LMFDB - Embedded newform 1008.3.m.e.127.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(127,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.127"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$-0.895644 + 1.09445i$ of defining polynomial
Character	$\chi$	$=$	1008.127
Dual form			1008.3.m.e.127.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{5} -2.64575i q^{7} +17.5112i q^{11} +20.3303 q^{13} -32.3303 q^{17} +28.0942i q^{19} -10.2016i q^{23} -21.0000 q^{25} -22.6606 q^{29} -27.7128i q^{31} -5.29150i q^{35} +62.6606 q^{37} +28.3303 q^{41} +48.8788i q^{43} +91.2108i q^{47} -7.00000 q^{49} -46.6606 q^{53} +35.0224i q^{55} +41.9506i q^{59} +12.3303 q^{61} +40.6606 q^{65} -55.4256i q^{67} +53.2964i q^{71} +21.3394 q^{73} +46.3303 q^{77} +96.9948i q^{79} +63.8794i q^{83} -64.6606 q^{85} +16.9909 q^{89} -53.7889i q^{91} +56.1884i q^{95} -95.3212 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 8 q^{13} - 56 q^{17} - 84 q^{25} + 56 q^{29} + 104 q^{37} + 40 q^{41} - 28 q^{49} - 40 q^{53} - 24 q^{61} + 16 q^{65} + 232 q^{73} + 112 q^{77} - 112 q^{85} - 152 q^{89} - 88 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 8 * q^13 - 56 * q^17 - 84 * q^25 + 56 * q^29 + 104 * q^37 + 40 * q^41 - 28 * q^49 - 40 * q^53 - 24 * q^61 + 16 * q^65 + 232 * q^73 + 112 * q^77 - 112 * q^85 - 152 * q^89 - 88 * q^97

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
$5$	2.00000	0.400000	0.200000	−	0.979796i	$-0.435906\pi$
$5$	2.00000	0.400000	0.200000	+	0.979796i	$0.435906\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	17.5112i	1.59193i	0.605344	+	0.795964i	$0.293036\pi$
$11$	17.5112i	1.59193i	−0.605344	+	0.795964i	$0.706964\pi$
$12$	0	0
$13$	20.3303	1.56387	0.781935	−	0.623360i	$-0.214233\pi$
$13$	20.3303	1.56387	0.781935	+	0.623360i	$0.214233\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−32.3303	−1.90178	−0.950891	−	0.309525i	$-0.899830\pi$
$17$	−32.3303	−1.90178	−0.950891	+	0.309525i	$0.899830\pi$
$18$	0	0
$19$	28.0942i	1.47864i	0.673353	+	0.739321i	$0.264854\pi$
$19$	28.0942i	1.47864i	−0.673353	+	0.739321i	$0.735146\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 10.2016i	− 0.443548i	−0.975098	−	0.221774i	$-0.928815\pi$
$23$	− 10.2016i	− 0.443548i	0.975098	−	0.221774i	$-0.0711847\pi$
$24$	0	0
$25$	−21.0000	−0.840000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−22.6606	−0.781400	−0.390700	−	0.920518i	$-0.627767\pi$
$29$	−22.6606	−0.781400	−0.390700	+	0.920518i	$0.627767\pi$
$30$	0	0
$31$	− 27.7128i	− 0.893962i	−0.894544	−	0.446981i	$-0.852499\pi$
$31$	− 27.7128i	− 0.893962i	0.894544	−	0.446981i	$-0.147501\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 5.29150i	− 0.151186i
$36$	0	0
$37$	62.6606	1.69353	0.846765	−	0.531967i	$-0.178547\pi$
$37$	62.6606	1.69353	0.846765	+	0.531967i	$0.178547\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	28.3303	0.690983	0.345491	−	0.938422i	$-0.387712\pi$
$41$	28.3303	0.690983	0.345491	+	0.938422i	$0.387712\pi$
$42$	0	0
$43$	48.8788i	1.13672i	0.822781	+	0.568358i	$0.192421\pi$
$43$	48.8788i	1.13672i	−0.822781	+	0.568358i	$0.807579\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	91.2108i	1.94066i	0.241792	+	0.970328i	$0.422265\pi$
$47$	91.2108i	1.94066i	−0.241792	+	0.970328i	$0.577735\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−46.6606	−0.880389	−0.440194	−	0.897902i	$-0.645090\pi$
$53$	−46.6606	−0.880389	−0.440194	+	0.897902i	$0.645090\pi$
$54$	0	0
$55$	35.0224i	0.636771i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	41.9506i	0.711027i	0.934671	+	0.355514i	$0.115694\pi$
$59$	41.9506i	0.711027i	−0.934671	+	0.355514i	$0.884306\pi$
$60$	0	0
$61$	12.3303	0.202136	0.101068	−	0.994880i	$-0.467774\pi$
$61$	12.3303	0.202136	0.101068	+	0.994880i	$0.467774\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	40.6606	0.625548
$66$	0	0
$67$	− 55.4256i	− 0.827248i	−0.910448	−	0.413624i	$-0.864263\pi$
$67$	− 55.4256i	− 0.827248i	0.910448	−	0.413624i	$-0.135737\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	53.2964i	0.750654i	0.926892	+	0.375327i	$0.122470\pi$
$71$	53.2964i	0.750654i	−0.926892	+	0.375327i	$0.877530\pi$
$72$	0	0
$73$	21.3394	0.292320	0.146160	−	0.989261i	$-0.453308\pi$
$73$	21.3394	0.292320	0.146160	+	0.989261i	$0.453308\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	46.3303	0.601692
$78$	0	0
$79$	96.9948i	1.22778i	0.789390	+	0.613891i	$0.210397\pi$
$79$	96.9948i	1.22778i	−0.789390	+	0.613891i	$0.789603\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	63.8794i	0.769632i	0.922993	+	0.384816i	$0.125735\pi$
