783i	− 0.0867618i
$894$	0	0
$895$	302.352	0.337824
$896$	0	0
$897$	0	0
$898$	0	0
$899$	413.231i	0.459656i
$900$	0	0
$901$	−504.518	−0.559953
$902$	0	0
$903$	0	0
$904$	0	0
$905$	374.713i	0.414048i
$906$	0	0
$907$	1071.42	1.18128	0.590639	−	0.806936i	$-0.298876\pi$
$907$	1071.42	1.18128	0.590639	+	0.806936i	$0.298876\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1217.59i	− 1.33654i	−0.743920	−	0.668269i	$-0.767035\pi$
$911$	− 1217.59i	− 1.33654i	0.743920	−	0.668269i	$-0.232965\pi$
$912$	0	0
$913$	38.6426	0.0423248
$914$	0	0
$915$	0	0
$916$	0	0
$917$	345.174i	0.376417i
$918$	0	0
$919$	−623.877	−0.678865	−0.339432	−	0.940630i	$-0.610235\pi$
$919$	−623.877	−0.678865	−0.339432	+	0.940630i	$0.610235\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	361.613i	0.391780i
$924$	0	0
$925$	323.478	0.349706
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1505.54i	1.62060i	0.586014	+	0.810301i	$0.300696\pi$
$929$	1505.54i	1.62060i	−0.586014	+	0.810301i	$0.699304\pi$
$930$	0	0
$931$	−19.6350	−0.0210902
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 498.603i	− 0.533266i
$936$	0	0
$937$	−1447.55	−1.54487	−0.772436	−	0.635093i	$-0.780962\pi$
$937$	−1447.55	−1.54487	−0.772436	+	0.635093i	$0.780962\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1283.40i	− 1.36387i	−0.731414	−	0.681933i	$-0.761139\pi$
$941$	− 1283.40i	− 1.36387i	0.731414	−	0.681933i	$-0.238861\pi$
$942$	0	0
$943$	−23.6861	−0.0251178
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.6890i	− 0.0176230i	−0.999961	−	0.00881150i	$-0.997195\pi$
$947$	− 16.6890i	− 0.0176230i	0.999961	−	0.00881150i	$-0.00280482\pi$
$948$	0	0
$949$	−745.260	−0.785311
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1171.67i	− 1.22946i	−0.788738	−	0.614729i	$-0.789265\pi$
$953$	− 1171.67i	− 1.22946i	0.788738	−	0.614729i	$-0.210735\pi$
$954$	0	0
$955$	−552.447	−0.578479
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 518.836i	− 0.541018i
$960$	0	0
$961$	−815.381	−0.848472
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 98.5093i	− 0.102082i
$966$	0	0
$967$	1103.34	1.14099	0.570496	−	0.821300i	$-0.306751\pi$
$967$	1103.34	1.14099	0.570496	+	0.821300i	$0.306751\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1059.56i	− 1.09120i	−0.838045	−	0.545602i	$-0.816301\pi$
$971$	− 1059.56i	− 1.09120i	0.838045	−	0.545602i	$-0.183699\pi$
$972$	0	0
$973$	−641.522	−0.659324
$974$	0	0
$975$	0	0
$976$	0	0
$977$	196.039i	0.200654i	0.994955	+	0.100327i	$0.0319889\pi$
$977$	196.039i	0.200654i	−0.994955	+	0.100327i	$0.968011\pi$
$978$	0	0
$979$	−23.9856	−0.0245001
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1428.15i	− 1.45285i	−0.687248	−	0.726423i	$-0.741181\pi$
$983$	− 1428.15i	− 1.45285i	0.687248	−	0.726423i	$-0.258819\pi$
$984$	0	0
$985$	−577.117	−0.585906
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 21.5827i	− 0.0218228i
$990$	0	0
$991$	−1800.34	−1.81669	−0.908347	−	0.418217i	$-0.862655\pi$
$991$	−1800.34	−1.81669	−0.908347	+	0.418217i	$0.862655\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1043.42i	− 1.04866i
$996$	0	0
$997$	1007.19	1.01022	0.505109	−	0.863056i	$-0.331452\pi$
$997$	1007.19	1.01022	0.505109	+	0.863056i	$0.331452\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.3.d.o.449.8		12
3.2	odd	2	inner	4032.3.d.o.449.5		12
4.3	odd	2		4032.3.d.n.449.8		12
8.3	odd	2		2016.3.d.f.449.5	yes	12
8.5	even	2		2016.3.d.e.449.5	✓	12
12.11	even	2		4032.3.d.n.449.5		12
24.5	odd	2		2016.3.d.e.449.8	yes	12
24.11	even	2		2016.3.d.f.449.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.3.d.e.449.5	✓	12	8.5	even	2
2016.3.d.e.449.8	yes	12	24.5	odd	2
2016.3.d.f.449.5	yes	12	8.3	odd	2
2016.3.d.f.449.8	yes	12	24.11	even	2
4032.3.d.n.449.5		12	12.11	even	2
4032.3.d.n.449.8		12	4.3	odd	2
4032.3.d.o.449.5		12	3.2	odd	2	inner
4032.3.d.o.449.8		12	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4032.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.88654i q^{5} +2.64575 q^{7} +O(q^{10})\)	\(q+2.88654i q^{5} +2.64575 q^{7} +18.0578i q^{11} +5.20437 q^{13} +9.56562i q^{17} -2.80500 q^{19} -0.616004i q^{23} +16.6679 q^{25} +34.2440i q^{29} +12.0673 q^{31} +7.63707i q^{35} +19.4073 q^{37} -38.4511i q^{41} +35.0366 q^{43} +27.6215i q^{47} +7.00000 q^{49} +52.7429i q^{53} -52.1245 q^{55} -30.7587i q^{59} +56.2655 q^{61} +15.0226i q^{65} -32.9285 q^{67} +69.4826i q^{71} -143.199 q^{73} +47.7764i q^{77} +10.9392 q^{79} -2.13994i q^{83} -27.6115 q^{85} +1.32827i q^{89} +13.7695 q^{91} -8.09675i q^{95} -10.2059 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 16 q^{13} + 64 q^{19} - 124 q^{25} - 160 q^{31} - 56 q^{37} - 64 q^{43} + 84 q^{49} + 160 q^{55} - 104 q^{61} - 64 q^{67} - 64 q^{73} - 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) \) 12 * q - 16 * q^13 + 64 * q^19 - 124 * q^25 - 160 * q^31 - 56 * q^37 - 64 * q^43 + 84 * q^49 + 160 * q^55 - 104 * q^61 - 64 * q^67 - 64 * q^73 - 32 * q^79 + 184 * q^85 - 96 * q^97

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