LMFDB - Embedded newform 8048.2.a.x.1.9

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:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-2.03324 q^{3} +4.46117 q^{5} -0.828843 q^{7} +1.13405 q^{9} +O(q^{10})$	$q-2.03324 q^{3} +4.46117 q^{5} -0.828843 q^{7} +1.13405 q^{9} -0.0966604 q^{11} +1.53910 q^{13} -9.07062 q^{15} -3.59316 q^{17} -8.33503 q^{19} +1.68523 q^{21} -2.93109 q^{23} +14.9021 q^{25} +3.79392 q^{27} +1.38083 q^{29} +2.48771 q^{31} +0.196533 q^{33} -3.69761 q^{35} -3.06621 q^{37} -3.12936 q^{39} -5.09004 q^{41} +12.0742 q^{43} +5.05920 q^{45} +3.75114 q^{47} -6.31302 q^{49} +7.30575 q^{51} +5.98285 q^{53} -0.431219 q^{55} +16.9471 q^{57} -1.55903 q^{59} -11.6668 q^{61} -0.939950 q^{63} +6.86620 q^{65} -1.19054 q^{67} +5.95959 q^{69} +5.48402 q^{71} +5.93015 q^{73} -30.2994 q^{75} +0.0801163 q^{77} -12.9836 q^{79} -11.1161 q^{81} -16.7451 q^{83} -16.0297 q^{85} -2.80755 q^{87} -7.41727 q^{89} -1.27567 q^{91} -5.05810 q^{93} -37.1840 q^{95} -14.8864 q^{97} -0.109618 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * 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
$3$	−2.03324	−1.17389	−0.586945	−	0.809627i	$-0.699669\pi$
$3$	−2.03324	−1.17389	−0.586945	+	0.809627i	$0.699669\pi$
$4$	0	0
$5$	4.46117	1.99510	0.997548	−	0.0699801i	$-0.0222936\pi$
$5$	4.46117	1.99510	0.997548	+	0.0699801i	$0.0222936\pi$
$6$	0	0
$7$	−0.828843	−0.313273	−0.156637	−	0.987656i	$-0.550065\pi$
$7$	−0.828843	−0.313273	−0.156637	+	0.987656i	$0.550065\pi$
$8$	0	0
$9$	1.13405	0.378017
$10$	0	0
$11$	−0.0966604	−0.0291442	−0.0145721	−	0.999894i	$-0.504639\pi$
$11$	−0.0966604	−0.0291442	−0.0145721	+	0.999894i	$0.504639\pi$
$12$	0	0
$13$	1.53910	0.426870	0.213435	−	0.976957i	$-0.431535\pi$
$13$	1.53910	0.426870	0.213435	+	0.976957i	$0.431535\pi$
$14$	0	0
$15$	−9.07062	−2.34202
$16$	0	0
$17$	−3.59316	−0.871470	−0.435735	−	0.900075i	$-0.643511\pi$
$17$	−3.59316	−0.871470	−0.435735	+	0.900075i	$0.643511\pi$
$18$	0	0
$19$	−8.33503	−1.91219	−0.956094	−	0.293061i	$-0.905326\pi$
$19$	−8.33503	−1.91219	−0.956094	+	0.293061i	$0.905326\pi$
$20$	0	0
$21$	1.68523	0.367748
$22$	0	0
$23$	−2.93109	−0.611174	−0.305587	−	0.952164i	$-0.598853\pi$
$23$	−2.93109	−0.611174	−0.305587	+	0.952164i	$0.598853\pi$
$24$	0	0
$25$	14.9021	2.98041
$26$	0	0
$27$	3.79392	0.730139
$28$	0	0
$29$	1.38083	0.256413	0.128207	−	0.991747i	$-0.459078\pi$
$29$	1.38083	0.256413	0.128207	+	0.991747i	$0.459078\pi$
$30$	0	0
$31$	2.48771	0.446805	0.223403	−	0.974726i	$-0.428283\pi$
$31$	2.48771	0.446805	0.223403	+	0.974726i	$0.428283\pi$
$32$	0	0
$33$	0.196533	0.0342121
$34$	0	0
$35$	−3.69761	−0.625010
$36$	0	0
