LMFDB - Embedded newform 1008.3.d.a.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(449,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	1008.449
Dual form			1008.3.d.a.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.89753i q^{5} -2.64575 q^{7} +O(q^{10})$	$q+8.89753i q^{5} -2.64575 q^{7} -17.7951i q^{11} +2.58301 q^{13} -25.8681i q^{17} -20.0000 q^{19} -17.7951i q^{23} -54.1660 q^{25} +11.9034i q^{29} +17.1660 q^{31} -23.5406i q^{35} +38.0000 q^{37} -15.7338i q^{41} +43.4980 q^{43} +16.9706i q^{47} +7.00000 q^{49} -85.5571i q^{53} +158.332 q^{55} -1.64899i q^{59} +100.332 q^{61} +22.9824i q^{65} -36.6640 q^{67} -17.7951i q^{71} +28.9150 q^{73} +47.0813i q^{77} -118.332 q^{79} +120.443i q^{83} +230.162 q^{85} -139.475i q^{89} -6.83399 q^{91} -177.951i q^{95} +44.4131 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 32 q^{13} - 80 q^{19} - 132 q^{25} - 16 q^{31} + 152 q^{37} - 80 q^{43} + 28 q^{49} + 464 q^{55} + 232 q^{61} + 192 q^{67} - 96 q^{73} - 304 q^{79} + 328 q^{85} - 112 q^{91} - 288 q^{97}+O(q^{100}) $ 4 * q - 32 * q^13 - 80 * q^19 - 132 * q^25 - 16 * q^31 + 152 * q^37 - 80 * q^43 + 28 * q^49 + 464 * q^55 + 232 * q^61 + 192 * q^67 - 96 * q^73 - 304 * q^79 + 328 * q^85 - 112 * q^91 - 288 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	8.89753i	1.77951i	0.456443	+	0.889753i	$0.349123\pi$
$5$	8.89753i	1.77951i	−0.456443	+	0.889753i	$0.650877\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 17.7951i	− 1.61773i	−0.587993	−	0.808866i	$-0.700082\pi$
$11$	− 17.7951i	− 1.61773i	0.587993	−	0.808866i	$-0.299918\pi$
$12$	0	0
$13$	2.58301	0.198693	0.0993464	−	0.995053i	$-0.468325\pi$
$13$	2.58301	0.198693	0.0993464	+	0.995053i	$0.468325\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 25.8681i	− 1.52165i	−0.648956	−	0.760826i	$-0.724794\pi$
$17$	− 25.8681i	− 1.52165i	0.648956	−	0.760826i	$-0.275206\pi$
$18$	0	0
$19$	−20.0000	−1.05263	−0.526316	−	0.850289i	$-0.676427\pi$
$19$	−20.0000	−1.05263	−0.526316	+	0.850289i	$0.676427\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 17.7951i	− 0.773698i	−0.922143	−	0.386849i	$-0.873563\pi$
$23$	− 17.7951i	− 0.773698i	0.922143	−	0.386849i	$-0.126437\pi$
$24$	0	0
$25$	−54.1660	−2.16664
$26$	0	0
$27$	0	0
$28$	0	0
$29$	11.9034i	0.410463i	0.978713	+	0.205232i	$0.0657947\pi$
$29$	11.9034i	0.410463i	−0.978713	+	0.205232i	$0.934205\pi$
$30$	0	0
$31$	17.1660	0.553742	0.276871	−	0.960907i	$-0.410702\pi$
$31$	17.1660	0.553742	0.276871	+	0.960907i	$0.410702\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 23.5406i	− 0.672590i
$36$	0	0
$37$	38.0000	1.02703	0.513514	−	0.858082i	$-0.328344\pi$
$37$	38.0000	1.02703	0.513514	+	0.858082i	$0.328344\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 15.7338i	− 0.383752i	−0.981419	−	0.191876i	$-0.938543\pi$
$41$	− 15.7338i	− 0.383752i	0.981419	−	0.191876i	$-0.0614571\pi$
$42$	0	0
$43$	43.4980	1.01158	0.505791	−	0.862656i	$-0.331201\pi$
$43$	43.4980	1.01158	0.505791	+	0.862656i	$0.331201\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16.9706i	0.361076i	0.983568	+	0.180538i	$0.0577838\pi$
$47$	16.9706i	0.361076i	−0.983568	+	0.180538i	$0.942216\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 85.5571i	− 1.61429i	−0.590356	−	0.807143i	$-0.701013\pi$
$53$	− 85.5571i	− 1.61429i	0.590356	−	0.807143i	$-0.298987\pi$
$54$	0	0
$55$	158.332	2.87876
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.64899i	− 0.0279489i	−0.999902	−	0.0139745i	$-0.995552\pi$
$59$	− 1.64899i	− 0.0279489i	0.999902	−	0.0139745i	$-0.00444836\pi$
$60$	0	0
$61$	100.332	1.64479	0.822394	−	0.568919i	$-0.192638\pi$
$61$	100.332	1.64479	0.822394	+	0.568919i	$0.192638\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	22.9824i	0.353575i
$66$	0	0
$67$	−36.6640	−0.547225	−0.273612	−	0.961840i	$-0.588219\pi$
$67$	−36.6640	−0.547225	−0.273612	+	0.961840i	$0.588219\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 17.7951i	− 0.250635i	−0.992117	−	0.125317i	$-0.960005\pi$
$71$	− 17.7951i	− 0.250635i	0.992117	−	0.125317i	$-0.0399949\pi$
$72$	0	0
$73$	28.9150	0.396096	0.198048	−	0.980192i	$-0.436540\pi$
$73$	28.9150	0.396096	0.198048	+	0.980192i	$0.436540\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	47.0813i	0.611445i
$78$	0	0
$79$	−118.332	−1.49787	−0.748937	−	0.662641i	$-0.769435\pi$
$79$	−118.332	−1.49787	−0.748937	+	0.662641i	$0.769435\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	120.443i	1.45112i	0.688159	+	0.725560i	$0.258419\pi$
$83$	120.443i	1.45112i	−0.688159	+	0.725560i	$0.741581\pi$
$84$	0	0
$85$	230.162	2.70779
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 139.475i	− 1.56713i	−0.621309	−	0.783566i	$-0.713399\pi$
