LMFDB - Embedded newform 8048.2.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.a (trivial)

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:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$1$
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} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.31191$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31191 q^{3} +0.313584 q^{5} +2.05708 q^{7} +2.34493 q^{9} +O(q^{10})$	$q-2.31191 q^{3} +0.313584 q^{5} +2.05708 q^{7} +2.34493 q^{9} -1.08501 q^{11} +2.96125 q^{13} -0.724979 q^{15} +5.04731 q^{17} +1.75954 q^{19} -4.75577 q^{21} -5.45732 q^{23} -4.90166 q^{25} +1.51446 q^{27} -4.82128 q^{29} -3.99225 q^{31} +2.50845 q^{33} +0.645067 q^{35} +11.2920 q^{37} -6.84614 q^{39} -8.82468 q^{41} -5.01513 q^{43} +0.735333 q^{45} +4.60784 q^{47} -2.76844 q^{49} -11.6689 q^{51} -7.97441 q^{53} -0.340242 q^{55} -4.06789 q^{57} +12.5908 q^{59} -14.2309 q^{61} +4.82370 q^{63} +0.928602 q^{65} -3.95648 q^{67} +12.6168 q^{69} +3.67602 q^{71} -0.0874702 q^{73} +11.3322 q^{75} -2.23195 q^{77} +6.33356 q^{79} -10.5361 q^{81} -12.1089 q^{83} +1.58276 q^{85} +11.1464 q^{87} -11.6351 q^{89} +6.09151 q^{91} +9.22973 q^{93} +0.551763 q^{95} -3.89026 q^{97} -2.54427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.31191	−1.33478	−0.667391	−	0.744707i	$-0.732589\pi$
$3$	−2.31191	−1.33478	−0.667391	+	0.744707i	$0.732589\pi$
$4$	0	0
$5$	0.313584	0.140239	0.0701196	−	0.997539i	$-0.477662\pi$
$5$	0.313584	0.140239	0.0701196	+	0.997539i	$0.477662\pi$
$6$	0	0
$7$	2.05708	0.777501	0.388751	−	0.921343i	$-0.372907\pi$
$7$	2.05708	0.777501	0.388751	+	0.921343i	$0.372907\pi$
$8$	0	0
$9$	2.34493	0.781643
$10$	0	0
$11$	−1.08501	−0.327143	−0.163571	−	0.986531i	$-0.552301\pi$
$11$	−1.08501	−0.327143	−0.163571	+	0.986531i	$0.552301\pi$
$12$	0	0
$13$	2.96125	0.821303	0.410651	−	0.911792i	$-0.365301\pi$
$13$	2.96125	0.821303	0.410651	+	0.911792i	$0.365301\pi$
$14$	0	0
$15$	−0.724979	−0.187189
$16$	0	0
$17$	5.04731	1.22415	0.612076	−	0.790799i	$-0.290335\pi$
$17$	5.04731	1.22415	0.612076	+	0.790799i	$0.290335\pi$
$18$	0	0
$19$	1.75954	0.403665	0.201833	−	0.979420i	$-0.435310\pi$
$19$	1.75954	0.403665	0.201833	+	0.979420i	$0.435310\pi$
$20$	0	0
$21$	−4.75577	−1.03779
$22$	0	0
$23$	−5.45732	−1.13793	−0.568965	−	0.822362i	$-0.692656\pi$
$23$	−5.45732	−1.13793	−0.568965	+	0.822362i	$0.692656\pi$
$24$	0	0
$25$	−4.90166	−0.980333
$26$	0	0
$27$	1.51446	0.291459
$28$	0	0
$29$	−4.82128	−0.895290	−0.447645	−	0.894211i	$-0.647737\pi$
$29$	−4.82128	−0.895290	−0.447645	+	0.894211i	$0.647737\pi$
$30$	0	0
$31$	−3.99225	−0.717029	−0.358515	−	0.933524i	$-0.616717\pi$
$31$	−3.99225	−0.717029	−0.358515	+	0.933524i	$0.616717\pi$
$32$	0	0
$33$	2.50845	0.436664
$34$	0	0
$35$	0.645067	0.109036
$36$	0	0
$37$	11.2920	1.85640	0.928200	−	0.372082i	$-0.121356\pi$
$37$	11.2920	1.85640	0.928200	+	0.372082i	$0.121356\pi$
$38$	0	0
$39$	−6.84614	−1.09626
$40$	0	0
$41$	−8.82468	−1.37818	−0.689092	−	0.724674i	$-0.741990\pi$
$41$	−8.82468	−1.37818	−0.689092	+	0.724674i	$0.741990\pi$
$42$	0	0
$43$	−5.01513	−0.764801	−0.382400	−	0.923997i	$-0.624902\pi$
$43$	−5.01513	−0.764801	−0.382400	+	0.923997i	$0.624902\pi$
$44$	0	0
$45$	0.735333	0.109617
$46$	0	0
$47$	4.60784	0.672122	0.336061	−	0.941840i	$-0.390905\pi$
$47$	4.60784	0.672122	0.336061	+	0.941840i	$0.390905\pi$
$48$	0	0
$49$	−2.76844	−0.395492
$50$	0	0
$51$	−11.6689	−1.63398
$52$	0	0
$53$	−7.97441	−1.09537	−0.547685	−	0.836685i	$-0.684491\pi$
$53$	−7.97441	−1.09537	−0.547685	+	0.836685i	$0.684491\pi$
$54$	0	0
$55$	−0.340242	−0.0458782
$56$	0	0
$57$	−4.06789	−0.538805
$58$	0	0
$59$	12.5908	1.63918	0.819591	−	0.572950i	$-0.194201\pi$
$59$	12.5908	1.63918	0.819591	+	0.572950i	$0.194201\pi$
$60$	0	0
$61$	−14.2309	−1.82209	−0.911043	−	0.412312i	$-0.864722\pi$
$61$	−14.2309	−1.82209	−0.911043	+	0.412312i	$0.864722\pi$
$62$	0	0
$63$	4.82370	0.607729
$64$	0	0
$65$	0.928602	0.115179
$66$	0	0
$67$	−3.95648	−0.483361	−0.241680	−	0.970356i	$-0.577699\pi$
$67$	−3.95648	−0.483361	−0.241680	+	0.970356i	$0.577699\pi$
$68$	0	0
$69$	12.6168	1.51889
$70$	0	0
$71$	3.67602	0.436263	0.218132	−	0.975919i	$-0.430004\pi$
$71$	3.67602	0.436263	0.218132	+	0.975919i	$0.430004\pi$
$72$	0	0
$73$	−0.0874702	−0.0102376	−0.00511881	−	0.999987i	$-0.501629\pi$
$73$	−0.0874702	−0.0102376	−0.00511881	+	0.999987i	$0.501629\pi$
$74$	0	0
$75$	11.3322	1.30853
$76$	0	0
$77$	−2.23195	−0.254354
$78$	0	0