$83$	63.8794i	0.769632i	−0.922993	+	0.384816i	$0.874265\pi$
$84$	0	0
$85$	−64.6606	−0.760713
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.9909	0.190909	0.0954545	−	0.995434i	$-0.469570\pi$
$89$	16.9909	0.190909	0.0954545	+	0.995434i	$0.469570\pi$
$90$	0	0
$91$	− 53.7889i	− 0.591087i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	56.1884i	0.591457i
$96$	0	0
$97$	−95.3212	−0.982693	−0.491346	−	0.870964i	$-0.663495\pi$
$97$	−95.3212	−0.982693	−0.491346	+	0.870964i	$0.663495\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.33939	−0.0132613	−0.00663067	−	0.999978i	$-0.502111\pi$
$101$	−1.33939	−0.0132613	−0.00663067	+	0.999978i	$0.502111\pi$
$102$	0	0
$103$	29.2384i	0.283868i	0.989876	+	0.141934i	$0.0453321\pi$
$103$	29.2384i	0.283868i	−0.989876	+	0.141934i	$0.954668\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 92.5772i	− 0.865208i	−0.901584	−	0.432604i	$-0.857595\pi$
$107$	− 92.5772i	− 0.865208i	0.901584	−	0.432604i	$-0.142405\pi$
$108$	0	0
$109$	−106.661	−0.978538	−0.489269	−	0.872133i	$-0.662736\pi$
$109$	−106.661	−0.978538	−0.489269	+	0.872133i	$0.662736\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	135.321	1.19753	0.598766	−	0.800924i	$-0.295658\pi$
$113$	135.321	1.19753	0.598766	+	0.800924i	$0.295658\pi$
$114$	0	0
$115$	− 20.4032i	− 0.177419i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	85.5379i	0.718806i
$120$	0	0
$121$	−185.642	−1.53423
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−92.0000	−0.736000
$126$	0	0
$127$	84.6640i	0.666646i	0.942813	+	0.333323i	$0.108170\pi$
$127$	84.6640i	0.666646i	−0.942813	+	0.333323i	$0.891830\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	75.4474i	0.575935i	0.957640	+	0.287967i	$0.0929794\pi$
$131$	75.4474i	0.575935i	−0.957640	+	0.287967i	$0.907021\pi$
$132$	0	0
$133$	74.3303	0.558874
$134$	0	0
$135$	0	0
$136$	0	0
$137$	173.303	1.26499	0.632493	−	0.774566i	$-0.282032\pi$
$137$	173.303	1.26499	0.632493	+	0.774566i	$0.282032\pi$
$138$	0	0
$139$	230.156i	1.65580i	0.560875	+	0.827900i	$0.310465\pi$
$139$	230.156i	1.65580i	−0.560875	+	0.827900i	$0.689535\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	356.008i	2.48957i
$144$	0	0
$145$	−45.3212	−0.312560
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−87.9818	−0.590482	−0.295241	−	0.955423i	$-0.595400\pi$
$149$	−87.9818	−0.590482	−0.295241	+	0.955423i	$0.595400\pi$
$150$	0	0
$151$	− 20.4032i	− 0.135121i	−0.997715	−	0.0675603i	$-0.978478\pi$
$151$	− 20.4032i	− 0.135121i	0.997715	−	0.0675603i	$-0.0215215\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 55.4256i	− 0.357585i
$156$	0	0
$157$	170.312	1.08479	0.542395	−	0.840123i	$-0.317518\pi$
$157$	170.312	1.08479	0.542395	+	0.840123i	$0.317518\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−26.9909	−0.167645
$162$	0	0
$163$	250.178i	1.53484i	0.641148	+	0.767418i	$0.278459\pi$
$163$	250.178i	1.53484i	−0.641148	+	0.767418i	$0.721541\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 228.249i	− 1.36676i	−0.730062	−	0.683381i	$-0.760509\pi$
$167$	− 228.249i	− 1.36676i	0.730062	−	0.683381i	$-0.239491\pi$
$168$	0	0
$169$	244.321	1.44569
$170$	0	0
$171$	0	0
$172$	0	0
$173$	39.3576	0.227500	0.113750	−	0.993509i	$-0.463714\pi$
$173$	39.3576	0.227500	0.113750	+	0.993509i	$0.463714\pi$
$174$	0	0
$175$	55.5608i	0.317490i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	64.8644i	0.362371i	0.983449	+	0.181186i	$0.0579935\pi$
$179$	64.8644i	0.362371i	−0.983449	+	0.181186i	$0.942007\pi$
$180$	0	0
$181$	13.6515	0.0754227	0.0377114	−	0.999289i	$-0.487993\pi$
$181$	13.6515	0.0754227	0.0377114	+	0.999289i	$0.487993\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	125.321	0.677412
$186$	0	0
$187$	− 566.143i	− 3.02750i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 238.451i	− 1.24843i	−0.781251	−	0.624217i	$-0.785418\pi$
$191$	− 238.451i	− 1.24843i	0.781251	−	0.624217i	$-0.214582\pi$
$192$	0	0
$193$	68.6424	0.355660	0.177830	−	0.984061i	$-0.443092\pi$
$193$	68.6424	0.355660	0.177830	+	0.984061i	$0.443092\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	211.982	1.07605	0.538025	−	0.842929i	$-0.319171\pi$
$197$	211.982	1.07605	0.538025	+	0.842929i	$0.319171\pi$
$198$	0	0
$199$	− 157.442i	− 0.791164i	−0.918431	−	0.395582i	$-0.870543\pi$
$199$	− 157.442i	− 0.791164i	0.918431	−	0.395582i	$-0.129457\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	59.9543i	0.295342i
$204$	0	0
$205$	56.6606	0.276393
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−491.964	−2.35389
$210$	0	0
$211$	− 205.876i	− 0.975716i	−0.872923	−	0.487858i	$-0.837778\pi$