$37$	−3.06621	−0.504082	−0.252041	−	0.967717i	$-0.581102\pi$
$37$	−3.06621	−0.504082	−0.252041	+	0.967717i	$0.581102\pi$
$38$	0	0
$39$	−3.12936	−0.501099
$40$	0	0
$41$	−5.09004	−0.794930	−0.397465	−	0.917617i	$-0.630110\pi$
$41$	−5.09004	−0.794930	−0.397465	+	0.917617i	$0.630110\pi$
$42$	0	0
$43$	12.0742	1.84130	0.920648	−	0.390394i	$-0.127661\pi$
$43$	12.0742	1.84130	0.920648	+	0.390394i	$0.127661\pi$
$44$	0	0
$45$	5.05920	0.754181
$46$	0	0
$47$	3.75114	0.547160	0.273580	−	0.961849i	$-0.411792\pi$
$47$	3.75114	0.547160	0.273580	+	0.961849i	$0.411792\pi$
$48$	0	0
$49$	−6.31302	−0.901860
$50$	0	0
$51$	7.30575	1.02301
$52$	0	0
$53$	5.98285	0.821807	0.410904	−	0.911679i	$-0.365213\pi$
$53$	5.98285	0.821807	0.410904	+	0.911679i	$0.365213\pi$
$54$	0	0
$55$	−0.431219	−0.0581455
$56$	0	0
$57$	16.9471	2.24470
$58$	0	0
$59$	−1.55903	−0.202968	−0.101484	−	0.994837i	$-0.532359\pi$
$59$	−1.55903	−0.202968	−0.101484	+	0.994837i	$0.532359\pi$
$60$	0	0
$61$	−11.6668	−1.49378	−0.746891	−	0.664947i	$-0.768454\pi$
$61$	−11.6668	−1.49378	−0.746891	+	0.664947i	$0.768454\pi$
$62$	0	0
$63$	−0.939950	−0.118423
$64$	0	0
$65$	6.86620	0.851648
$66$	0	0
$67$	−1.19054	−0.145448	−0.0727239	−	0.997352i	$-0.523169\pi$
$67$	−1.19054	−0.145448	−0.0727239	+	0.997352i	$0.523169\pi$
$68$	0	0
$69$	5.95959	0.717450
$70$	0	0
$71$	5.48402	0.650834	0.325417	−	0.945571i	$-0.394495\pi$
$71$	5.48402	0.650834	0.325417	+	0.945571i	$0.394495\pi$
$72$	0	0
$73$	5.93015	0.694072	0.347036	−	0.937852i	$-0.387188\pi$
$73$	5.93015	0.694072	0.347036	+	0.937852i	$0.387188\pi$
$74$	0	0
$75$	−30.2994	−3.49867
$76$	0	0
$77$	0.0801163	0.00913010
$78$	0	0
$79$	−12.9836	−1.46077	−0.730384	−	0.683036i	$-0.760659\pi$
$79$	−12.9836	−1.46077	−0.730384	+	0.683036i	$0.760659\pi$
$80$	0	0
$81$	−11.1161	−1.23512
$82$	0	0
$83$	−16.7451	−1.83801	−0.919005	−	0.394245i	$-0.871006\pi$
$83$	−16.7451	−1.83801	−0.919005	+	0.394245i	$0.871006\pi$
$84$	0	0
$85$	−16.0297	−1.73867
$86$	0	0
$87$	−2.80755	−0.301001
$88$	0	0
$89$	−7.41727	−0.786229	−0.393114	−	0.919490i	$-0.628602\pi$
$89$	−7.41727	−0.786229	−0.393114	+	0.919490i	$0.628602\pi$
$90$	0	0
$91$	−1.27567	−0.133727
$92$	0	0
$93$	−5.05810	−0.524500
$94$	0	0
$95$	−37.1840	−3.81500
$96$	0	0
$97$	−14.8864	−1.51149	−0.755744	−	0.654867i	$-0.772724\pi$
$97$	−14.8864	−1.51149	−0.755744	+	0.654867i	$0.772724\pi$
$98$	0	0
$99$	−0.109618	−0.0110170
$100$	0	0
$101$	−1.27897	−0.127262	−0.0636312	−	0.997973i	$-0.520268\pi$
$101$	−1.27897	−0.127262	−0.0636312	+	0.997973i	$0.520268\pi$
$102$	0	0
$103$	−7.74066	−0.762710	−0.381355	−	0.924429i	$-0.624543\pi$
$103$	−7.74066	−0.762710	−0.381355	+	0.924429i	$0.624543\pi$