$79$	6.33356	0.712581	0.356291	−	0.934375i	$-0.384041\pi$
$79$	6.33356	0.712581	0.356291	+	0.934375i	$0.384041\pi$
$80$	0	0
$81$	−10.5361	−1.17068
$82$	0	0
$83$	−12.1089	−1.32912	−0.664560	−	0.747235i	$-0.731381\pi$
$83$	−12.1089	−1.32912	−0.664560	+	0.747235i	$0.731381\pi$
$84$	0	0
$85$	1.58276	0.171674
$86$	0	0
$87$	11.1464	1.19502
$88$	0	0
$89$	−11.6351	−1.23332	−0.616661	−	0.787229i	$-0.711515\pi$
$89$	−11.6351	−1.23332	−0.616661	+	0.787229i	$0.711515\pi$
$90$	0	0
$91$	6.09151	0.638564
$92$	0	0
$93$	9.22973	0.957078
$94$	0	0
$95$	0.551763	0.0566097
$96$	0	0
$97$	−3.89026	−0.394996	−0.197498	−	0.980303i	$-0.563282\pi$
$97$	−3.89026	−0.394996	−0.197498	+	0.980303i	$0.563282\pi$
$98$	0	0
$99$	−2.54427	−0.255709
$100$	0	0
$101$	16.1957	1.61154	0.805768	−	0.592231i	$-0.201753\pi$
$101$	16.1957	1.61154	0.805768	+	0.592231i	$0.201753\pi$
$102$	0	0
$103$	0.396116	0.0390305	0.0195152	−	0.999810i	$-0.493788\pi$
$103$	0.396116	0.0390305	0.0195152	+	0.999810i	$0.493788\pi$
$104$	0	0
$105$	−1.49134	−0.145540
$106$	0	0
$107$	15.5233	1.50070	0.750349	−	0.661042i	$-0.229886\pi$
$107$	15.5233	1.50070	0.750349	+	0.661042i	$0.229886\pi$
$108$	0	0
$109$	−14.5641	−1.39499	−0.697494	−	0.716591i	$-0.745701\pi$
$109$	−14.5641	−1.39499	−0.697494	+	0.716591i	$0.745701\pi$
$110$	0	0
$111$	−26.1062	−2.47789
$112$	0	0
$113$	21.0421	1.97948	0.989738	−	0.142896i	$-0.0456415\pi$
$113$	21.0421	1.97948	0.989738	+	0.142896i	$0.0456415\pi$
$114$	0	0
$115$	−1.71133	−0.159582
$116$	0	0
$117$	6.94392	0.641966
$118$	0	0
$119$	10.3827	0.951779
$120$	0	0
$121$	−9.82275	−0.892978
$122$	0	0
$123$	20.4019	1.83958
$124$	0	0
$125$	−3.10501	−0.277720
$126$	0	0
$127$	−0.165017	−0.0146429	−0.00732146	−	0.999973i	$-0.502331\pi$
$127$	−0.165017	−0.0146429	−0.00732146	+	0.999973i	$0.502331\pi$
$128$	0	0
$129$	11.5945	1.02084
$130$	0	0
$131$	−15.6782	−1.36981	−0.684907	−	0.728631i	$-0.740157\pi$
$131$	−15.6782	−1.36981	−0.684907	+	0.728631i	$0.740157\pi$
$132$	0	0
$133$	3.61950	0.313850
$134$	0	0
$135$	0.474912	0.0408739
$136$	0	0
$137$	−2.34813	−0.200614	−0.100307	−	0.994957i	$-0.531982\pi$
$137$	−2.34813	−0.200614	−0.100307	+	0.994957i	$0.531982\pi$
$138$	0	0
$139$	−3.78482	−0.321024	−0.160512	−	0.987034i	$-0.551315\pi$
$139$	−3.78482	−0.321024	−0.160512	+	0.987034i	$0.551315\pi$
$140$	0	0
$141$	−10.6529	−0.897137
$142$	0	0
$143$	−3.21299	−0.268683
$144$	0	0
$145$	−1.51188	−0.125555
$146$	0	0
$147$	6.40039	0.527895
$148$	0	0
$149$	16.7918	1.37564	0.687819	−	0.725882i	$-0.258568\pi$
$149$	16.7918	1.37564	0.687819	+	0.725882i	$0.258568\pi$
$150$	0	0
$151$	−12.4633	−1.01425	−0.507125	−	0.861872i	$-0.669292\pi$
$151$	−12.4633	−1.01425	−0.507125	+	0.861872i	$0.669292\pi$
$152$	0	0
$153$	11.8356	0.956850
$154$	0	0
$155$	−1.25191	−0.100556
$156$	0	0
$157$	9.95206	0.794261	0.397130	−	0.917762i	$-0.370006\pi$
$157$	9.95206	0.794261	0.397130	+	0.917762i	$0.370006\pi$
$158$	0	0
$159$	18.4361	1.46208
$160$	0	0
$161$	−11.2261	−0.884742
$162$	0	0
$163$	0.573840	0.0449466	0.0224733	−	0.999747i	$-0.492846\pi$
$163$	0.573840	0.0449466	0.0224733	+	0.999747i	$0.492846\pi$
$164$	0	0
$165$	0.786610	0.0612375
$166$	0	0
$167$	−18.1228	−1.40239	−0.701193	−	0.712971i	$-0.747349\pi$
$167$	−18.1228	−1.40239	−0.701193	+	0.712971i	$0.747349\pi$
$168$	0	0
$169$	−4.23100	−0.325462
$170$	0	0
$171$	4.12599	0.315522
$172$	0	0
$173$	14.1417	1.07517	0.537585	−	0.843209i	$-0.319337\pi$
$173$	14.1417	1.07517	0.537585	+	0.843209i	$0.319337\pi$
$174$	0	0
$175$	−10.0831	−0.762210
$176$	0	0
$177$	−29.1088	−2.18795
$178$	0	0
$179$	16.9686	1.26829	0.634147	−	0.773212i	$-0.281351\pi$
$179$	16.9686	1.26829	0.634147	+	0.773212i	$0.281351\pi$
$180$	0	0
$181$	−14.6765	−1.09090	−0.545449	−	0.838144i	$-0.683641\pi$
$181$	−14.6765	−1.09090	−0.545449	+	0.838144i	$0.683641\pi$
$182$	0	0
$183$	32.9007	2.43209
$184$	0	0
$185$	3.54101	0.260340
$186$	0	0
$187$	−5.47638	−0.400472
$188$	0	0
$189$	3.11536	0.226609
$190$	0	0
$191$	8.84162	0.639757	0.319879	−	0.947459i	$-0.396358\pi$
$191$	8.84162	0.639757	0.319879	+	0.947459i	$0.396358\pi$
$192$	0	0
$193$	21.8647	1.57385	0.786927	−	0.617046i	$-0.211671\pi$
$193$	21.8647	1.57385	0.786927	+	0.617046i	$0.211671\pi$
$194$	0	0
$195$	−2.14684	−0.153739
$196$	0	0
$197$	9.39851	0.669616	0.334808	−	0.942286i	$-0.391329\pi$
$197$	9.39851	0.669616	0.334808	+	0.942286i	$0.391329\pi$
$198$	0	0
$199$	−20.1117	−1.42568	−0.712840	−	0.701327i	$-0.752591\pi$
$199$	−20.1117	−1.42568	−0.712840	+	0.701327i	$0.752591\pi$
$200$	0	0