$89$	− 139.475i	− 1.56713i	0.621309	−	0.783566i	$-0.286601\pi$
$90$	0	0
$91$	−6.83399	−0.0750988
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 177.951i	− 1.87316i
$96$	0	0
$97$	44.4131	0.457867	0.228933	−	0.973442i	$-0.426476\pi$
$97$	44.4131	0.457867	0.228933	+	0.973442i	$0.426476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	31.8799i	0.315642i	0.987468	+	0.157821i	$0.0504470\pi$
$101$	31.8799i	0.315642i	−0.987468	+	0.157821i	$0.949553\pi$
$102$	0	0
$103$	−4.50197	−0.0437084	−0.0218542	−	0.999761i	$-0.506957\pi$
$103$	−4.50197	−0.0437084	−0.0218542	+	0.999761i	$0.506957\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 172.179i	− 1.60915i	−0.593851	−	0.804575i	$-0.702393\pi$
$107$	− 172.179i	− 1.60915i	0.593851	−	0.804575i	$-0.297607\pi$
$108$	0	0
$109$	−177.830	−1.63147	−0.815734	−	0.578427i	$-0.803667\pi$
$109$	−177.830	−1.63147	−0.815734	+	0.578427i	$0.803667\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	31.3475i	0.277411i	0.990334	+	0.138706i	$0.0442942\pi$
$113$	31.3475i	0.277411i	−0.990334	+	0.138706i	$0.955706\pi$
$114$	0	0
$115$	158.332	1.37680
$116$	0	0
$117$	0	0
$118$	0	0
$119$	68.4405i	0.575131i
$120$	0	0
$121$	−195.664	−1.61706
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 259.505i	− 2.07604i
$126$	0	0
$127$	−214.332	−1.68765	−0.843827	−	0.536616i	$-0.819703\pi$
$127$	−214.332	−1.68765	−0.843827	+	0.536616i	$0.819703\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 91.4488i	− 0.698082i	−0.937107	−	0.349041i	$-0.886507\pi$
$131$	− 91.4488i	− 0.698082i	0.937107	−	0.349041i	$-0.113493\pi$
$132$	0	0
$133$	52.9150	0.397857
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 106.891i	− 0.780223i	−0.920768	−	0.390111i	$-0.872437\pi$
$137$	− 106.891i	− 0.780223i	0.920768	−	0.390111i	$-0.127563\pi$
$138$	0	0
$139$	121.328	0.872864	0.436432	−	0.899737i	$-0.356242\pi$
$139$	121.328	0.872864	0.436432	+	0.899737i	$0.356242\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 45.9647i	− 0.321432i
$144$	0	0
$145$	−105.911	−0.730421
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 17.6749i	− 0.118623i	−0.998240	−	0.0593117i	$-0.981109\pi$
$149$	− 17.6749i	− 0.118623i	0.998240	−	0.0593117i	$-0.0188906\pi$
$150$	0	0
$151$	50.8340	0.336649	0.168324	−	0.985732i	$-0.446164\pi$
$151$	50.8340	0.336649	0.168324	+	0.985732i	$0.446164\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	152.735i	0.985388i
$156$	0	0
$157$	68.9961	0.439465	0.219733	−	0.975560i	$-0.429481\pi$
$157$	68.9961	0.439465	0.219733	+	0.975560i	$0.429481\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	47.0813i	0.292430i
$162$	0	0
$163$	166.996	1.02452	0.512258	−	0.858832i	$-0.328809\pi$
$163$	166.996	1.02452	0.512258	+	0.858832i	$0.328809\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 120.443i	− 0.721215i	−0.932718	−	0.360608i	$-0.882569\pi$
$167$	− 120.443i	− 0.721215i	0.932718	−	0.360608i	$-0.117431\pi$
$168$	0	0
$169$	−162.328	−0.960521
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 91.8610i	− 0.530989i	−0.964112	−	0.265494i	$-0.914465\pi$
$173$	− 91.8610i	− 0.530989i	0.964112	−	0.265494i	$-0.0855351\pi$
$174$	0	0
$175$	143.310	0.818913
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 133.291i	− 0.744643i	−0.928104	−	0.372321i	$-0.878562\pi$
$179$	− 133.291i	− 0.744643i	0.928104	−	0.372321i	$-0.121438\pi$
$180$	0	0
$181$	83.0850	0.459033	0.229517	−	0.973305i	$-0.426286\pi$
$181$	83.0850	0.459033	0.229517	+	0.973305i	$0.426286\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	338.106i	1.82760i
$186$	0	0
$187$	−460.324	−2.46163
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 41.3616i	− 0.216553i	−0.994121	−	0.108276i	$-0.965467\pi$
$191$	− 41.3616i	− 0.216553i	0.994121	−	0.108276i	$-0.0345331\pi$
$192$	0	0
$193$	134.000	0.694301	0.347150	−	0.937810i	$-0.387149\pi$
$193$	134.000	0.694301	0.347150	+	0.937810i	$0.387149\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 68.8269i	− 0.349375i	−0.984624	−	0.174688i	$-0.944108\pi$
$197$	− 68.8269i	− 0.349375i	0.984624	−	0.174688i	$-0.0558916\pi$
$198$	0	0
$199$	278.494	1.39947	0.699734	−	0.714404i	$-0.253302\pi$
$199$	278.494	1.39947	0.699734	+	0.714404i	$0.253302\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 31.4935i	− 0.155140i
$204$	0	0
$205$	139.992	0.682888
$206$	0	0
$207$	0	0
$208$	0	0
$209$	355.901i	1.70288i
$210$	0	0
$211$	211.498	1.00236	0.501180	−	0.865343i	$-0.332899\pi$
$211$	211.498	1.00236	0.501180	+	0.865343i	$0.332899\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	387.025i	1.80012i
$216$	0	0
$217$	−45.4170	−0.209295
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 66.8174i	− 0.302341i
$222$	0	0
$223$	−222.494	−0.997731	−0.498866	−	0.866679i	$-0.666250\pi$