$104$	0	0
$105$	7.51812	0.733693
$106$	0	0
$107$	1.14319	0.110517	0.0552583	−	0.998472i	$-0.482402\pi$
$107$	1.14319	0.110517	0.0552583	+	0.998472i	$0.482402\pi$
$108$	0	0
$109$	12.5095	1.19819	0.599094	−	0.800679i	$-0.295527\pi$
$109$	12.5095	1.19819	0.599094	+	0.800679i	$0.295527\pi$
$110$	0	0
$111$	6.23434	0.591737
$112$	0	0
$113$	7.08097	0.666122	0.333061	−	0.942905i	$-0.391919\pi$
$113$	7.08097	0.666122	0.333061	+	0.942905i	$0.391919\pi$
$114$	0	0
$115$	−13.0761	−1.21935
$116$	0	0
$117$	1.74542	0.161364
$118$	0	0
$119$	2.97817	0.273008
$120$	0	0
$121$	−10.9907	−0.999151
$122$	0	0
$123$	10.3493	0.933161
$124$	0	0
$125$	44.1748	3.95111
$126$	0	0
$127$	7.72311	0.685315	0.342657	−	0.939460i	$-0.388673\pi$
$127$	7.72311	0.685315	0.342657	+	0.939460i	$0.388673\pi$
$128$	0	0
$129$	−24.5497	−2.16148
$130$	0	0
$131$	10.8006	0.943651	0.471826	−	0.881692i	$-0.343595\pi$
$131$	10.8006	0.943651	0.471826	+	0.881692i	$0.343595\pi$
$132$	0	0
$133$	6.90843	0.599037
$134$	0	0
$135$	16.9253	1.45670
$136$	0	0
$137$	−17.6756	−1.51012	−0.755062	−	0.655653i	$-0.772393\pi$
$137$	−17.6756	−1.51012	−0.755062	+	0.655653i	$0.772393\pi$
$138$	0	0
$139$	15.3348	1.30068	0.650341	−	0.759642i	$-0.274626\pi$
$139$	15.3348	1.30068	0.650341	+	0.759642i	$0.274626\pi$
$140$	0	0
$141$	−7.62696	−0.642306
$142$	0	0
$143$	−0.148770	−0.0124408
$144$	0	0
$145$	6.16011	0.511570
$146$	0	0
$147$	12.8359	1.05868
$148$	0	0
$149$	4.40314	0.360719	0.180360	−	0.983601i	$-0.442274\pi$
$149$	4.40314	0.360719	0.180360	+	0.983601i	$0.442274\pi$
$150$	0	0
$151$	−17.2749	−1.40581	−0.702904	−	0.711285i	$-0.748114\pi$
$151$	−17.2749	−1.40581	−0.702904	+	0.711285i	$0.748114\pi$
$152$	0	0
$153$	−4.07483	−0.329430
$154$	0	0
$155$	11.0981	0.891420
$156$	0	0
$157$	−10.9732	−0.875755	−0.437877	−	0.899035i	$-0.644270\pi$
$157$	−10.9732	−0.875755	−0.437877	+	0.899035i	$0.644270\pi$
$158$	0	0
$159$	−12.1645	−0.964711
$160$	0	0
$161$	2.42941	0.191464
$162$	0	0
$163$	−14.5974	−1.14336	−0.571678	−	0.820478i	$-0.693707\pi$
$163$	−14.5974	−1.14336	−0.571678	+	0.820478i	$0.693707\pi$
$164$	0	0
$165$	0.876769	0.0682564
$166$	0	0
$167$	18.8031	1.45502	0.727512	−	0.686094i	$-0.240676\pi$
$167$	18.8031	1.45502	0.727512	+	0.686094i	$0.240676\pi$
$168$	0	0
$169$	−10.6312	−0.817782
$170$	0	0
$171$	−9.45235	−0.722839
$172$	0	0
$173$	5.93420	0.451169	0.225584	−	0.974224i	$-0.427571\pi$
$173$	5.93420	0.451169	0.225584	+	0.974224i	$0.427571\pi$
$174$	0	0
$175$	−12.3515	−0.933683
$176$	0	0
$177$	3.16987	0.238262
$178$	0	0
$179$	−16.2703	−1.21610	−0.608048	−	0.793900i	$-0.708047\pi$
$179$	−16.2703	−1.21610	−0.608048	+	0.793900i	$0.708047\pi$