$201$	9.14703	0.645182
$202$	0	0
$203$	−9.91774	−0.696089
$204$	0	0
$205$	−2.76728	−0.193275
$206$	0	0
$207$	−12.7970	−0.889455
$208$	0	0
$209$	−1.90911	−0.132056
$210$	0	0
$211$	19.4562	1.33942	0.669711	−	0.742622i	$-0.266418\pi$
$211$	19.4562	1.33942	0.669711	+	0.742622i	$0.266418\pi$
$212$	0	0
$213$	−8.49863	−0.582316
$214$	0	0
$215$	−1.57267	−0.107255
$216$	0	0
$217$	−8.21236	−0.557491
$218$	0	0
$219$	0.202223	0.0136650
$220$	0	0
$221$	14.9463	1.00540
$222$	0	0
$223$	−3.84963	−0.257790	−0.128895	−	0.991658i	$-0.541143\pi$
$223$	−3.84963	−0.257790	−0.128895	+	0.991658i	$0.541143\pi$
$224$	0	0
$225$	−11.4941	−0.766271
$226$	0	0
$227$	21.8944	1.45318	0.726592	−	0.687069i	$-0.241103\pi$
$227$	21.8944	1.45318	0.726592	+	0.687069i	$0.241103\pi$
$228$	0	0
$229$	−5.79770	−0.383123	−0.191561	−	0.981481i	$-0.561355\pi$
$229$	−5.79770	−0.383123	−0.191561	+	0.981481i	$0.561355\pi$
$230$	0	0
$231$	5.16006	0.339507
$232$	0	0
$233$	−8.75044	−0.573260	−0.286630	−	0.958041i	$-0.592535\pi$
$233$	−8.75044	−0.573260	−0.286630	+	0.958041i	$0.592535\pi$
$234$	0	0
$235$	1.44495	0.0942579
$236$	0	0
$237$	−14.6426	−0.951141
$238$	0	0
$239$	−20.4156	−1.32057	−0.660287	−	0.751013i	$-0.729565\pi$
$239$	−20.4156	−1.32057	−0.660287	+	0.751013i	$0.729565\pi$
$240$	0	0
$241$	0.651713	0.0419805	0.0209903	−	0.999780i	$-0.493318\pi$
$241$	0.651713	0.0419805	0.0209903	+	0.999780i	$0.493318\pi$
$242$	0	0
$243$	19.8151	1.27114
$244$	0	0
$245$	−0.868140	−0.0554634
$246$	0	0
$247$	5.21043	0.331531
$248$	0	0
$249$	27.9946	1.77409
$250$	0	0
$251$	−13.8281	−0.872824	−0.436412	−	0.899747i	$-0.643751\pi$
$251$	−13.8281	−0.872824	−0.436412	+	0.899747i	$0.643751\pi$
$252$	0	0
$253$	5.92125	0.372266
$254$	0	0
$255$	−3.65919	−0.229147
$256$	0	0
$257$	−18.2409	−1.13783	−0.568917	−	0.822395i	$-0.692637\pi$
$257$	−18.2409	−1.13783	−0.568917	+	0.822395i	$0.692637\pi$
$258$	0	0
$259$	23.2286	1.44335
$260$	0	0
$261$	−11.3056	−0.699797
$262$	0	0
$263$	−9.43545	−0.581815	−0.290907	−	0.956751i	$-0.593957\pi$
$263$	−9.43545	−0.581815	−0.290907	+	0.956751i	$0.593957\pi$
$264$	0	0
$265$	−2.50065	−0.153614
$266$	0	0
$267$	26.8994	1.64622
$268$	0	0
$269$	9.43395	0.575198	0.287599	−	0.957751i	$-0.407143\pi$
$269$	9.43395	0.575198	0.287599	+	0.957751i	$0.407143\pi$
$270$	0	0
$271$	−27.9870	−1.70009	−0.850045	−	0.526710i	$-0.823425\pi$
$271$	−27.9870	−1.70009	−0.850045	+	0.526710i	$0.823425\pi$
$272$	0	0
$273$	−14.0830	−0.852344
$274$	0	0
$275$	5.31836	0.320709
$276$	0	0
$277$	−13.4031	−0.805313	−0.402657	−	0.915351i	$-0.631913\pi$
$277$	−13.4031	−0.805313	−0.402657	+	0.915351i	$0.631913\pi$
$278$	0	0
$279$	−9.36155	−0.560461
$280$	0	0
$281$	14.8067	0.883296	0.441648	−	0.897188i	$-0.354394\pi$
$281$	14.8067	0.883296	0.441648	+	0.897188i	$0.354394\pi$
$282$	0	0
$283$	14.2952	0.849761	0.424881	−	0.905249i	$-0.360316\pi$
$283$	14.2952	0.849761	0.424881	+	0.905249i	$0.360316\pi$
$284$	0	0
$285$	−1.27563	−0.0755616
$286$	0	0
$287$	−18.1530	−1.07154
$288$	0	0
$289$	8.47530	0.498547
$290$	0	0
$291$	8.99393	0.527233
$292$	0	0
$293$	−22.1450	−1.29373	−0.646864	−	0.762606i	$-0.723920\pi$
$293$	−22.1450	−1.29373	−0.646864	+	0.762606i	$0.723920\pi$
$294$	0	0
$295$	3.94827	0.229877
$296$	0	0
$297$	−1.64321	−0.0953486
$298$	0	0
$299$	−16.1605	−0.934585
$300$	0	0
$301$	−10.3165	−0.594633
$302$	0	0
$303$	−37.4431	−2.15105
$304$	0	0
$305$	−4.46260	−0.255528
$306$	0	0
$307$	10.0601	0.574158	0.287079	−	0.957907i	$-0.407316\pi$
$307$	10.0601	0.574158	0.287079	+	0.957907i	$0.407316\pi$
$308$	0	0
$309$	−0.915785	−0.0520972
$310$	0	0
$311$	−23.1243	−1.31126	−0.655629	−	0.755083i	$-0.727597\pi$
$311$	−23.1243	−1.31126	−0.655629	+	0.755083i	$0.727597\pi$
$312$	0	0
$313$	−13.6493	−0.771502	−0.385751	−	0.922603i	$-0.626058\pi$
$313$	−13.6493	−0.771502	−0.385751	+	0.922603i	$0.626058\pi$
$314$	0	0
$315$	1.51264	0.0852274
$316$	0	0
$317$	10.9027	0.612356	0.306178	−	0.951974i	$-0.400950\pi$
$317$	10.9027	0.612356	0.306178	+	0.951974i	$0.400950\pi$
$318$	0	0
$319$	5.23114	0.292888
$320$	0	0
$321$	−35.8886	−2.00310
$322$	0	0
$323$	8.88092	0.494148
$324$	0	0
$325$	−14.5151	−0.805150
$326$	0	0
$327$	33.6709	1.86200
$328$	0	0
$329$	9.47867	0.522576
$330$	0	0
$331$	−13.6866	−0.752284	−0.376142	−	0.926562i	$-0.622750\pi$
$331$	−13.6866	−0.752284	−0.376142	+	0.926562i	$0.622750\pi$
$332$	0	0
$333$	26.4790	1.45104
$334$	0	0
$335$	−1.24069	−0.0677861
$336$	0	0
$337$	−27.2125	−1.48236	−0.741179	−	0.671307i	$-0.765733\pi$