$223$	−222.494	−0.997731	−0.498866	+	0.866679i	$0.666250\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 101.823i	− 0.448561i	−0.974525	−	0.224281i	$-0.927997\pi$
$227$	− 101.823i	− 0.448561i	0.974525	−	0.224281i	$-0.0720032\pi$
$228$	0	0
$229$	−163.085	−0.712161	−0.356081	−	0.934455i	$-0.615887\pi$
$229$	−163.085	−0.712161	−0.356081	+	0.934455i	$0.615887\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	362.858i	1.55733i	0.627441	+	0.778664i	$0.284102\pi$
$233$	362.858i	1.55733i	−0.627441	+	0.778664i	$0.715898\pi$
$234$	0	0
$235$	−150.996	−0.642536
$236$	0	0
$237$	0	0
$238$	0	0
$239$	177.126i	0.741113i	0.928810	+	0.370557i	$0.120833\pi$
$239$	177.126i	0.741113i	−0.928810	+	0.370557i	$0.879167\pi$
$240$	0	0
$241$	152.753	0.633830	0.316915	−	0.948454i	$-0.397353\pi$
$241$	152.753	0.633830	0.316915	+	0.948454i	$0.397353\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	62.2827i	0.254215i
$246$	0	0
$247$	−51.6601	−0.209150
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 356.382i	− 1.41985i	−0.704278	−	0.709924i	$-0.748729\pi$
$251$	− 356.382i	− 1.41985i	0.704278	−	0.709924i	$-0.251271\pi$
$252$	0	0
$253$	−316.664	−1.25164
$254$	0	0
$255$	0	0
$256$	0	0
$257$	59.5689i	0.231786i	0.993262	+	0.115893i	$0.0369729\pi$
$257$	59.5689i	0.231786i	−0.993262	+	0.115893i	$0.963027\pi$
$258$	0	0
$259$	−100.539	−0.388180
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.42045i	0.0282146i	0.999900	+	0.0141073i	$0.00449065\pi$
$263$	7.42045i	0.0282146i	−0.999900	+	0.0141073i	$0.995509\pi$
$264$	0	0
$265$	761.247	2.87263
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 430.207i	− 1.59928i	−0.600477	−	0.799642i	$-0.705023\pi$
$269$	− 430.207i	− 1.59928i	0.600477	−	0.799642i	$-0.294977\pi$
$270$	0	0
$271$	41.1660	0.151904	0.0759520	−	0.997111i	$-0.475800\pi$
$271$	41.1660	0.151904	0.0759520	+	0.997111i	$0.475800\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	963.887i	3.50504i
$276$	0	0
$277$	32.0000	0.115523	0.0577617	−	0.998330i	$-0.481604\pi$
$277$	32.0000	0.115523	0.0577617	+	0.998330i	$0.481604\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.0907i	0.0608211i	0.999537	+	0.0304106i	$0.00968147\pi$
$281$	17.0907i	0.0608211i	−0.999537	+	0.0304106i	$0.990319\pi$
$282$	0	0
$283$	−439.660	−1.55357	−0.776785	−	0.629766i	$-0.783151\pi$
$283$	−439.660	−1.55357	−0.776785	+	0.629766i	$0.783151\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	41.6278i	0.145045i
$288$	0	0
$289$	−380.158	−1.31543
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 394.377i	− 1.34600i	−0.739644	−	0.672998i	$-0.765006\pi$
$293$	− 394.377i	− 1.34600i	0.739644	−	0.672998i	$-0.234994\pi$
$294$	0	0
$295$	14.6719	0.0497353
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 45.9647i	− 0.153728i
$300$	0	0
$301$	−115.085	−0.382342
$302$	0	0
$303$	0	0
$304$	0	0
$305$	892.707i	2.92691i
$306$	0	0
$307$	23.3360	0.0760129	0.0380064	−	0.999277i	$-0.487899\pi$
$307$	23.3360	0.0760129	0.0380064	+	0.999277i	$0.487899\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	527.256i	1.69536i	0.530511	+	0.847678i	$0.322000\pi$
$311$	527.256i	1.69536i	−0.530511	+	0.847678i	$0.678000\pi$
$312$	0	0
$313$	−295.328	−0.943540	−0.471770	−	0.881722i	$-0.656385\pi$
$313$	−295.328	−0.943540	−0.471770	+	0.881722i	$0.656385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	107.475i	0.339037i	0.985527	+	0.169518i	$0.0542212\pi$
$317$	107.475i	0.339037i	−0.985527	+	0.169518i	$0.945779\pi$
$318$	0	0
$319$	211.822	0.664019
$320$	0	0
$321$	0	0
$322$	0	0
$323$	517.362i	1.60174i
$324$	0	0
$325$	−139.911	−0.430496
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 44.8999i	− 0.136474i
$330$	0	0
$331$	273.490	0.826254	0.413127	−	0.910673i	$-0.364437\pi$
$331$	273.490	0.826254	0.413127	+	0.910673i	$0.364437\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 326.219i	− 0.973789i
$336$	0	0
$337$	−341.166	−1.01236	−0.506181	−	0.862427i	$-0.668943\pi$
$337$	−341.166	−1.01236	−0.506181	+	0.862427i	$0.668943\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 305.470i	− 0.895807i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 116.320i	− 0.335217i	−0.985854	−	0.167609i	$-0.946395\pi$
$347$	− 116.320i	− 0.335217i	0.985854	−	0.167609i	$-0.0536045\pi$
$348$	0	0
$349$	−158.324	−0.453651	−0.226825	−	0.973935i	$-0.572835\pi$
$349$	−158.324	−0.453651	−0.226825	+	0.973935i	$0.572835\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 230.339i	− 0.652519i	−0.945280	−	0.326260i	$-0.894212\pi$
$353$	− 230.339i	− 0.652519i	0.945280	−	0.326260i	$-0.105788\pi$
$354$	0	0
$355$	158.332	0.446006
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 171.698i	− 0.478269i	−0.970987	−	0.239134i	$-0.923136\pi$