$180$	0	0
$181$	8.76453	0.651463	0.325731	−	0.945462i	$-0.394390\pi$
$181$	8.76453	0.651463	0.325731	+	0.945462i	$0.394390\pi$
$182$	0	0
$183$	23.7214	1.75354
$184$	0	0
$185$	−13.6789	−1.00569
$186$	0	0
$187$	0.347316	0.0253983
$188$	0	0
$189$	−3.14456	−0.228733
$190$	0	0
$191$	−26.0947	−1.88815	−0.944073	−	0.329737i	$-0.893040\pi$
$191$	−26.0947	−1.88815	−0.944073	+	0.329737i	$0.893040\pi$
$192$	0	0
$193$	0.0816559	0.00587772	0.00293886	−	0.999996i	$-0.499065\pi$
$193$	0.0816559	0.00587772	0.00293886	+	0.999996i	$0.499065\pi$
$194$	0	0
$195$	−13.9606	−0.999741
$196$	0	0
$197$	−18.2323	−1.29899	−0.649497	−	0.760364i	$-0.725021\pi$
$197$	−18.2323	−1.29899	−0.649497	+	0.760364i	$0.725021\pi$
$198$	0	0
$199$	9.13270	0.647400	0.323700	−	0.946160i	$-0.395073\pi$
$199$	9.13270	0.647400	0.323700	+	0.946160i	$0.395073\pi$
$200$	0	0
$201$	2.42065	0.170740
$202$	0	0
$203$	−1.14449	−0.0803274
$204$	0	0
$205$	−22.7075	−1.58596
$206$	0	0
$207$	−3.32400	−0.231034
$208$	0	0
$209$	0.805667	0.0557292
$210$	0	0
$211$	−10.4387	−0.718629	−0.359315	−	0.933217i	$-0.616989\pi$
$211$	−10.4387	−0.718629	−0.359315	+	0.933217i	$0.616989\pi$
$212$	0	0
$213$	−11.1503	−0.764007
$214$	0	0
$215$	53.8650	3.67356
$216$	0	0
$217$	−2.06192	−0.139972
$218$	0	0
$219$	−12.0574	−0.814764
$220$	0	0
$221$	−5.53025	−0.372005
$222$	0	0
$223$	14.9343	1.00008	0.500039	−	0.866003i	$-0.333319\pi$
$223$	14.9343	1.00008	0.500039	+	0.866003i	$0.333319\pi$
$224$	0	0
$225$	16.8997	1.12665
$226$	0	0
$227$	−9.60679	−0.637625	−0.318813	−	0.947818i	$-0.603284\pi$
$227$	−9.60679	−0.637625	−0.318813	+	0.947818i	$0.603284\pi$
$228$	0	0
$229$	−10.0213	−0.662227	−0.331113	−	0.943591i	$-0.607424\pi$
$229$	−10.0213	−0.662227	−0.331113	+	0.943591i	$0.607424\pi$
$230$	0	0
$231$	−0.162895	−0.0107177
$232$	0	0
$233$	−4.67799	−0.306466	−0.153233	−	0.988190i	$-0.548968\pi$
$233$	−4.67799	−0.306466	−0.153233	+	0.988190i	$0.548968\pi$
$234$	0	0
$235$	16.7345	1.09164
$236$	0	0
$237$	26.3987	1.71478
$238$	0	0
$239$	−21.0064	−1.35879	−0.679394	−	0.733773i	$-0.737757\pi$
$239$	−21.0064	−1.35879	−0.679394	+	0.733773i	$0.737757\pi$
$240$	0	0
$241$	28.3671	1.82729	0.913644	−	0.406515i	$-0.133256\pi$
$241$	28.3671	1.82729	0.913644	+	0.406515i	$0.133256\pi$
$242$	0	0
$243$	11.2199	0.719755
$244$	0	0
$245$	−28.1635	−1.79930
$246$	0	0
$247$	−12.8285	−0.816256
$248$	0	0
$249$	34.0467	2.15762
$250$	0	0
$251$	2.90011	0.183053	0.0915267	−	0.995803i	$-0.470825\pi$
$251$	2.90011	0.183053	0.0915267	+	0.995803i	$0.470825\pi$
$252$	0	0
$253$	0.283320	0.0178122
$254$	0	0
$255$	32.5922	2.04100
$256$	0	0
$257$	−15.5426	−0.969522	−0.484761	−	0.874647i	$-0.661093\pi$