$337$	−27.2125	−1.48236	−0.741179	+	0.671307i	$0.765733\pi$
$338$	0	0
$339$	−48.6475	−2.64217
$340$	0	0
$341$	4.33163	0.234571
$342$	0	0
$343$	−20.0944	−1.08500
$344$	0	0
$345$	3.95644	0.213008
$346$	0	0
$347$	−26.0157	−1.39660	−0.698298	−	0.715807i	$-0.746059\pi$
$347$	−26.0157	−1.39660	−0.698298	+	0.715807i	$0.746059\pi$
$348$	0	0
$349$	−12.4207	−0.664863	−0.332431	−	0.943127i	$-0.607869\pi$
$349$	−12.4207	−0.664863	−0.332431	+	0.943127i	$0.607869\pi$
$350$	0	0
$351$	4.48470	0.239376
$352$	0	0
$353$	1.91690	0.102026	0.0510131	−	0.998698i	$-0.483755\pi$
$353$	1.91690	0.102026	0.0510131	+	0.998698i	$0.483755\pi$
$354$	0	0
$355$	1.15274	0.0611812
$356$	0	0
$357$	−24.0038	−1.27042
$358$	0	0
$359$	1.77423	0.0936404	0.0468202	−	0.998903i	$-0.485091\pi$
$359$	1.77423	0.0936404	0.0468202	+	0.998903i	$0.485091\pi$
$360$	0	0
$361$	−15.9040	−0.837054
$362$	0	0
$363$	22.7093	1.19193
$364$	0	0
$365$	−0.0274293	−0.00143571
$366$	0	0
$367$	−5.74270	−0.299766	−0.149883	−	0.988704i	$-0.547890\pi$
$367$	−5.74270	−0.299766	−0.149883	+	0.988704i	$0.547890\pi$
$368$	0	0
$369$	−20.6933	−1.07725
$370$	0	0
$371$	−16.4040	−0.851651
$372$	0	0
$373$	1.13054	0.0585371	0.0292686	−	0.999572i	$-0.490682\pi$
$373$	1.13054	0.0585371	0.0292686	+	0.999572i	$0.490682\pi$
$374$	0	0
$375$	7.17850	0.370696
$376$	0	0
$377$	−14.2770	−0.735304
$378$	0	0
$379$	26.7808	1.37564	0.687818	−	0.725883i	$-0.258569\pi$
$379$	26.7808	1.37564	0.687818	+	0.725883i	$0.258569\pi$
$380$	0	0
$381$	0.381505	0.0195451
$382$	0	0
$383$	17.2752	0.882720	0.441360	−	0.897330i	$-0.354496\pi$
$383$	17.2752	0.882720	0.441360	+	0.897330i	$0.354496\pi$
$384$	0	0
$385$	−0.699904	−0.0356704
$386$	0	0
$387$	−11.7601	−0.597801
$388$	0	0
$389$	9.30090	0.471574	0.235787	−	0.971805i	$-0.424233\pi$
$389$	9.30090	0.471574	0.235787	+	0.971805i	$0.424233\pi$
$390$	0	0
$391$	−27.5448	−1.39300
$392$	0	0
$393$	36.2467	1.82840
$394$	0	0
$395$	1.98611	0.0999318
$396$	0	0
$397$	−23.8089	−1.19493	−0.597466	−	0.801894i	$-0.703826\pi$
$397$	−23.8089	−1.19493	−0.597466	+	0.801894i	$0.703826\pi$
$398$	0	0
$399$	−8.36796	−0.418922
$400$	0	0
$401$	−12.5745	−0.627940	−0.313970	−	0.949433i	$-0.601659\pi$
$401$	−12.5745	−0.627940	−0.313970	+	0.949433i	$0.601659\pi$
$402$	0	0
$403$	−11.8221	−0.588898
$404$	0	0
$405$	−3.30395	−0.164175
$406$	0	0
$407$	−12.2520	−0.607308
$408$	0	0
$409$	1.19394	0.0590366	0.0295183	−	0.999564i	$-0.490603\pi$
$409$	1.19394	0.0590366	0.0295183	+	0.999564i	$0.490603\pi$
$410$	0	0
$411$	5.42866	0.267776
$412$	0	0
$413$	25.9002	1.27447
$414$	0	0
$415$	−3.79715	−0.186395
$416$	0	0
$417$	8.75016	0.428497
$418$	0	0
$419$	22.1433	1.08177	0.540886	−	0.841096i	$-0.318089\pi$
$419$	22.1433	1.08177	0.540886	+	0.841096i	$0.318089\pi$
$420$	0	0
$421$	−0.885112	−0.0431377	−0.0215689	−	0.999767i	$-0.506866\pi$
$421$	−0.885112	−0.0431377	−0.0215689	+	0.999767i	$0.506866\pi$
$422$	0	0
$423$	10.8051	0.525360
$424$	0	0
$425$	−24.7402	−1.20008
$426$	0	0
$427$	−29.2741	−1.41667
$428$	0	0
$429$	7.42814	0.358634
$430$	0	0
$431$	−6.37737	−0.307187	−0.153594	−	0.988134i	$-0.549085\pi$
$431$	−6.37737	−0.307187	−0.153594	+	0.988134i	$0.549085\pi$
$432$	0	0
$433$	13.4133	0.644601	0.322300	−	0.946637i	$-0.395544\pi$
$433$	13.4133	0.644601	0.322300	+	0.946637i	$0.395544\pi$
$434$	0	0
$435$	3.49533	0.167588
$436$	0	0
$437$	−9.60235	−0.459343
$438$	0	0
$439$	21.6740	1.03444	0.517222	−	0.855851i	$-0.326966\pi$
$439$	21.6740	1.03444	0.517222	+	0.855851i	$0.326966\pi$
$440$	0	0
$441$	−6.49180	−0.309133
$442$	0	0
$443$	−22.7776	−1.08220	−0.541098	−	0.840960i	$-0.681991\pi$
$443$	−22.7776	−1.08220	−0.541098	+	0.840960i	$0.681991\pi$
$444$	0	0
$445$	−3.64860	−0.172960
$446$	0	0
$447$	−38.8212	−1.83618
$448$	0	0
$449$	27.0661	1.27733	0.638664	−	0.769486i	$-0.279488\pi$
$449$	27.0661	1.27733	0.638664	+	0.769486i	$0.279488\pi$
$450$	0	0
$451$	9.57487	0.450863
$452$	0	0
$453$	28.8141	1.35380
$454$	0	0
$455$	1.91020	0.0895517
$456$	0	0
$457$	22.6730	1.06060	0.530300	−	0.847810i	$-0.322079\pi$
$457$	22.6730	1.06060	0.530300	+	0.847810i	$0.322079\pi$
$458$	0	0
$459$	7.64396	0.356789
$460$	0	0
$461$	26.3466	1.22708	0.613541	−	0.789663i	$-0.289745\pi$
$461$	26.3466	1.22708	0.613541	+	0.789663i	$0.289745\pi$
$462$	0	0
$463$	−21.6735	−1.00725	−0.503626	−	0.863922i	$-0.668001\pi$
$463$	−21.6735	−1.00725	−0.503626	+	0.863922i	$0.668001\pi$
$464$	0	0
$465$	2.89430	0.134220
$466$	0	0
$467$	−14.0501	−0.650162	−0.325081	−	0.945686i	$-0.605392\pi$