$211$	− 205.876i	− 0.975716i	0.872923	−	0.487858i	$-0.162222\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	97.7576i	0.454687i
$216$	0	0
$217$	−73.3212	−0.337886
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−657.285	−2.97414
$222$	0	0
$223$	− 164.307i	− 0.736802i	−0.929667	−	0.368401i	$-0.879905\pi$
$223$	− 164.307i	− 0.736802i	0.929667	−	0.368401i	$-0.120095\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 282.086i	− 1.24267i	−0.783545	−	0.621336i	$-0.786590\pi$
$227$	− 282.086i	− 1.24267i	0.783545	−	0.621336i	$-0.213410\pi$
$228$	0	0
$229$	93.6515	0.408959	0.204479	−	0.978871i	$-0.434450\pi$
$229$	93.6515	0.408959	0.204479	+	0.978871i	$0.434450\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	225.339	0.967122	0.483561	−	0.875311i	$-0.339343\pi$
$233$	225.339	0.967122	0.483561	+	0.875311i	$0.339343\pi$
$234$	0	0
$235$	182.422i	0.776263i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 81.7720i	− 0.342142i	−0.985259	−	0.171071i	$-0.945277\pi$
$239$	− 81.7720i	− 0.342142i	0.985259	−	0.171071i	$-0.0547228\pi$
$240$	0	0
$241$	−326.000	−1.35270	−0.676349	−	0.736582i	$-0.736439\pi$
$241$	−326.000	−1.35270	−0.676349	+	0.736582i	$0.736439\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−14.0000	−0.0571429
$246$	0	0
$247$	571.164i	2.31240i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 255.136i	− 1.01648i	−0.861216	−	0.508240i	$-0.830296\pi$
$251$	− 255.136i	− 1.01648i	0.861216	−	0.508240i	$-0.169704\pi$
$252$	0	0
$253$	178.642	0.706097
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−129.652	−0.504481	−0.252240	−	0.967665i	$-0.581167\pi$
$257$	−129.652	−0.504481	−0.252240	+	0.967665i	$0.581167\pi$
$258$	0	0
$259$	− 165.784i	− 0.640094i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	394.685i	1.50070i	0.661038	+	0.750352i	$0.270116\pi$
$263$	394.685i	1.50070i	−0.661038	+	0.750352i	$0.729884\pi$
$264$	0	0
$265$	−93.3212	−0.352156
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−27.9818	−0.104022	−0.0520108	−	0.998647i	$-0.516563\pi$
$269$	−27.9818	−0.104022	−0.0520108	+	0.998647i	$0.516563\pi$
$270$	0	0
$271$	− 26.9500i	− 0.0994465i	−0.998763	−	0.0497233i	$-0.984166\pi$
$271$	− 26.9500i	− 0.0994465i	0.998763	−	0.0497233i	$-0.0158339\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 367.735i	− 1.33722i
$276$	0	0
$277$	48.6788	0.175736	0.0878678	−	0.996132i	$-0.471995\pi$
$277$	48.6788	0.175736	0.0878678	+	0.996132i	$0.471995\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−113.303	−0.403214	−0.201607	−	0.979467i	$-0.564616\pi$
$281$	−113.303	−0.403214	−0.201607	+	0.979467i	$0.564616\pi$
$282$	0	0
$283$	− 238.992i	− 0.844493i	−0.906481	−	0.422246i	$-0.861242\pi$
$283$	− 238.992i	− 0.844493i	0.906481	−	0.422246i	$-0.138758\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 74.9549i	− 0.261167i
$288$	0	0
$289$	756.248	2.61678
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.37576	−0.0183473	−0.00917365	−	0.999958i	$-0.502920\pi$
$293$	−5.37576	−0.0183473	−0.00917365	+	0.999958i	$0.502920\pi$
$294$	0	0
$295$	83.9012i	0.284411i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 207.402i	− 0.693651i
$300$	0	0
$301$	129.321	0.429639
$302$	0	0
$303$	0	0
$304$	0	0
$305$	24.6606	0.0808544
$306$	0	0
$307$	− 12.7122i	− 0.0414078i	−0.999786	−	0.0207039i	$-0.993409\pi$
$307$	− 12.7122i	− 0.0414078i	0.999786	−	0.0207039i	$-0.00659073\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	172.061i	0.553250i	0.960978	+	0.276625i	$0.0892161\pi$
$311$	172.061i	0.553250i	−0.960978	+	0.276625i	$0.910784\pi$
$312$	0	0
$313$	6.66061	0.0212799	0.0106399	−	0.999943i	$-0.496613\pi$
$313$	6.66061	0.0212799	0.0106399	+	0.999943i	$0.496613\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	598.624	1.88840	0.944202	−	0.329366i	$-0.106835\pi$
$317$	598.624	1.88840	0.944202	+	0.329366i	$0.106835\pi$
$318$	0	0
$319$	− 396.815i	− 1.24393i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 908.294i	− 2.81206i
$324$	0	0
$325$	−426.936	−1.31365
$326$	0	0
$327$	0	0
$328$	0	0
$329$	241.321	0.733499
$330$	0	0
$331$	− 446.012i	− 1.34747i	−0.738974	−	0.673734i	$-0.764689\pi$
$331$	− 446.012i	− 1.34747i	0.738974	−	0.673734i	$-0.235311\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 110.851i	− 0.330899i
$336$	0	0
$337$	6.03637	0.0179121	0.00895603	−	0.999960i	$-0.497149\pi$
$337$	6.03637	0.0179121	0.00895603	+	0.999960i	$0.497149\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	485.285	1.42312
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 439.306i	− 1.26601i	−0.774147	−	0.633005i	$-0.781821\pi$