$359$	− 171.698i	− 0.478269i	0.970987	−	0.239134i	$-0.0768636\pi$
$360$	0	0
$361$	39.0000	0.108033
$362$	0	0
$363$	0	0
$364$	0	0
$365$	257.272i	0.704856i
$366$	0	0
$367$	−517.490	−1.41005	−0.705027	−	0.709180i	$-0.749065\pi$
$367$	−517.490	−1.41005	−0.705027	+	0.709180i	$0.749065\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	226.363i	0.610143i
$372$	0	0
$373$	−233.336	−0.625566	−0.312783	−	0.949825i	$-0.601261\pi$
$373$	−233.336	−0.625566	−0.312783	+	0.949825i	$0.601261\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	30.7466i	0.0815560i
$378$	0	0
$379$	−441.166	−1.16403	−0.582013	−	0.813179i	$-0.697735\pi$
$379$	−441.166	−1.16403	−0.582013	+	0.813179i	$0.697735\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	213.060i	0.556292i	0.960539	+	0.278146i	$0.0897200\pi$
$383$	213.060i	0.556292i	−0.960539	+	0.278146i	$0.910280\pi$
$384$	0	0
$385$	−418.907	−1.08807
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 565.096i	− 1.45269i	−0.687331	−	0.726344i	$-0.741218\pi$
$389$	− 565.096i	− 1.45269i	0.687331	−	0.726344i	$-0.258782\pi$
$390$	0	0
$391$	−460.324	−1.17730
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1052.86i	− 2.66547i
$396$	0	0
$397$	498.324	1.25522	0.627612	−	0.778526i	$-0.284032\pi$
$397$	498.324	1.25522	0.627612	+	0.778526i	$0.284032\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 193.392i	− 0.482275i	−0.970491	−	0.241138i	$-0.922480\pi$
$401$	− 193.392i	− 0.482275i	0.970491	−	0.241138i	$-0.0775205\pi$
$402$	0	0
$403$	44.3399	0.110025
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 676.212i	− 1.66145i
$408$	0	0
$409$	−454.243	−1.11062	−0.555309	−	0.831644i	$-0.687400\pi$
$409$	−454.243	−1.11062	−0.555309	+	0.831644i	$0.687400\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.36281i	0.0105637i
$414$	0	0
$415$	−1071.64	−2.58228
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 339.411i	− 0.810051i	−0.914305	−	0.405025i	$-0.867263\pi$
$419$	− 339.411i	− 0.810051i	0.914305	−	0.405025i	$-0.132737\pi$
$420$	0	0
$421$	247.320	0.587459	0.293729	−	0.955889i	$-0.405104\pi$
$421$	247.320	0.587459	0.293729	+	0.955889i	$0.405104\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1401.17i	3.29687i
$426$	0	0
$427$	−265.454	−0.621671
$428$	0	0
$429$	0	0
$430$	0	0
$431$	456.419i	1.05898i	0.848317	+	0.529489i	$0.177616\pi$
$431$	456.419i	1.05898i	−0.848317	+	0.529489i	$0.822384\pi$
$432$	0	0
$433$	−637.984	−1.47340	−0.736702	−	0.676217i	$-0.763618\pi$
$433$	−637.984	−1.47340	−0.736702	+	0.676217i	$0.763618\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	355.901i	0.814419i
$438$	0	0
$439$	784.146	1.78621	0.893105	−	0.449848i	$-0.148522\pi$
$439$	784.146	1.78621	0.893105	+	0.449848i	$0.148522\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	472.222i	1.06596i	0.846127	+	0.532981i	$0.178928\pi$
$443$	472.222i	1.06596i	−0.846127	+	0.532981i	$0.821072\pi$
$444$	0	0
$445$	1240.98	2.78872
$446$	0	0
$447$	0	0
$448$	0	0
$449$	739.852i	1.64778i	0.566752	+	0.823888i	$0.308200\pi$
$449$	739.852i	1.64778i	−0.566752	+	0.823888i	$0.691800\pi$
$450$	0	0
$451$	−279.984	−0.620808
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 60.8056i	− 0.133639i
$456$	0	0
$457$	−248.324	−0.543379	−0.271689	−	0.962385i	$-0.587582\pi$
$457$	−248.324	−0.543379	−0.271689	+	0.962385i	$0.587582\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 355.970i	− 0.772168i	−0.922464	−	0.386084i	$-0.873827\pi$
$461$	− 355.970i	− 0.772168i	0.922464	−	0.386084i	$-0.126173\pi$
$462$	0	0
$463$	6.33202	0.0136761	0.00683804	−	0.999977i	$-0.497823\pi$
$463$	6.33202	0.0136761	0.00683804	+	0.999977i	$0.497823\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	878.691i	1.88156i	0.339011	+	0.940782i	$0.389907\pi$
$467$	878.691i	1.88156i	−0.339011	+	0.940782i	$0.610093\pi$
$468$	0	0
$469$	97.0039	0.206831
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 774.050i	− 1.63647i
$474$	0	0
$475$	1083.32	2.28067
$476$	0	0
$477$	0	0
$478$	0	0
$479$	224.396i	0.468468i	0.972180	+	0.234234i	$0.0752581\pi$
$479$	224.396i	0.468468i	−0.972180	+	0.234234i	$0.924742\pi$
$480$	0	0
$481$	98.1542	0.204063
$482$	0	0
$483$	0	0
$484$	0	0
$485$	395.166i	0.814776i
$486$	0	0
$487$	717.490	1.47329	0.736643	−	0.676282i	$-0.236410\pi$
$487$	717.490	1.47329	0.736643	+	0.676282i	$0.236410\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	274.002i	0.558050i	0.960284	+	0.279025i	$0.0900112\pi$
$491$	274.002i	0.558050i	−0.960284	+	0.279025i	$0.909989\pi$
$492$	0	0
$493$	307.919	0.624582
$494$	0	0
$495$	0	0
$496$	0	0
$497$	47.0813i	0.0947310i
$498$	0	0
$499$	728.810	1.46054	0.730271	−	0.683158i	$-0.239394\pi$