$467$	−14.0501	−0.650162	−0.325081	+	0.945686i	$0.605392\pi$
$468$	0	0
$469$	−8.13878	−0.375814
$470$	0	0
$471$	−23.0083	−1.06017
$472$	0	0
$473$	5.44147	0.250199
$474$	0	0
$475$	−8.62466	−0.395726
$476$	0	0
$477$	−18.6994	−0.856188
$478$	0	0
$479$	−3.03697	−0.138763	−0.0693813	−	0.997590i	$-0.522103\pi$
$479$	−3.03697	−0.138763	−0.0693813	+	0.997590i	$0.522103\pi$
$480$	0	0
$481$	33.4385	1.52467
$482$	0	0
$483$	25.9538	1.18094
$484$	0	0
$485$	−1.21992	−0.0553939
$486$	0	0
$487$	23.8023	1.07858	0.539292	−	0.842119i	$-0.318692\pi$
$487$	23.8023	1.07858	0.539292	+	0.842119i	$0.318692\pi$
$488$	0	0
$489$	−1.32667	−0.0599940
$490$	0	0
$491$	9.32412	0.420792	0.210396	−	0.977616i	$-0.432525\pi$
$491$	9.32412	0.420792	0.210396	+	0.977616i	$0.432525\pi$
$492$	0	0
$493$	−24.3345	−1.09597
$494$	0	0
$495$	−0.797844	−0.0358604
$496$	0	0
$497$	7.56185	0.339195
$498$	0	0
$499$	−28.7311	−1.28618	−0.643089	−	0.765791i	$-0.722348\pi$
$499$	−28.7311	−1.28618	−0.643089	+	0.765791i	$0.722348\pi$
$500$	0	0
$501$	41.8984	1.87188
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	5.07873	0.226001
$506$	0	0
$507$	9.78169	0.434420
$508$	0	0
$509$	−10.2027	−0.452227	−0.226113	−	0.974101i	$-0.572602\pi$
$509$	−10.2027	−0.452227	−0.226113	+	0.974101i	$0.572602\pi$
$510$	0	0
$511$	−0.179933	−0.00795976
$512$	0	0
$513$	2.66475	0.117652
$514$	0	0
$515$	0.124216	0.00547360
$516$	0	0
$517$	−4.99955	−0.219880
$518$	0	0
$519$	−32.6943	−1.43512
$520$	0	0
$521$	10.3231	0.452265	0.226133	−	0.974097i	$-0.427392\pi$
$521$	10.3231	0.452265	0.226133	+	0.974097i	$0.427392\pi$
$522$	0	0
$523$	−30.6867	−1.34184	−0.670918	−	0.741532i	$-0.734100\pi$
$523$	−30.6867	−1.34184	−0.670918	+	0.741532i	$0.734100\pi$
$524$	0	0
$525$	23.3112	1.01738
$526$	0	0
$527$	−20.1501	−0.877753
$528$	0	0
$529$	6.78235	0.294885
$530$	0	0
$531$	29.5245	1.28125
$532$	0	0
$533$	−26.1321	−1.13191
$534$	0	0
$535$	4.86788	0.210457
$536$	0	0
$537$	−39.2299	−1.69290
$538$	0	0
$539$	3.00379	0.129382
$540$	0	0
$541$	−8.48911	−0.364975	−0.182488	−	0.983208i	$-0.558415\pi$
$541$	−8.48911	−0.364975	−0.182488	+	0.983208i	$0.558415\pi$
$542$	0	0
$543$	33.9308	1.45611
$544$	0	0
$545$	−4.56707	−0.195632
$546$	0	0
$547$	−11.3150	−0.483796	−0.241898	−	0.970302i	$-0.577770\pi$
$547$	−11.3150	−0.483796	−0.241898	+	0.970302i	$0.577770\pi$
$548$	0	0
$549$	−33.3706	−1.42422
$550$	0	0
$551$	−8.48322	−0.361397
$552$	0	0
$553$	13.0286	0.554033
$554$	0	0
$555$	−8.18649	−0.347497
$556$	0	0
$557$	18.9841	0.804384	0.402192	−	0.915555i	$-0.368248\pi$
$557$	18.9841	0.804384	0.402192	+	0.915555i	$0.368248\pi$
$558$	0	0
$559$	−14.8511	−0.628133
$560$	0	0
$561$	12.6609	0.534543
$562$	0	0
$563$	−31.0339	−1.30792	−0.653961	−	0.756528i	$-0.726894\pi$
$563$	−31.0339	−1.30792	−0.653961	+	0.756528i	$0.726894\pi$
$564$	0	0
$565$	6.59848	0.277600
$566$	0	0
$567$	−21.6735	−0.910203
$568$	0	0
$569$	−7.73591	−0.324306	−0.162153	−	0.986766i	$-0.551844\pi$
$569$	−7.73591	−0.324306	−0.162153	+	0.986766i	$0.551844\pi$
$570$	0	0
$571$	−15.3422	−0.642050	−0.321025	−	0.947071i	$-0.604027\pi$
$571$	−15.3422	−0.642050	−0.321025	+	0.947071i	$0.604027\pi$
$572$	0	0
$573$	−20.4410	−0.853937
$574$	0	0
$575$	26.7500	1.11555
$576$	0	0
$577$	32.3069	1.34495	0.672476	−	0.740119i	$-0.265231\pi$
$577$	32.3069	1.34495	0.672476	+	0.740119i	$0.265231\pi$
$578$	0	0
$579$	−50.5492	−2.10075
$580$	0	0
$581$	−24.9088	−1.03339
$582$	0	0
$583$	8.65231	0.358342
$584$	0	0
$585$	2.17751	0.0900288
$586$	0	0
$587$	−32.1738	−1.32796	−0.663978	−	0.747752i	$-0.731133\pi$
$587$	−32.1738	−1.32796	−0.663978	+	0.747752i	$0.731133\pi$
$588$	0	0
$589$	−7.02451	−0.289440
$590$	0	0
$591$	−21.7285	−0.893792
$592$	0	0
$593$	13.5037	0.554529	0.277264	−	0.960794i	$-0.410572\pi$
$593$	13.5037	0.554529	0.277264	+	0.960794i	$0.410572\pi$
$594$	0	0
$595$	3.25585	0.133477
$596$	0	0
$597$	46.4964	1.90297
$598$	0	0
$599$	−41.1016	−1.67937	−0.839683	−	0.543077i	$-0.817259\pi$
$599$	−41.1016	−1.67937	−0.839683	+	0.543077i	$0.817259\pi$
$600$	0	0
$601$	−41.9216	−1.71002	−0.855009	−	0.518613i	$-0.826448\pi$
$601$	−41.9216	−1.71002	−0.855009	+	0.518613i	$0.826448\pi$
$602$	0	0
$603$	−9.27767	−0.377816
$604$	0	0
$605$	−3.08026	−0.125230
$606$	0	0
$607$	−30.0202	−1.21848	−0.609241	−	0.792985i	$-0.708526\pi$
$607$	−30.0202	−1.21848	−0.609241	+	0.792985i	$0.708526\pi$
$608$	0	0
$609$	22.9289	0.929127
$610$	0	0
$611$	13.6450	0.552016
$612$	0	0
$613$	−5.13831	−0.207535	−0.103767	−	0.994602i	$-0.533090\pi$
$613$	−5.13831	−0.207535	−0.103767	+	0.994602i	$0.533090\pi$