$347$	− 439.306i	− 1.26601i	0.774147	−	0.633005i	$-0.218179\pi$
$348$	0	0
$349$	−223.009	−0.638995	−0.319497	−	0.947587i	$-0.603514\pi$
$349$	−223.009	−0.638995	−0.319497	+	0.947587i	$0.603514\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−624.955	−1.77041	−0.885205	−	0.465202i	$-0.845982\pi$
$353$	−624.955	−1.77041	−0.885205	+	0.465202i	$0.845982\pi$
$354$	0	0
$355$	106.593i	0.300262i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 112.218i	− 0.312584i	−0.987711	−	0.156292i	$-0.950046\pi$
$359$	− 112.218i	− 0.312584i	0.987711	−	0.156292i	$-0.0499541\pi$
$360$	0	0
$361$	−428.285	−1.18638
$362$	0	0
$363$	0	0
$364$	0	0
$365$	42.6788	0.116928
$366$	0	0
$367$	− 288.252i	− 0.785427i	−0.919661	−	0.392714i	$-0.871536\pi$
$367$	− 288.252i	− 0.785427i	0.919661	−	0.392714i	$-0.128464\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	123.452i	0.332756i
$372$	0	0
$373$	−391.248	−1.04892	−0.524462	−	0.851434i	$-0.675733\pi$
$373$	−391.248	−1.04892	−0.524462	+	0.851434i	$0.675733\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−460.697	−1.22201
$378$	0	0
$379$	412.197i	1.08759i	0.839218	+	0.543795i	$0.183013\pi$
$379$	412.197i	1.08759i	−0.839218	+	0.543795i	$0.816987\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	56.1884i	0.146706i	0.997306	+	0.0733530i	$0.0233700\pi$
$383$	56.1884i	0.146706i	−0.997306	+	0.0733530i	$0.976630\pi$
$384$	0	0
$385$	92.6606	0.240677
$386$	0	0
$387$	0	0
$388$	0	0
$389$	401.267	1.03153	0.515767	−	0.856729i	$-0.327507\pi$
$389$	401.267	1.03153	0.515767	+	0.856729i	$0.327507\pi$
$390$	0	0
$391$	329.821i	0.843532i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	193.990i	0.491113i
$396$	0	0
$397$	−278.936	−0.702610	−0.351305	−	0.936261i	$-0.614262\pi$
$397$	−278.936	−0.702610	−0.351305	+	0.936261i	$0.614262\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	467.982	1.16704	0.583518	−	0.812100i	$-0.301676\pi$
$401$	467.982	1.16704	0.583518	+	0.812100i	$0.301676\pi$
$402$	0	0
$403$	− 563.410i	− 1.39804i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1097.26i	2.69598i
$408$	0	0
$409$	69.9636	0.171060	0.0855301	−	0.996336i	$-0.472742\pi$
$409$	69.9636	0.171060	0.0855301	+	0.996336i	$0.472742\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	110.991	0.268743
$414$	0	0
$415$	127.759i	0.307853i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	111.233i	0.265472i	0.991151	+	0.132736i	$0.0423762\pi$
$419$	111.233i	0.265472i	−0.991151	+	0.132736i	$0.957624\pi$
$420$	0	0
$421$	−436.606	−1.03707	−0.518535	−	0.855057i	$-0.673522\pi$
$421$	−436.606	−1.03707	−0.518535	+	0.855057i	$0.673522\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	678.936	1.59750
$426$	0	0
$427$	− 32.6229i	− 0.0764003i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 664.060i	− 1.54074i	−0.637596	−	0.770371i	$-0.720071\pi$
$431$	− 664.060i	− 1.54074i	0.637596	−	0.770371i	$-0.279929\pi$
$432$	0	0
$433$	−116.055	−0.268024	−0.134012	−	0.990980i	$-0.542786\pi$
$433$	−116.055	−0.268024	−0.134012	+	0.990980i	$0.542786\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	286.606	0.655849
$438$	0	0
$439$	363.762i	0.828615i	0.910137	+	0.414308i	$0.135976\pi$
$439$	363.762i	0.828615i	−0.910137	+	0.414308i	$0.864024\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 723.299i	− 1.63273i	−0.577537	−	0.816365i	$-0.695986\pi$
$443$	− 723.299i	− 1.63273i	0.577537	−	0.816365i	$-0.304014\pi$
$444$	0	0
$445$	33.9818	0.0763636
$446$	0	0
$447$	0	0
$448$	0	0
$449$	264.715	0.589566	0.294783	−	0.955564i	$-0.404753\pi$
$449$	264.715	0.589566	0.294783	+	0.955564i	$0.404753\pi$
$450$	0	0
$451$	496.098i	1.10000i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 107.578i	− 0.236435i
$456$	0	0
$457$	−277.927	−0.608156	−0.304078	−	0.952647i	$-0.598348\pi$
$457$	−277.927	−0.608156	−0.304078	+	0.952647i	$0.598348\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−758.000	−1.64425	−0.822126	−	0.569306i	$-0.807212\pi$
$461$	−758.000	−1.64425	−0.822126	+	0.569306i	$0.807212\pi$
$462$	0	0
$463$	− 499.593i	− 1.07904i	−0.841974	−	0.539518i	$-0.818607\pi$
$463$	− 499.593i	− 1.07904i	0.841974	−	0.539518i	$-0.181393\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	419.888i	0.899117i	0.893251	+	0.449558i	$0.148419\pi$
$467$	419.888i	0.899117i	−0.893251	+	0.449558i	$0.851581\pi$
$468$	0	0
$469$	−146.642	−0.312670
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−855.927	−1.80957
$474$	0	0
$475$	− 589.978i	− 1.24206i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 690.850i	− 1.44228i	−0.692791	−	0.721138i	$-0.743619\pi$