$499$	728.810	1.46054	0.730271	+	0.683158i	$0.239394\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 594.657i	− 1.18222i	−0.806590	−	0.591111i	$-0.798690\pi$
$503$	− 594.657i	− 1.18222i	0.806590	−	0.591111i	$-0.201310\pi$
$504$	0	0
$505$	−283.652	−0.561688
$506$	0	0
$507$	0	0
$508$	0	0
$509$	994.015i	1.95288i	0.215795	+	0.976439i	$0.430766\pi$
$509$	994.015i	1.95288i	−0.215795	+	0.976439i	$0.569234\pi$
$510$	0	0
$511$	−76.5020	−0.149710
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 40.0564i	− 0.0777794i
$516$	0	0
$517$	301.992	0.584124
$518$	0	0
$519$	0	0
$520$	0	0
$521$	40.8459i	0.0783990i	0.999231	+	0.0391995i	$0.0124808\pi$
$521$	40.8459i	0.0783990i	−0.999231	+	0.0391995i	$0.987519\pi$
$522$	0	0
$523$	232.000	0.443595	0.221797	−	0.975093i	$-0.428808\pi$
$523$	232.000	0.443595	0.221797	+	0.975093i	$0.428808\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 444.052i	− 0.842603i
$528$	0	0
$529$	212.336	0.401391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 40.6405i	− 0.0762487i
$534$	0	0
$535$	1531.97	2.86349
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 124.565i	− 0.231105i
$540$	0	0
$541$	250.332	0.462721	0.231360	−	0.972868i	$-0.425682\pi$
$541$	250.332	0.462721	0.231360	+	0.972868i	$0.425682\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1582.25i	− 2.90321i
$546$	0	0
$547$	−888.324	−1.62399	−0.811996	−	0.583662i	$-0.801619\pi$
$547$	−888.324	−1.62399	−0.811996	+	0.583662i	$0.801619\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 238.069i	− 0.432066i
$552$	0	0
$553$	313.077	0.566143
$554$	0	0
$555$	0	0
$556$	0	0
$557$	316.309i	0.567879i	0.958842	+	0.283940i	$0.0916415\pi$
$557$	316.309i	0.567879i	−0.958842	+	0.283940i	$0.908358\pi$
$558$	0	0
$559$	112.356	0.200994
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 58.4690i	− 0.103853i	−0.998651	−	0.0519263i	$-0.983464\pi$
$563$	− 58.4690i	− 0.103853i	0.998651	−	0.0519263i	$-0.0165361\pi$
$564$	0	0
$565$	−278.915	−0.493655
$566$	0	0
$567$	0	0
$568$	0	0
$569$	221.665i	0.389570i	0.980846	+	0.194785i	$0.0624009\pi$
$569$	221.665i	0.389570i	−0.980846	+	0.194785i	$0.937599\pi$
$570$	0	0
$571$	487.644	0.854018	0.427009	−	0.904247i	$-0.359567\pi$
$571$	487.644	0.854018	0.427009	+	0.904247i	$0.359567\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	963.887i	1.67633i
$576$	0	0
$577$	−487.328	−0.844589	−0.422295	−	0.906459i	$-0.638775\pi$
$577$	−487.328	−0.844589	−0.422295	+	0.906459i	$0.638775\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 318.662i	− 0.548472i
$582$	0	0
$583$	−1522.49	−2.61148
$584$	0	0
$585$	0	0
$586$	0	0
$587$	445.701i	0.759286i	0.925133	+	0.379643i	$0.123953\pi$
$587$	445.701i	0.759286i	−0.925133	+	0.379643i	$0.876047\pi$
$588$	0	0
$589$	−343.320	−0.582887
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 276.648i	− 0.466523i	−0.972414	−	0.233261i	$-0.925060\pi$
$593$	− 276.648i	− 0.466523i	0.972414	−	0.233261i	$-0.0749397\pi$
$594$	0	0
$595$	−608.952	−1.02345
$596$	0	0
$597$	0	0
$598$	0	0
$599$	82.3793i	0.137528i	0.997633	+	0.0687641i	$0.0219056\pi$
$599$	82.3793i	0.137528i	−0.997633	+	0.0687641i	$0.978094\pi$
$600$	0	0
$601$	−418.000	−0.695507	−0.347754	−	0.937586i	$-0.613055\pi$
$601$	−418.000	−0.695507	−0.347754	+	0.937586i	$0.613055\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1740.93i	− 2.87756i
$606$	0	0
$607$	−76.8419	−0.126593	−0.0632964	−	0.997995i	$-0.520161\pi$
$607$	−76.8419	−0.126593	−0.0632964	+	0.997995i	$0.520161\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	43.8351i	0.0717431i
$612$	0	0
$613$	59.3281	0.0967832	0.0483916	−	0.998828i	$-0.484590\pi$
$613$	59.3281	0.0967832	0.0483916	+	0.998828i	$0.484590\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.1143i	0.0471869i	0.999722	+	0.0235935i	$0.00751073\pi$
$617$	29.1143i	0.0471869i	−0.999722	+	0.0235935i	$0.992489\pi$
$618$	0	0
$619$	−455.644	−0.736098	−0.368049	−	0.929806i	$-0.619974\pi$
$619$	−455.644	−0.736098	−0.368049	+	0.929806i	$0.619974\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	369.015i	0.592320i
$624$	0	0
$625$	954.806	1.52769
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 982.987i	− 1.56278i
$630$	0	0
$631$	45.0039	0.0713216	0.0356608	−	0.999364i	$-0.488646\pi$
$631$	45.0039	0.0713216	0.0356608	+	0.999364i	$0.488646\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1907.03i	− 3.00319i
$636$	0	0
$637$	18.0810	0.0283847
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 641.223i	− 1.00035i	−0.865925	−	0.500174i	$-0.833269\pi$
$641$	− 641.223i	− 1.00035i	0.865925	−	0.500174i	$-0.166731\pi$
$642$	0	0
$643$	604.000	0.939347	0.469673	−	0.882840i	$-0.344372\pi$
$643$	604.000	0.939347	0.469673	+	0.882840i	$0.344372\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	179.600i	0.277588i	0.990321	+	0.138794i	$0.0443226\pi$