$614$	0	0
$615$	6.39771	0.257981
$616$	0	0
$617$	−28.5920	−1.15107	−0.575535	−	0.817777i	$-0.695206\pi$
$617$	−28.5920	−1.15107	−0.575535	+	0.817777i	$0.695206\pi$
$618$	0	0
$619$	17.4136	0.699912	0.349956	−	0.936766i	$-0.386197\pi$
$619$	17.4136	0.699912	0.349956	+	0.936766i	$0.386197\pi$
$620$	0	0
$621$	−8.26491	−0.331659
$622$	0	0
$623$	−23.9344	−0.958910
$624$	0	0
$625$	23.5346	0.941386
$626$	0	0
$627$	4.41370	0.176266
$628$	0	0
$629$	56.9944	2.27251
$630$	0	0
$631$	7.43247	0.295882	0.147941	−	0.988996i	$-0.452735\pi$
$631$	7.43247	0.295882	0.147941	+	0.988996i	$0.452735\pi$
$632$	0	0
$633$	−44.9810	−1.78784
$634$	0	0
$635$	−0.0517469	−0.00205351
$636$	0	0
$637$	−8.19805	−0.324819
$638$	0	0
$639$	8.62001	0.341002
$640$	0	0
$641$	−16.5290	−0.652857	−0.326429	−	0.945222i	$-0.605845\pi$
$641$	−16.5290	−0.652857	−0.326429	+	0.945222i	$0.605845\pi$
$642$	0	0
$643$	−30.4545	−1.20101	−0.600505	−	0.799621i	$-0.705034\pi$
$643$	−30.4545	−1.20101	−0.600505	+	0.799621i	$0.705034\pi$
$644$	0	0
$645$	3.63587	0.143162
$646$	0	0
$647$	47.4767	1.86650	0.933251	−	0.359225i	$-0.116959\pi$
$647$	47.4767	1.86650	0.933251	+	0.359225i	$0.116959\pi$
$648$	0	0
$649$	−13.6611	−0.536246
$650$	0	0
$651$	18.9862	0.744129
$652$	0	0
$653$	−20.2956	−0.794229	−0.397115	−	0.917769i	$-0.629988\pi$
$653$	−20.2956	−0.794229	−0.397115	+	0.917769i	$0.629988\pi$
$654$	0	0
$655$	−4.91645	−0.192102
$656$	0	0
$657$	−0.205112	−0.00800216
$658$	0	0
$659$	−17.1982	−0.669948	−0.334974	−	0.942227i	$-0.608728\pi$
$659$	−17.1982	−0.669948	−0.334974	+	0.942227i	$0.608728\pi$
$660$	0	0
$661$	−30.1298	−1.17191	−0.585956	−	0.810343i	$-0.699281\pi$
$661$	−30.1298	−1.17191	−0.585956	+	0.810343i	$0.699281\pi$
$662$	0	0
$663$	−34.5546	−1.34199
$664$	0	0
$665$	1.13502	0.0440141
$666$	0	0
$667$	26.3113	1.01878
$668$	0	0
$669$	8.90000	0.344094
$670$	0	0
$671$	15.4407	0.596082
$672$	0	0
$673$	−17.0150	−0.655879	−0.327939	−	0.944699i	$-0.606354\pi$
$673$	−17.0150	−0.655879	−0.327939	+	0.944699i	$0.606354\pi$
$674$	0	0
$675$	−7.42339	−0.285726
$676$	0	0
$677$	−28.2993	−1.08763	−0.543815	−	0.839205i	$-0.683021\pi$
$677$	−28.2993	−1.08763	−0.543815	+	0.839205i	$0.683021\pi$
$678$	0	0
$679$	−8.00255	−0.307110
$680$	0	0
$681$	−50.6179	−1.93968
$682$	0	0
$683$	−23.5315	−0.900407	−0.450204	−	0.892926i	$-0.648649\pi$
$683$	−23.5315	−0.900407	−0.450204	+	0.892926i	$0.648649\pi$
$684$	0	0
$685$	−0.736335	−0.0281339
$686$	0	0
$687$	13.4038	0.511385
$688$	0	0
$689$	−23.6142	−0.899630
$690$	0	0
$691$	15.6433	0.595097	0.297549	−	0.954707i	$-0.403831\pi$
$691$	15.6433	0.595097	0.297549	+	0.954707i	$0.403831\pi$
$692$	0	0
$693$	−5.23376	−0.198814
$694$	0	0
$695$	−1.18686	−0.0450201
$696$	0	0
$697$	−44.5409	−1.68711
$698$	0	0
$699$	20.2302	0.765178
$700$	0	0
$701$	−15.1173	−0.570973	−0.285486	−	0.958383i	$-0.592155\pi$
$701$	−15.1173	−0.570973	−0.285486	+	0.958383i	$0.592155\pi$
$702$	0	0
$703$	19.8687	0.749364
$704$	0	0
$705$	−3.34059	−0.125814
$706$	0	0
$707$	33.3159	1.25297
$708$	0	0
$709$	−25.1426	−0.944252	−0.472126	−	0.881531i	$-0.656513\pi$
$709$	−25.1426	−0.944252	−0.472126	+	0.881531i	$0.656513\pi$
$710$	0	0
$711$	14.8518	0.556984
$712$	0	0
$713$	21.7870	0.815929
$714$	0	0
$715$	−1.00754	−0.0376799
$716$	0	0
$717$	47.1990	1.76268
$718$	0	0
$719$	−47.4630	−1.77007	−0.885036	−	0.465523i	$-0.845866\pi$
$719$	−47.4630	−1.77007	−0.885036	+	0.465523i	$0.845866\pi$
$720$	0	0
$721$	0.814840	0.0303462
$722$	0	0
$723$	−1.50670	−0.0560348
$724$	0	0
$725$	23.6323	0.877682
$726$	0	0
$727$	33.1208	1.22838	0.614190	−	0.789158i	$-0.289483\pi$
$727$	33.1208	1.22838	0.614190	+	0.789158i	$0.289483\pi$
$728$	0	0
$729$	−14.2025	−0.526018
$730$	0	0
$731$	−25.3129	−0.936232
$732$	0	0
$733$	11.3389	0.418811	0.209405	−	0.977829i	$-0.432847\pi$
$733$	11.3389	0.418811	0.209405	+	0.977829i	$0.432847\pi$
$734$	0	0
$735$	2.00706	0.0740316
$736$	0	0
$737$	4.29282	0.158128
$738$	0	0
$739$	−24.5679	−0.903744	−0.451872	−	0.892083i	$-0.649244\pi$
$739$	−24.5679	−0.903744	−0.451872	+	0.892083i	$0.649244\pi$
$740$	0	0
$741$	−12.0460	−0.442522
$742$	0	0
$743$	−42.9122	−1.57430	−0.787148	−	0.616765i	$-0.788443\pi$
$743$	−42.9122	−1.57430	−0.787148	+	0.616765i	$0.788443\pi$
$744$	0	0
$745$	5.26565	0.192918
$746$	0	0
$747$	−28.3944	−1.03890
$748$	0	0
$749$	31.9327	1.16679
$750$	0	0
$751$	−33.4029	−1.21889	−0.609445	−	0.792828i	$-0.708608\pi$
$751$	−33.4029	−1.21889	−0.609445	+	0.792828i	$0.708608\pi$
$752$	0	0