$479$	− 690.850i	− 1.44228i	0.692791	−	0.721138i	$-0.256381\pi$
$480$	0	0
$481$	1273.91	2.64846
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−190.642	−0.393077
$486$	0	0
$487$	163.544i	0.335819i	0.985802	+	0.167910i	$0.0537017\pi$
$487$	163.544i	0.335819i	−0.985802	+	0.167910i	$0.946298\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 184.551i	− 0.375867i	−0.982182	−	0.187934i	$-0.939821\pi$
$491$	− 184.551i	− 0.375867i	0.982182	−	0.187934i	$-0.0601790\pi$
$492$	0	0
$493$	732.624	1.48605
$494$	0	0
$495$	0	0
$496$	0	0
$497$	141.009	0.283721
$498$	0	0
$499$	− 574.659i	− 1.15162i	−0.817583	−	0.575811i	$-0.804686\pi$
$499$	− 574.659i	− 1.15162i	0.817583	−	0.575811i	$-0.195314\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	198.567i	0.394764i	0.980327	+	0.197382i	$0.0632440\pi$
$503$	198.567i	0.394764i	−0.980327	+	0.197382i	$0.936756\pi$
$504$	0	0
$505$	−2.67879	−0.00530453
$506$	0	0
$507$	0	0
$508$	0	0
$509$	209.376	0.411347	0.205674	−	0.978621i	$-0.434061\pi$
$509$	209.376	0.411347	0.205674	+	0.978621i	$0.434061\pi$
$510$	0	0
$511$	− 56.4587i	− 0.110487i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	58.4768i	0.113547i
$516$	0	0
$517$	−1597.21	−3.08939
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−340.367	−0.653295	−0.326647	−	0.945146i	$-0.605919\pi$
$521$	−340.367	−0.653295	−0.326647	+	0.945146i	$0.605919\pi$
$522$	0	0
$523$	237.022i	0.453196i	0.973988	+	0.226598i	$0.0727604\pi$
$523$	237.022i	0.453196i	−0.973988	+	0.226598i	$0.927240\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	895.964i	1.70012i
$528$	0	0
$529$	424.927	0.803265
$530$	0	0
$531$	0	0
$532$	0	0
$533$	575.964	1.08061
$534$	0	0
$535$	− 185.154i	− 0.346083i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 122.578i	− 0.227418i
$540$	0	0
$541$	692.570	1.28017	0.640083	−	0.768306i	$-0.278900\pi$
$541$	692.570	1.28017	0.640083	+	0.768306i	$0.278900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−213.321	−0.391415
$546$	0	0
$547$	301.027i	0.550323i	0.961398	+	0.275162i	$0.0887314\pi$
$547$	301.027i	0.550323i	−0.961398	+	0.275162i	$0.911269\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 636.632i	− 1.15541i
$552$	0	0
$553$	256.624	0.464058
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−82.6970	−0.148469	−0.0742343	−	0.997241i	$-0.523651\pi$
$557$	−82.6970	−0.148469	−0.0742343	+	0.997241i	$0.523651\pi$
$558$	0	0
$559$	993.721i	1.77768i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 75.1290i	− 0.133444i	−0.997772	−	0.0667220i	$-0.978746\pi$
$563$	− 75.1290i	− 0.133444i	0.997772	−	0.0667220i	$-0.0212541\pi$
$564$	0	0
$565$	270.642	0.479013
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−255.909	−0.449752	−0.224876	−	0.974387i	$-0.572198\pi$
$569$	−255.909	−0.449752	−0.224876	+	0.974387i	$0.572198\pi$
$570$	0	0
$571$	791.341i	1.38589i	0.720992	+	0.692943i	$0.243686\pi$
$571$	791.341i	1.38589i	−0.720992	+	0.692943i	$0.756314\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	214.234i	0.372580i
$576$	0	0
$577$	−384.642	−0.666625	−0.333312	−	0.942816i	$-0.608166\pi$
$577$	−384.642	−0.666625	−0.333312	+	0.942816i	$0.608166\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	169.009	0.290893
$582$	0	0
$583$	− 817.084i	− 1.40152i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	429.167i	0.731120i	0.930788	+	0.365560i	$0.119122\pi$
$587$	429.167i	0.731120i	−0.930788	+	0.365560i	$0.880878\pi$
$588$	0	0
$589$	778.570	1.32185
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−859.524	−1.44945	−0.724725	−	0.689038i	$-0.758033\pi$
$593$	−859.524	−1.44945	−0.724725	+	0.689038i	$0.758033\pi$
$594$	0	0
$595$	171.076i	0.287522i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	745.228i	1.24412i	0.782970	+	0.622060i	$0.213704\pi$
$599$	745.228i	1.24412i	−0.782970	+	0.622060i	$0.786296\pi$
$600$	0	0
$601$	52.6424	0.0875914	0.0437957	−	0.999041i	$-0.486055\pi$
$601$	52.6424	0.0875914	0.0437957	+	0.999041i	$0.486055\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−371.285	−0.613694
$606$	0	0
$607$	635.425i	1.04683i	0.852078	+	0.523414i	$0.175342\pi$
$607$	635.425i	1.04683i	−0.852078	+	0.523414i	$0.824658\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1854.34i	3.03493i
$612$	0	0
$613$	71.9818	0.117425	0.0587127	−	0.998275i	$-0.481300\pi$
$613$	71.9818	0.117425	0.0587127	+	0.998275i	$0.481300\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1173.23	1.90151	0.950754	−	0.309947i	$-0.100311\pi$
$617$	1173.23	1.90151	0.950754	+	0.309947i	$0.100311\pi$
$618$	0	0