$647$	179.600i	0.277588i	−0.990321	+	0.138794i	$0.955677\pi$
$648$	0	0
$649$	−29.3438	−0.0452139
$650$	0	0
$651$	0	0
$652$	0	0
$653$	392.092i	0.600447i	0.953869	+	0.300224i	$0.0970613\pi$
$653$	392.092i	0.600447i	−0.953869	+	0.300224i	$0.902939\pi$
$654$	0	0
$655$	813.668	1.24224
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1266.54i	− 1.92191i	−0.276701	−	0.960956i	$-0.589241\pi$
$659$	− 1266.54i	− 1.92191i	0.276701	−	0.960956i	$-0.410759\pi$
$660$	0	0
$661$	917.644	1.38827	0.694133	−	0.719846i	$-0.255788\pi$
$661$	917.644	1.38827	0.694133	+	0.719846i	$0.255788\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	470.813i	0.707989i
$666$	0	0
$667$	211.822	0.317574
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1785.41i	− 2.66083i
$672$	0	0
$673$	−152.008	−0.225866	−0.112933	−	0.993603i	$-0.536025\pi$
$673$	−152.008	−0.225866	−0.112933	+	0.993603i	$0.536025\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	163.178i	0.241031i	0.992711	+	0.120516i	$0.0384548\pi$
$677$	163.178i	0.241031i	−0.992711	+	0.120516i	$0.961545\pi$
$678$	0	0
$679$	−117.506	−0.173057
$680$	0	0
$681$	0	0
$682$	0	0
$683$	324.914i	0.475716i	0.971300	+	0.237858i	$0.0764453\pi$
$683$	324.914i	0.475716i	−0.971300	+	0.237858i	$0.923555\pi$
$684$	0	0
$685$	951.061	1.38841
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 220.995i	− 0.320747i
$690$	0	0
$691$	−1218.98	−1.76408	−0.882041	−	0.471173i	$-0.843831\pi$
$691$	−1218.98	−1.76408	−0.882041	+	0.471173i	$0.843831\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1079.52i	1.55327i
$696$	0	0
$697$	−407.004	−0.583937
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 427.202i	− 0.609417i	−0.952446	−	0.304709i	$-0.901441\pi$
$701$	− 427.202i	− 0.609417i	0.952446	−	0.304709i	$-0.0985591\pi$
$702$	0	0
$703$	−760.000	−1.08108
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 84.3463i	− 0.119302i
$708$	0	0
$709$	71.4980	0.100843	0.0504217	−	0.998728i	$-0.483943\pi$
$709$	71.4980	0.100843	0.0504217	+	0.998728i	$0.483943\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 305.470i	− 0.428429i
$714$	0	0
$715$	408.972	0.571989
$716$	0	0
$717$	0	0
$718$	0	0
$719$	111.030i	0.154422i	0.997015	+	0.0772112i	$0.0246016\pi$
$719$	111.030i	0.154422i	−0.997015	+	0.0772112i	$0.975398\pi$
$720$	0	0
$721$	11.9111	0.0165202
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 644.761i	− 0.889326i
$726$	0	0
$727$	1338.82	1.84157	0.920783	−	0.390076i	$-0.127551\pi$
$727$	1338.82	1.84157	0.920783	+	0.390076i	$0.127551\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1125.21i	− 1.53928i
$732$	0	0
$733$	49.0771	0.0669538	0.0334769	−	0.999439i	$-0.489342\pi$
$733$	49.0771	0.0669538	0.0334769	+	0.999439i	$0.489342\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	652.439i	0.885263i
$738$	0	0
$739$	−1430.32	−1.93548	−0.967738	−	0.251960i	$-0.918925\pi$
$739$	−1430.32	−1.93548	−0.967738	+	0.251960i	$0.918925\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 875.736i	− 1.17865i	−0.807896	−	0.589325i	$-0.799394\pi$
$743$	− 875.736i	− 1.17865i	0.807896	−	0.589325i	$-0.200606\pi$
$744$	0	0
$745$	157.263	0.211091
$746$	0	0
$747$	0	0
$748$	0	0
$749$	455.543i	0.608202i
$750$	0	0
$751$	−320.826	−0.427199	−0.213599	−	0.976921i	$-0.568519\pi$
$751$	−320.826	−0.427199	−0.213599	+	0.976921i	$0.568519\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	452.297i	0.599069i
$756$	0	0
$757$	289.830	0.382867	0.191433	−	0.981506i	$-0.438686\pi$
$757$	289.830	0.382867	0.191433	+	0.981506i	$0.438686\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 704.657i	− 0.925962i	−0.886368	−	0.462981i	$-0.846780\pi$
$761$	− 704.657i	− 0.925962i	0.886368	−	0.462981i	$-0.153220\pi$
$762$	0	0
$763$	470.494	0.616637
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.25934i	− 0.00555325i
$768$	0	0
$769$	−117.320	−0.152562	−0.0762810	−	0.997086i	$-0.524305\pi$
$769$	−117.320	−0.152562	−0.0762810	+	0.997086i	$0.524305\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	658.005i	0.851235i	0.904903	+	0.425618i	$0.139943\pi$
$773$	658.005i	0.851235i	−0.904903	+	0.425618i	$0.860057\pi$
$774$	0	0
$775$	−929.814	−1.19976
$776$	0	0
$777$	0	0
$778$	0	0
$779$	314.676i	0.403949i
$780$	0	0
$781$	−316.664	−0.405460
$782$	0	0
$783$	0	0
$784$	0	0
$785$	613.894i	0.782031i
$786$	0	0
$787$	1200.65	1.52560	0.762801	−	0.646634i	$-0.223824\pi$
$787$	1200.65	1.52560	0.762801	+	0.646634i	$0.223824\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 82.9376i	− 0.104852i
$792$	0	0
$793$	259.158	0.326807
$794$	0	0
$795$	0	0
$796$	0	0
$797$	797.411i	1.00052i	0.865876	+	0.500258i	$0.166761\pi$
$797$	797.411i	1.00052i	−0.865876	+	0.500258i	$0.833239\pi$