$753$	31.9694	1.16503
$754$	0	0
$755$	−3.90830	−0.142238
$756$	0	0
$757$	15.7657	0.573014	0.286507	−	0.958078i	$-0.407506\pi$
$757$	15.7657	0.573014	0.286507	+	0.958078i	$0.407506\pi$
$758$	0	0
$759$	−13.6894	−0.496894
$760$	0	0
$761$	−26.9916	−0.978445	−0.489223	−	0.872159i	$-0.662719\pi$
$761$	−26.9916	−0.978445	−0.489223	+	0.872159i	$0.662719\pi$
$762$	0	0
$763$	−29.9594	−1.08460
$764$	0	0
$765$	3.71145	0.134188
$766$	0	0
$767$	37.2845	1.34626
$768$	0	0
$769$	−42.2575	−1.52385	−0.761923	−	0.647668i	$-0.775745\pi$
$769$	−42.2575	−1.52385	−0.761923	+	0.647668i	$0.775745\pi$
$770$	0	0
$771$	42.1712	1.51876
$772$	0	0
$773$	−9.46197	−0.340324	−0.170162	−	0.985416i	$-0.554429\pi$
$773$	−9.46197	−0.340324	−0.170162	+	0.985416i	$0.554429\pi$
$774$	0	0
$775$	19.5687	0.702928
$776$	0	0
$777$	−53.7024	−1.92656
$778$	0	0
$779$	−15.5274	−0.556325
$780$	0	0
$781$	−3.98852	−0.142720
$782$	0	0
$783$	−7.30165	−0.260940
$784$	0	0
$785$	3.12081	0.111387
$786$	0	0
$787$	4.14712	0.147829	0.0739144	−	0.997265i	$-0.476451\pi$
$787$	4.14712	0.147829	0.0739144	+	0.997265i	$0.476451\pi$
$788$	0	0
$789$	21.8139	0.776596
$790$	0	0
$791$	43.2852	1.53904
$792$	0	0
$793$	−42.1414	−1.49648
$794$	0	0
$795$	5.78128	0.205041
$796$	0	0
$797$	38.8666	1.37672	0.688362	−	0.725367i	$-0.258330\pi$
$797$	38.8666	1.37672	0.688362	+	0.725367i	$0.258330\pi$
$798$	0	0
$799$	23.2572	0.822780
$800$	0	0
$801$	−27.2836	−0.964018
$802$	0	0
$803$	0.0949061	0.00334916
$804$	0	0
$805$	−3.52034	−0.124076
$806$	0	0
$807$	−21.8104	−0.767764
$808$	0	0
$809$	11.9348	0.419606	0.209803	−	0.977744i	$-0.432718\pi$
$809$	11.9348	0.419606	0.209803	+	0.977744i	$0.432718\pi$
$810$	0	0
$811$	37.3838	1.31272	0.656361	−	0.754447i	$-0.272095\pi$
$811$	37.3838	1.31272	0.656361	+	0.754447i	$0.272095\pi$
$812$	0	0
$813$	64.7035	2.26925
$814$	0	0
$815$	0.179947	0.00630328
$816$	0	0
$817$	−8.82431	−0.308723
$818$	0	0
$819$	14.2842	0.499129
$820$	0	0
$821$	31.8685	1.11222	0.556109	−	0.831109i	$-0.312294\pi$
$821$	31.8685	1.11222	0.556109	+	0.831109i	$0.312294\pi$
$822$	0	0
$823$	−21.6594	−0.754998	−0.377499	−	0.926010i	$-0.623216\pi$
$823$	−21.6594	−0.754998	−0.377499	+	0.926010i	$0.623216\pi$
$824$	0	0
$825$	−12.2956	−0.428077
$826$	0	0
$827$	−17.3472	−0.603222	−0.301611	−	0.953431i	$-0.597524\pi$
$827$	−17.3472	−0.603222	−0.301611	+	0.953431i	$0.597524\pi$
$828$	0	0
$829$	−22.8312	−0.792960	−0.396480	−	0.918043i	$-0.629768\pi$
$829$	−22.8312	−0.792960	−0.396480	+	0.918043i	$0.629768\pi$
$830$	0	0
$831$	30.9867	1.07492
$832$	0	0
$833$	−13.9732	−0.484142
$834$	0	0
$835$	−5.68303	−0.196670
$836$	0	0
$837$	−6.04612	−0.208984
$838$	0	0
$839$	1.11414	0.0384643	0.0192322	−	0.999815i	$-0.493878\pi$
$839$	1.11414	0.0384643	0.0192322	+	0.999815i	$0.493878\pi$
$840$	0	0
$841$	−5.75524	−0.198457
$842$	0	0
$843$	−34.2319	−1.17901
$844$	0	0
$845$	−1.32678	−0.0456425
$846$	0	0
$847$	−20.2061	−0.694291
$848$	0	0
$849$	−33.0492	−1.13425
$850$	0	0
$851$	−61.6243	−2.11245
$852$	0	0
$853$	56.4461	1.93268	0.966339	−	0.257271i	$-0.0828234\pi$
$853$	56.4461	1.93268	0.966339	+	0.257271i	$0.0828234\pi$
$854$	0	0
$855$	1.29385	0.0442486
$856$	0	0
$857$	44.3950	1.51650	0.758252	−	0.651961i	$-0.226053\pi$
$857$	44.3950	1.51650	0.758252	+	0.651961i	$0.226053\pi$
$858$	0	0
$859$	−12.3720	−0.422128	−0.211064	−	0.977472i	$-0.567693\pi$
$859$	−12.3720	−0.422128	−0.211064	+	0.977472i	$0.567693\pi$
$860$	0	0
$861$	41.9682	1.43027
$862$	0	0
$863$	10.4584	0.356007	0.178003	−	0.984030i	$-0.443036\pi$
$863$	10.4584	0.356007	0.178003	+	0.984030i	$0.443036\pi$
$864$	0	0
$865$	4.43460	0.150781
$866$	0	0
$867$	−19.5941	−0.665452
$868$	0	0
$869$	−6.87198	−0.233116
$870$	0	0
$871$	−11.7161	−0.396986
$872$	0	0
$873$	−9.12238	−0.308746
$874$	0	0
$875$	−6.38723	−0.215928
$876$	0	0
$877$	−17.1800	−0.580127	−0.290064	−	0.957007i	$-0.593676\pi$
$877$	−17.1800	−0.580127	−0.290064	+	0.957007i	$0.593676\pi$
$878$	0	0
$879$	51.1974	1.72684
$880$	0	0
$881$	−21.7701	−0.733455	−0.366727	−	0.930328i	$-0.619522\pi$
$881$	−21.7701	−0.733455	−0.366727	+	0.930328i	$0.619522\pi$
$882$	0	0
$883$	51.8136	1.74366	0.871832	−	0.489804i	$-0.162932\pi$
$883$	51.8136	1.74366	0.871832	+	0.489804i	$0.162932\pi$
$884$	0	0
$885$	−9.12806	−0.306836
$886$	0	0
$887$	30.4190	1.02137	0.510685	−	0.859768i	$-0.329392\pi$
$887$	30.4190	1.02137	0.510685	+	0.859768i	$0.329392\pi$
$888$	0	0
$889$	−0.339453	−0.0113849
$890$	0	0
$891$	11.4318	0.382979
$892$	0	0
$893$	8.10766	0.271313