$619$	− 97.6947i	− 0.157827i	−0.996881	−	0.0789133i	$-0.974855\pi$
$619$	− 97.6947i	− 0.157827i	0.996881	−	0.0789133i	$-0.0251450\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 44.9537i	− 0.0721569i
$624$	0	0
$625$	341.000	0.545600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2025.84	−3.22073
$630$	0	0
$631$	− 198.692i	− 0.314885i	−0.987528	−	0.157443i	$-0.949675\pi$
$631$	− 198.692i	− 0.314885i	0.987528	−	0.157443i	$-0.0503249\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	169.328i	0.266658i
$636$	0	0
$637$	−142.312	−0.223410
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−342.588	−0.534458	−0.267229	−	0.963633i	$-0.586108\pi$
$641$	−342.588	−0.534458	−0.267229	+	0.963633i	$0.586108\pi$
$642$	0	0
$643$	− 776.340i	− 1.20737i	−0.797222	−	0.603686i	$-0.793698\pi$
$643$	− 776.340i	− 1.20737i	0.797222	−	0.603686i	$-0.206302\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	755.430i	1.16759i	0.811902	+	0.583794i	$0.198432\pi$
$647$	755.430i	1.16759i	−0.811902	+	0.583794i	$0.801568\pi$
$648$	0	0
$649$	−734.606	−1.13190
$650$	0	0
$651$	0	0
$652$	0	0
$653$	681.339	1.04340	0.521699	−	0.853129i	$-0.325298\pi$
$653$	681.339	1.04340	0.521699	+	0.853129i	$0.325298\pi$
$654$	0	0
$655$	150.895i	0.230374i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	65.9456i	0.100069i	0.998747	+	0.0500346i	$0.0159332\pi$
$659$	65.9456i	0.100069i	−0.998747	+	0.0500346i	$0.984067\pi$
$660$	0	0
$661$	161.615	0.244501	0.122250	−	0.992499i	$-0.460989\pi$
$661$	161.615	0.244501	0.122250	+	0.992499i	$0.460989\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	148.661	0.223550
$666$	0	0
$667$	231.175i	0.346589i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	215.919i	0.321786i
$672$	0	0
$673$	451.248	0.670503	0.335251	−	0.942129i	$-0.391179\pi$
$673$	451.248	0.670503	0.335251	+	0.942129i	$0.391179\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	117.964	0.174245	0.0871223	−	0.996198i	$-0.472233\pi$
$677$	117.964	0.174245	0.0871223	+	0.996198i	$0.472233\pi$
$678$	0	0
$679$	252.196i	0.371423i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 49.9268i	− 0.0730993i	−0.999332	−	0.0365496i	$-0.988363\pi$
$683$	− 49.9268i	− 0.0730993i	0.999332	−	0.0365496i	$-0.0116367\pi$
$684$	0	0
$685$	346.606	0.505994
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−948.624	−1.37681
$690$	0	0
$691$	− 725.047i	− 1.04927i	−0.851327	−	0.524636i	$-0.824201\pi$
$691$	− 725.047i	− 1.04927i	0.851327	−	0.524636i	$-0.175799\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	460.313i	0.662320i
$696$	0	0
$697$	−915.927	−1.31410
$698$	0	0
$699$	0	0
$700$	0	0
$701$	587.982	0.838776	0.419388	−	0.907807i	$-0.362245\pi$
$701$	587.982	0.838776	0.419388	+	0.907807i	$0.362245\pi$
$702$	0	0
$703$	1760.40i	2.50413i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.54370i	0.00501231i
$708$	0	0
$709$	801.303	1.13019	0.565094	−	0.825027i	$-0.308840\pi$
$709$	801.303	1.13019	0.565094	+	0.825027i	$0.308840\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−282.715	−0.396515
$714$	0	0
$715$	712.016i	0.995827i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 328.740i	− 0.457218i	−0.973518	−	0.228609i	$-0.926582\pi$
$719$	− 328.740i	− 0.457218i	0.973518	−	0.228609i	$-0.0734177\pi$
$720$	0	0
$721$	77.3576	0.107292
$722$	0	0
$723$	0	0
$724$	0	0
$725$	475.873	0.656376
$726$	0	0
$727$	− 767.442i	− 1.05563i	−0.849360	−	0.527814i	$-0.823012\pi$
$727$	− 767.442i	− 1.05563i	0.849360	−	0.527814i	$-0.176988\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1580.27i	− 2.16179i
$732$	0	0
$733$	−60.2939	−0.0822564	−0.0411282	−	0.999154i	$-0.513095\pi$
$733$	−60.2939	−0.0822564	−0.0411282	+	0.999154i	$0.513095\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	970.570	1.31692
$738$	0	0
$739$	920.499i	1.24560i	0.782381	+	0.622801i	$0.214005\pi$
$739$	920.499i	1.24560i	−0.782381	+	0.622801i	$0.785995\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	598.273i	0.805213i	0.915373	+	0.402606i	$0.131896\pi$
$743$	598.273i	0.805213i	−0.915373	+	0.402606i	$0.868104\pi$
$744$	0	0
$745$	−175.964	−0.236193
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−244.936	−0.327018
$750$	0	0
$751$	150.450i	0.200334i	0.994971	+	0.100167i	$0.0319376\pi$
$751$	150.450i	0.200334i	−0.994971	+	0.100167i	$0.968062\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 40.8064i	− 0.0540482i
$756$	0	0
$757$	773.339	1.02158	0.510792	−	0.859704i	$-0.329352\pi$
$757$	773.339	1.02158	0.510792	+	0.859704i	$0.329352\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	389.652	0.512026	0.256013	−	0.966673i	$-0.417591\pi$