$798$	0	0
$799$	438.996	0.549432
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 514.545i	− 0.640778i
$804$	0	0
$805$	−418.907	−0.520382
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 156.016i	− 0.192851i	−0.995340	−	0.0964254i	$-0.969259\pi$
$809$	− 156.016i	− 0.192851i	0.995340	−	0.0964254i	$-0.0307409\pi$
$810$	0	0
$811$	598.316	0.737751	0.368876	−	0.929479i	$-0.379743\pi$
$811$	598.316	0.737751	0.368876	+	0.929479i	$0.379743\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1485.85i	1.82313i
$816$	0	0
$817$	−869.961	−1.06482
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 980.978i	− 1.19486i	−0.801922	−	0.597429i	$-0.796189\pi$
$821$	− 980.978i	− 1.19486i	0.801922	−	0.597429i	$-0.203811\pi$
$822$	0	0
$823$	431.336	0.524102	0.262051	−	0.965054i	$-0.415601\pi$
$823$	431.336	0.524102	0.262051	+	0.965054i	$0.415601\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1219.41i	1.47449i	0.675623	+	0.737247i	$0.263875\pi$
$827$	1219.41i	1.47449i	−0.675623	+	0.737247i	$0.736125\pi$
$828$	0	0
$829$	−770.081	−0.928928	−0.464464	−	0.885592i	$-0.653753\pi$
$829$	−770.081	−0.928928	−0.464464	+	0.885592i	$0.653753\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 181.077i	− 0.217379i
$834$	0	0
$835$	1071.64	1.28341
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1310.03i	− 1.56142i	−0.624894	−	0.780710i	$-0.714858\pi$
$839$	− 1310.03i	− 1.56142i	0.624894	−	0.780710i	$-0.285142\pi$
$840$	0	0
$841$	699.308	0.831520
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1444.32i	− 1.70925i
$846$	0	0
$847$	517.678	0.611191
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 676.212i	− 0.794609i
$852$	0	0
$853$	898.988	1.05391	0.526957	−	0.849892i	$-0.323333\pi$
$853$	898.988	1.05391	0.526957	+	0.849892i	$0.323333\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 746.156i	− 0.870660i	−0.900271	−	0.435330i	$-0.856632\pi$
$857$	− 746.156i	− 0.870660i	0.900271	−	0.435330i	$-0.143368\pi$
$858$	0	0
$859$	991.984	1.15481	0.577406	−	0.816457i	$-0.304065\pi$
$859$	991.984	1.15481	0.577406	+	0.816457i	$0.304065\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	209.418i	0.242663i	0.992612	+	0.121332i	$0.0387164\pi$
$863$	209.418i	0.242663i	−0.992612	+	0.121332i	$0.961284\pi$
$864$	0	0
$865$	817.336	0.944897
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2105.73i	2.42316i
$870$	0	0
$871$	−94.7034	−0.108730
$872$	0	0
$873$	0	0
$874$	0	0
$875$	686.587i	0.784671i
$876$	0	0
$877$	−865.304	−0.986664	−0.493332	−	0.869841i	$-0.664221\pi$
$877$	−865.304	−0.986664	−0.493332	+	0.869841i	$0.664221\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 995.046i	− 1.12945i	−0.825279	−	0.564725i	$-0.808982\pi$
$881$	− 995.046i	− 1.12945i	0.825279	−	0.564725i	$-0.191018\pi$
$882$	0	0
$883$	−101.474	−0.114920	−0.0574600	−	0.998348i	$-0.518300\pi$
$883$	−101.474	−0.114920	−0.0574600	+	0.998348i	$0.518300\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1074.09i	1.21093i	0.795873	+	0.605464i	$0.207012\pi$
$887$	1074.09i	1.21093i	−0.795873	+	0.605464i	$0.792988\pi$
$888$	0	0
$889$	567.069	0.637873
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 339.411i	− 0.380080i
$894$	0	0
$895$	1185.96	1.32510
$896$	0	0
$897$	0	0
$898$	0	0
$899$	204.334i	0.227291i
$900$	0	0
$901$	−2213.20	−2.45638
$902$	0	0
$903$	0	0
$904$	0	0
$905$	739.251i	0.816852i
$906$	0	0
$907$	432.162	0.476474	0.238237	−	0.971207i	$-0.423430\pi$
$907$	432.162	0.476474	0.238237	+	0.971207i	$0.423430\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 104.984i	− 0.115241i	−0.998339	−	0.0576205i	$-0.981649\pi$
$911$	− 104.984i	− 0.115241i	0.998339	−	0.0576205i	$-0.0183513\pi$
$912$	0	0
$913$	2143.29	2.34752
$914$	0	0
$915$	0	0
$916$	0	0
$917$	241.951i	0.263850i
$918$	0	0
$919$	91.8379	0.0999325	0.0499662	−	0.998751i	$-0.484089\pi$
$919$	91.8379	0.0999325	0.0499662	+	0.998751i	$0.484089\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 45.9647i	− 0.0497993i
$924$	0	0
$925$	−2058.31	−2.22520
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1309.00i	1.40904i	0.709682	+	0.704522i	$0.248838\pi$
$929$	1309.00i	1.40904i	−0.709682	+	0.704522i	$0.751162\pi$
$930$	0	0
$931$	−140.000	−0.150376
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 4095.75i	− 4.38048i
$936$	0	0
$937$	1262.00	1.34685	0.673426	−	0.739255i	$-0.264822\pi$
$937$	1262.00	1.34685	0.673426	+	0.739255i	$0.264822\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1315.97i	1.39849i	0.714884	+	0.699243i	$0.246479\pi$
$941$	1315.97i	1.39849i	−0.714884	+	0.699243i	$0.753521\pi$
$942$	0	0
$943$	−279.984	−0.296908
$944$	0	0
$945$	0	0
$946$	0	0
$947$	486.582i	0.513814i	0.966436	+	0.256907i	$0.0827034\pi$