$894$	0	0
$895$	5.32109	0.177865
$896$	0	0
$897$	37.3616	1.24747
$898$	0	0
$899$	19.2478	0.641949
$900$	0	0
$901$	−40.2493	−1.34090
$902$	0	0
$903$	23.8508	0.793706
$904$	0	0
$905$	−4.60233	−0.152987
$906$	0	0
$907$	−18.8270	−0.625140	−0.312570	−	0.949895i	$-0.601190\pi$
$907$	−18.8270	−0.625140	−0.312570	+	0.949895i	$0.601190\pi$
$908$	0	0
$909$	37.9779	1.25965
$910$	0	0
$911$	−17.8657	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$911$	−17.8657	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$912$	0	0
$913$	13.1382	0.434812
$914$	0	0
$915$	10.3171	0.341074
$916$	0	0
$917$	−32.2513	−1.06503
$918$	0	0
$919$	7.19078	0.237202	0.118601	−	0.992942i	$-0.462159\pi$
$919$	7.19078	0.237202	0.118601	+	0.992942i	$0.462159\pi$
$920$	0	0
$921$	−23.2580	−0.766376
$922$	0	0
$923$	10.8856	0.358304
$924$	0	0
$925$	−55.3498	−1.81989
$926$	0	0
$927$	0.928864	0.0305079
$928$	0	0
$929$	5.00013	0.164049	0.0820246	−	0.996630i	$-0.473861\pi$
$929$	5.00013	0.164049	0.0820246	+	0.996630i	$0.473861\pi$
$930$	0	0
$931$	−4.87117	−0.159646
$932$	0	0
$933$	53.4613	1.75024
$934$	0	0
$935$	−1.71731	−0.0561619
$936$	0	0
$937$	−26.3052	−0.859353	−0.429677	−	0.902983i	$-0.641372\pi$
$937$	−26.3052	−0.859353	−0.429677	+	0.902983i	$0.641372\pi$
$938$	0	0
$939$	31.5559	1.02979
$940$	0	0
$941$	−36.4265	−1.18747	−0.593734	−	0.804661i	$-0.702347\pi$
$941$	−36.4265	−1.18747	−0.593734	+	0.804661i	$0.702347\pi$
$942$	0	0
$943$	48.1591	1.56828
$944$	0	0
$945$	0.976930	0.0317795
$946$	0	0
$947$	2.39473	0.0778182	0.0389091	−	0.999243i	$-0.487612\pi$
$947$	2.39473	0.0778182	0.0389091	+	0.999243i	$0.487612\pi$
$948$	0	0
$949$	−0.259021	−0.00840818
$950$	0	0
$951$	−25.2060	−0.817361
$952$	0	0
$953$	−39.2429	−1.27120	−0.635601	−	0.772017i	$-0.719248\pi$
$953$	−39.2429	−1.27120	−0.635601	+	0.772017i	$0.719248\pi$
$954$	0	0
$955$	2.77259	0.0897191
$956$	0	0
$957$	−12.0939	−0.390941
$958$	0	0
$959$	−4.83027	−0.155978
$960$	0	0
$961$	−15.0619	−0.485869
$962$	0	0
$963$	36.4011	1.17301
$964$	0	0
$965$	6.85643	0.220716
$966$	0	0
$967$	36.1119	1.16128	0.580640	−	0.814160i	$-0.302802\pi$
$967$	36.1119	1.16128	0.580640	+	0.814160i	$0.302802\pi$
$968$	0	0
$969$	−20.5319	−0.659579
$970$	0	0
$971$	39.9564	1.28226	0.641130	−	0.767432i	$-0.278466\pi$
$971$	39.9564	1.28226	0.641130	+	0.767432i	$0.278466\pi$
$972$	0	0
$973$	−7.78565	−0.249597
$974$	0	0
$975$	33.5575	1.07470
$976$	0	0
$977$	16.2872	0.521072	0.260536	−	0.965464i	$-0.416101\pi$
$977$	16.2872	0.521072	0.260536	+	0.965464i	$0.416101\pi$
$978$	0	0
$979$	12.6242	0.403473
$980$	0	0
$981$	−34.1518	−1.09038
$982$	0	0
$983$	−29.0951	−0.927988	−0.463994	−	0.885838i	$-0.653584\pi$
$983$	−29.0951	−0.927988	−0.463994	+	0.885838i	$0.653584\pi$
$984$	0	0
$985$	2.94723	0.0939064
$986$	0	0
$987$	−21.9138	−0.697525
$988$	0	0
$989$	27.3692	0.870289
$990$	0	0
$991$	−23.1783	−0.736284	−0.368142	−	0.929770i	$-0.620006\pi$
$991$	−23.1783	−0.736284	−0.368142	+	0.929770i	$0.620006\pi$
$992$	0	0
$993$	31.6422	1.00414
$994$	0	0
$995$	−6.30671	−0.199936
$996$	0	0
$997$	52.3558	1.65813	0.829063	−	0.559155i	$-0.188874\pi$
$997$	52.3558	1.65813	0.829063	+	0.559155i	$0.188874\pi$
$998$	0	0
$999$	17.1014	0.541064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.4		12
4.3	odd	2		1006.2.a.j.1.9	✓	12
12.11	even	2		9054.2.a.bi.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.9	✓	12	4.3	odd	2
8048.2.a.q.1.4		12	1.1	even	1	trivial
9054.2.a.bi.1.7		12	12.11	even	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31191 q^{3} +0.313584 q^{5} +2.05708 q^{7} +2.34493 q^{9} +O(q^{10})\)	\(q-2.31191 q^{3} +0.313584 q^{5} +2.05708 q^{7} +2.34493 q^{9} -1.08501 q^{11} +2.96125 q^{13} -0.724979 q^{15} +5.04731 q^{17} +1.75954 q^{19} -4.75577 q^{21} -5.45732 q^{23} -4.90166 q^{25} +1.51446 q^{27} -4.82128 q^{29} -3.99225 q^{31} +2.50845 q^{33} +0.645067 q^{35} +11.2920 q^{37} -6.84614 q^{39} -8.82468 q^{41} -5.01513 q^{43} +0.735333 q^{45} +4.60784 q^{47} -2.76844 q^{49} -11.6689 q^{51} -7.97441 q^{53} -0.340242 q^{55} -4.06789 q^{57} +12.5908 q^{59} -14.2309 q^{61} +4.82370 q^{63} +0.928602 q^{65} -3.95648 q^{67} +12.6168 q^{69} +3.67602 q^{71} -0.0874702 q^{73} +11.3322 q^{75} -2.23195 q^{77} +6.33356 q^{79} -10.5361 q^{81} -12.1089 q^{83} +1.58276 q^{85} +11.1464 q^{87} -11.6351 q^{89} +6.09151 q^{91} +9.22973 q^{93} +0.551763 q^{95} -3.89026 q^{97} -2.54427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

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