$761$	389.652	0.512026	0.256013	+	0.966673i	$0.417591\pi$
$762$	0	0
$763$	282.197i	0.369852i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	852.869i	1.11195i
$768$	0	0
$769$	15.9818	0.0207826	0.0103913	−	0.999946i	$-0.496692\pi$
$769$	15.9818	0.0207826	0.0103913	+	0.999946i	$0.496692\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	833.303	1.07801	0.539006	−	0.842302i	$-0.318800\pi$
$773$	833.303	1.07801	0.539006	+	0.842302i	$0.318800\pi$
$774$	0	0
$775$	581.969i	0.750928i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	795.918i	1.02172i
$780$	0	0
$781$	−933.285	−1.19499
$782$	0	0
$783$	0	0
$784$	0	0
$785$	340.624	0.433916
$786$	0	0
$787$	1498.72i	1.90434i	0.305565	+	0.952171i	$0.401155\pi$
$787$	1498.72i	1.90434i	−0.305565	+	0.952171i	$0.598845\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 358.026i	− 0.452625i
$792$	0	0
$793$	250.679	0.316114
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−800.715	−1.00466	−0.502331	−	0.864676i	$-0.667524\pi$
$797$	−800.715	−1.00466	−0.502331	+	0.864676i	$0.667524\pi$
$798$	0	0
$799$	− 2948.87i	− 3.69071i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	373.679i	0.465353i
$804$	0	0
$805$	−53.9818	−0.0670582
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−243.394	−0.300858	−0.150429	−	0.988621i	$-0.548065\pi$
$809$	−243.394	−0.300858	−0.150429	+	0.988621i	$0.548065\pi$
$810$	0	0
$811$	− 750.597i	− 0.925521i	−0.886483	−	0.462760i	$-0.846859\pi$
$811$	− 750.597i	− 0.925521i	0.886483	−	0.462760i	$-0.153141\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	500.356i	0.613934i
$816$	0	0
$817$	−1373.21	−1.68080
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1203.98	1.46648	0.733241	−	0.679969i	$-0.238007\pi$
$821$	1203.98	1.46648	0.733241	+	0.679969i	$0.238007\pi$
$822$	0	0
$823$	1385.51i	1.68349i	0.539873	+	0.841746i	$0.318472\pi$
$823$	1385.51i	1.68349i	−0.539873	+	0.841746i	$0.681528\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	120.608i	0.145839i	0.997338	+	0.0729193i	$0.0232315\pi$
$827$	120.608i	0.145839i	−0.997338	+	0.0729193i	$0.976768\pi$
$828$	0	0
$829$	−814.936	−0.983035	−0.491518	−	0.870868i	$-0.663558\pi$
$829$	−814.936	−0.983035	−0.491518	+	0.870868i	$0.663558\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	226.312	0.271683
$834$	0	0
$835$	− 456.499i	− 0.546705i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1267.35i	1.51055i	0.655406	+	0.755276i	$0.272497\pi$
$839$	1267.35i	1.51055i	−0.655406	+	0.755276i	$0.727503\pi$
$840$	0	0
$841$	−327.497	−0.389414
$842$	0	0
$843$	0	0
$844$	0	0
$845$	488.642	0.578275
$846$	0	0
$847$	491.164i	0.579886i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 639.239i	− 0.751162i
$852$	0	0
$853$	−824.330	−0.966390	−0.483195	−	0.875513i	$-0.660524\pi$
$853$	−824.330	−0.966390	−0.483195	+	0.875513i	$0.660524\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1054.83	1.23084	0.615418	−	0.788201i	$-0.288987\pi$
$857$	1054.83	1.23084	0.615418	+	0.788201i	$0.288987\pi$
$858$	0	0
$859$	− 288.189i	− 0.335493i	−0.985830	−	0.167747i	$-0.946351\pi$
$859$	− 288.189i	− 0.335493i	0.985830	−	0.167747i	$-0.0536491\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 497.338i	− 0.576290i	−0.957587	−	0.288145i	$-0.906961\pi$
$863$	− 497.338i	− 0.576290i	0.957587	−	0.288145i	$-0.0930385\pi$
$864$	0	0
$865$	78.7152	0.0910002
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1698.50	−1.95454
$870$	0	0
$871$	− 1126.82i	− 1.29371i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	243.409i	0.278182i
$876$	0	0
$877$	−822.624	−0.937998	−0.468999	−	0.883199i	$-0.655385\pi$
$877$	−822.624	−0.937998	−0.468999	+	0.883199i	$0.655385\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1017.47	1.15490	0.577452	−	0.816425i	$-0.304047\pi$
$881$	1017.47	1.15490	0.577452	+	0.816425i	$0.304047\pi$
$882$	0	0
$883$	1333.58i	1.51029i	0.655559	+	0.755144i	$0.272433\pi$
$883$	1333.58i	1.51029i	−0.655559	+	0.755144i	$0.727567\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1422.95i	− 1.60423i	−0.597170	−	0.802115i	$-0.703708\pi$
$887$	− 1422.95i	− 1.60423i	0.597170	−	0.802115i	$-0.296292\pi$
$888$	0	0
$889$	224.000	0.251969
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2562.50	−2.86954
$894$	0	0
$895$	129.729i	0.144948i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	627.989i	0.698542i
$900$	0	0
$901$	1508.55	1.67431
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.3030	0.0301691
$906$	0	0
$907$	− 1055.57i	− 1.16380i	−0.813260	−	0.581901i	$-0.802309\pi$