$947$	486.582i	0.513814i	−0.966436	+	0.256907i	$0.917297\pi$
$948$	0	0
$949$	74.6877	0.0787014
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 43.3711i	− 0.0455100i	−0.999741	−	0.0227550i	$-0.992756\pi$
$953$	− 43.3711i	− 0.0455100i	0.999741	−	0.0227550i	$-0.00724377\pi$
$954$	0	0
$955$	368.016	0.385357
$956$	0	0
$957$	0	0
$958$	0	0
$959$	282.806i	0.294896i
$960$	0	0
$961$	−666.328	−0.693369
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1192.27i	1.23551i
$966$	0	0
$967$	−1648.99	−1.70526	−0.852631	−	0.522514i	$-0.824994\pi$
$967$	−1648.99	−1.70526	−0.852631	+	0.522514i	$0.824994\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	518.323i	0.533803i	0.963724	+	0.266902i	$0.0859999\pi$
$971$	518.323i	0.533803i	−0.963724	+	0.266902i	$0.914000\pi$
$972$	0	0
$973$	−321.004	−0.329912
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 109.948i	− 0.112536i	−0.998416	−	0.0562682i	$-0.982080\pi$
$977$	− 109.948i	− 0.112536i	0.998416	−	0.0562682i	$-0.0179202\pi$
$978$	0	0
$979$	−2481.96	−2.53520
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 589.710i	− 0.599909i	−0.953954	−	0.299954i	$-0.903029\pi$
$983$	− 589.710i	− 0.599909i	0.953954	−	0.299954i	$-0.0969715\pi$
$984$	0	0
$985$	612.389	0.621715
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 774.050i	− 0.782659i
$990$	0	0
$991$	−713.474	−0.719954	−0.359977	−	0.932961i	$-0.617215\pi$
$991$	−713.474	−0.719954	−0.359977	+	0.932961i	$0.617215\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2477.91i	2.49036i
$996$	0	0
$997$	1222.99	1.22667	0.613334	−	0.789824i	$-0.289828\pi$
$997$	1222.99	1.22667	0.613334	+	0.789824i	$0.289828\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.d.a.449.4		4
3.2	odd	2	inner	1008.3.d.a.449.1		4
4.3	odd	2		126.3.b.a.71.2	✓	4
8.3	odd	2		4032.3.d.i.449.1		4
8.5	even	2		4032.3.d.j.449.1		4
12.11	even	2		126.3.b.a.71.3	yes	4
20.3	even	4		3150.3.c.b.449.2		8
20.7	even	4		3150.3.c.b.449.8		8
20.19	odd	2		3150.3.e.e.701.3		4
24.5	odd	2		4032.3.d.j.449.4		4
24.11	even	2		4032.3.d.i.449.4		4
28.3	even	6		882.3.s.i.863.4		8
28.11	odd	6		882.3.s.e.863.3		8
28.19	even	6		882.3.s.i.557.1		8
28.23	odd	6		882.3.s.e.557.2		8
28.27	even	2		882.3.b.f.197.1		4
36.7	odd	6		1134.3.q.c.701.1		8
36.11	even	6		1134.3.q.c.701.4		8
36.23	even	6		1134.3.q.c.1079.1		8
36.31	odd	6		1134.3.q.c.1079.4		8
60.23	odd	4		3150.3.c.b.449.5		8
60.47	odd	4		3150.3.c.b.449.3		8
60.59	even	2		3150.3.e.e.701.1		4
84.11	even	6		882.3.s.e.863.2		8
84.23	even	6		882.3.s.e.557.3		8
84.47	odd	6		882.3.s.i.557.4		8
84.59	odd	6		882.3.s.i.863.1		8
84.83	odd	2		882.3.b.f.197.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.b.a.71.2	✓	4	4.3	odd	2
126.3.b.a.71.3	yes	4	12.11	even	2
882.3.b.f.197.1		4	28.27	even	2
882.3.b.f.197.4		4	84.83	odd	2
882.3.s.e.557.2		8	28.23	odd	6
882.3.s.e.557.3		8	84.23	even	6
882.3.s.e.863.2		8	84.11	even	6
882.3.s.e.863.3		8	28.11	odd	6
882.3.s.i.557.1		8	28.19	even	6
882.3.s.i.557.4		8	84.47	odd	6
882.3.s.i.863.1		8	84.59	odd	6
882.3.s.i.863.4		8	28.3	even	6
1008.3.d.a.449.1		4	3.2	odd	2	inner
1008.3.d.a.449.4		4	1.1	even	1	trivial
1134.3.q.c.701.1		8	36.7	odd	6
1134.3.q.c.701.4		8	36.11	even	6
1134.3.q.c.1079.1		8	36.23	even	6
1134.3.q.c.1079.4		8	36.31	odd	6
3150.3.c.b.449.2		8	20.3	even	4
3150.3.c.b.449.3		8	60.47	odd	4
3150.3.c.b.449.5		8	60.23	odd	4
3150.3.c.b.449.8		8	20.7	even	4
3150.3.e.e.701.1		4	60.59	even	2
3150.3.e.e.701.3		4	20.19	odd	2
4032.3.d.i.449.1		4	8.3	odd	2
4032.3.d.i.449.4		4	24.11	even	2
4032.3.d.j.449.1		4	8.5	even	2
4032.3.d.j.449.4		4	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.89753i q^{5} -2.64575 q^{7} +O(q^{10})\)	\(q+8.89753i q^{5} -2.64575 q^{7} -17.7951i q^{11} +2.58301 q^{13} -25.8681i q^{17} -20.0000 q^{19} -17.7951i q^{23} -54.1660 q^{25} +11.9034i q^{29} +17.1660 q^{31} -23.5406i q^{35} +38.0000 q^{37} -15.7338i q^{41} +43.4980 q^{43} +16.9706i q^{47} +7.00000 q^{49} -85.5571i q^{53} +158.332 q^{55} -1.64899i q^{59} +100.332 q^{61} +22.9824i q^{65} -36.6640 q^{67} -17.7951i q^{71} +28.9150 q^{73} +47.0813i q^{77} -118.332 q^{79} +120.443i q^{83} +230.162 q^{85} -139.475i q^{89} -6.83399 q^{91} -177.951i q^{95} +44.4131 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 32 q^{13} - 80 q^{19} - 132 q^{25} - 16 q^{31} + 152 q^{37} - 80 q^{43} + 28 q^{49} + 464 q^{55} + 232 q^{61} + 192 q^{67} - 96 q^{73} - 304 q^{79} + 328 q^{85} - 112 q^{91} - 288 q^{97}+O(q^{100}) \) 4 * q - 32 * q^13 - 80 * q^19 - 132 * q^25 - 16 * q^31 + 152 * q^37 - 80 * q^43 + 28 * q^49 + 464 * q^55 + 232 * q^61 + 192 * q^67 - 96 * q^73 - 304 * q^79 + 328 * q^85 - 112 * q^91 - 288 * q^97

Introduction

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