$907$	− 1055.57i	− 1.16380i	0.813260	−	0.581901i	$-0.197691\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 239.851i	− 0.263283i	−0.991297	−	0.131641i	$-0.957975\pi$
$911$	− 239.851i	− 0.263283i	0.991297	−	0.131641i	$-0.0420247\pi$
$912$	0	0
$913$	−1118.61	−1.22520
$914$	0	0
$915$	0	0
$916$	0	0
$917$	199.615	0.217683
$918$	0	0
$919$	− 690.850i	− 0.751741i	−0.926672	−	0.375871i	$-0.877344\pi$
$919$	− 690.850i	− 0.751741i	0.926672	−	0.375871i	$-0.122656\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1083.53i	1.17392i
$924$	0	0
$925$	−1315.87	−1.42257
$926$	0	0
$927$	0	0
$928$	0	0
$929$	918.900	0.989128	0.494564	−	0.869141i	$-0.335328\pi$
$929$	918.900	0.989128	0.494564	+	0.869141i	$0.335328\pi$
$930$	0	0
$931$	− 196.659i	− 0.211235i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1132.29i	− 1.21100i
$936$	0	0
$937$	−513.891	−0.548443	−0.274221	−	0.961667i	$-0.588420\pi$
$937$	−513.891	−0.548443	−0.274221	+	0.961667i	$0.588420\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−525.855	−0.558825	−0.279413	−	0.960171i	$-0.590140\pi$
$941$	−525.855	−0.558825	−0.279413	+	0.960171i	$0.590140\pi$
$942$	0	0
$943$	− 289.015i	− 0.306484i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1822.53i	1.92453i	0.272110	+	0.962266i	$0.412279\pi$
$947$	1822.53i	1.92453i	−0.272110	+	0.962266i	$0.587721\pi$
$948$	0	0
$949$	433.836	0.457151
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1394.04	1.46279	0.731394	−	0.681955i	$-0.238870\pi$
$953$	1394.04	1.46279	0.731394	+	0.681955i	$0.238870\pi$
$954$	0	0
$955$	− 476.902i	− 0.499374i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 458.517i	− 0.478120i
$960$	0	0
$961$	193.000	0.200832
$962$	0	0
$963$	0	0
$964$	0	0
$965$	137.285	0.142264
$966$	0	0
$967$	1199.92i	1.24086i	0.784260	+	0.620432i	$0.213043\pi$
$967$	1199.92i	1.24086i	−0.784260	+	0.620432i	$0.786957\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1067.52i	− 1.09940i	−0.835362	−	0.549700i	$-0.814742\pi$
$971$	− 1067.52i	− 1.09940i	0.835362	−	0.549700i	$-0.185258\pi$
$972$	0	0
$973$	608.936	0.625834
$974$	0	0
$975$	0	0
$976$	0	0
$977$	399.945	0.409361	0.204680	−	0.978829i	$-0.434385\pi$
$977$	399.945	0.409361	0.204680	+	0.978829i	$0.434385\pi$
$978$	0	0
$979$	297.531i	0.303914i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.3820i	0.0156480i	0.999969	+	0.00782401i	$0.00249049\pi$
$983$	15.3820i	0.0156480i	−0.999969	+	0.00782401i	$0.997510\pi$
$984$	0	0
$985$	423.964	0.430420
$986$	0	0
$987$	0	0
$988$	0	0
$989$	498.642	0.504188
$990$	0	0
$991$	− 1242.69i	− 1.25398i	−0.779028	−	0.626989i	$-0.784287\pi$
$991$	− 1242.69i	− 1.25398i	0.779028	−	0.626989i	$-0.215713\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 314.883i	− 0.316466i
$996$	0	0
$997$	604.330	0.606149	0.303074	−	0.952967i	$-0.401987\pi$
$997$	604.330	0.606149	0.303074	+	0.952967i	$0.401987\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.m.e.127.2		4
3.2	odd	2		336.3.m.b.127.3	yes	4
4.3	odd	2	inner	1008.3.m.e.127.3		4
12.11	even	2		336.3.m.b.127.2	✓	4
21.20	even	2		2352.3.m.i.1471.1		4
24.5	odd	2		1344.3.m.b.127.1		4
24.11	even	2		1344.3.m.b.127.4		4
84.83	odd	2		2352.3.m.i.1471.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.m.b.127.2	✓	4	12.11	even	2
336.3.m.b.127.3	yes	4	3.2	odd	2
1008.3.m.e.127.2		4	1.1	even	1	trivial
1008.3.m.e.127.3		4	4.3	odd	2	inner
1344.3.m.b.127.1		4	24.5	odd	2
1344.3.m.b.127.4		4	24.11	even	2
2352.3.m.i.1471.1		4	21.20	even	2
2352.3.m.i.1471.4		4	84.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{5} -2.64575i q^{7} +17.5112i q^{11} +20.3303 q^{13} -32.3303 q^{17} +28.0942i q^{19} -10.2016i q^{23} -21.0000 q^{25} -22.6606 q^{29} -27.7128i q^{31} -5.29150i q^{35} +62.6606 q^{37} +28.3303 q^{41} +48.8788i q^{43} +91.2108i q^{47} -7.00000 q^{49} -46.6606 q^{53} +35.0224i q^{55} +41.9506i q^{59} +12.3303 q^{61} +40.6606 q^{65} -55.4256i q^{67} +53.2964i q^{71} +21.3394 q^{73} +46.3303 q^{77} +96.9948i q^{79} +63.8794i q^{83} -64.6606 q^{85} +16.9909 q^{89} -53.7889i q^{91} +56.1884i q^{95} -95.3212 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 8 q^{13} - 56 q^{17} - 84 q^{25} + 56 q^{29} + 104 q^{37} + 40 q^{41} - 28 q^{49} - 40 q^{53} - 24 q^{61} + 16 q^{65} + 232 q^{73} + 112 q^{77} - 112 q^{85} - 152 q^{89} - 88 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 8 * q^13 - 56 * q^17 - 84 * q^25 + 56 * q^29 + 104 * q^37 + 40 * q^41 - 28 * q^49 - 40 * q^53 - 24 * q^61 + 16 * q^65 + 232 * q^73 + 112 * q^77 - 112 * q^85 - 152 * q^89 - 88 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more