$257$	−15.5426	−0.969522	−0.484761	+	0.874647i	$0.661093\pi$
$258$	0	0
$259$	2.54141	0.157915
$260$	0	0
$261$	1.56593	0.0969286
$262$	0	0
$263$	−3.41655	−0.210673	−0.105337	−	0.994437i	$-0.533592\pi$
$263$	−3.41655	−0.210673	−0.105337	+	0.994437i	$0.533592\pi$
$264$	0	0
$265$	26.6905	1.63959
$266$	0	0
$267$	15.0811	0.922946
$268$	0	0
$269$	7.20309	0.439180	0.219590	−	0.975592i	$-0.429528\pi$
$269$	7.20309	0.439180	0.219590	+	0.975592i	$0.429528\pi$
$270$	0	0
$271$	32.2336	1.95805	0.979025	−	0.203739i	$-0.0653094\pi$
$271$	32.2336	1.95805	0.979025	+	0.203739i	$0.0653094\pi$
$272$	0	0
$273$	2.59375	0.156981
$274$	0	0
$275$	−1.44044	−0.0868617
$276$	0	0
$277$	20.8997	1.25574	0.627871	−	0.778318i	$-0.283927\pi$
$277$	20.8997	1.25574	0.627871	+	0.778318i	$0.283927\pi$
$278$	0	0
$279$	2.82119	0.168900
$280$	0	0
$281$	11.9037	0.710116	0.355058	−	0.934844i	$-0.384461\pi$
$281$	11.9037	0.710116	0.355058	+	0.934844i	$0.384461\pi$
$282$	0	0
$283$	−3.15617	−0.187615	−0.0938074	−	0.995590i	$-0.529904\pi$
$283$	−3.15617	−0.187615	−0.0938074	+	0.995590i	$0.529904\pi$
$284$	0	0
$285$	75.6039	4.47839
$286$	0	0
$287$	4.21884	0.249030
$288$	0	0
$289$	−4.08919	−0.240540
$290$	0	0
$291$	30.2676	1.77432
$292$	0	0
$293$	−20.7626	−1.21297	−0.606483	−	0.795097i	$-0.707420\pi$
$293$	−20.7626	−1.21297	−0.606483	+	0.795097i	$0.707420\pi$
$294$	0	0
$295$	−6.95510	−0.404941
$296$	0	0
$297$	−0.366721	−0.0212793
$298$	0	0
$299$	−4.51124	−0.260892
$300$	0	0
$301$	−10.0076	−0.576829
$302$	0	0
$303$	2.60045	0.149392
$304$	0	0
$305$	−52.0476	−2.98024
$306$	0	0
$307$	−25.1602	−1.43597	−0.717983	−	0.696061i	$-0.754934\pi$
$307$	−25.1602	−1.43597	−0.717983	+	0.696061i	$0.754934\pi$
$308$	0	0
$309$	15.7386	0.895338
$310$	0	0
$311$	24.0755	1.36520	0.682598	−	0.730794i	$-0.260850\pi$
$311$	24.0755	1.36520	0.682598	+	0.730794i	$0.260850\pi$
$312$	0	0
$313$	−18.8139	−1.06342	−0.531712	−	0.846925i	$-0.678451\pi$
$313$	−18.8139	−1.06342	−0.531712	+	0.846925i	$0.678451\pi$
$314$	0	0
$315$	−4.19328	−0.236265
$316$	0	0
$317$	20.2446	1.13705	0.568525	−	0.822666i	$-0.307514\pi$
$317$	20.2446	1.13705	0.568525	+	0.822666i	$0.307514\pi$
$318$	0	0
$319$	−0.133471	−0.00747296
$320$	0	0
$321$	−2.32438	−0.129734
$322$	0	0
$323$	29.9491	1.66641
$324$	0	0
$325$	22.9358	1.27225
$326$	0	0
$327$	−25.4347	−1.40654
$328$	0	0
$329$	−3.10911	−0.171411
$330$	0	0
$331$	−30.0247	−1.65030	−0.825152	−	0.564910i	$-0.808911\pi$
$331$	−30.0247	−1.65030	−0.825152	+	0.564910i	$0.808911\pi$
$332$	0	0
$333$	−3.47724	−0.190552
$334$	0	0
$335$	−5.31121	−0.290182
$336$	0	0
$337$	−30.1862	−1.64435	−0.822174	−	0.569236i	$-0.807239\pi$
$337$	−30.1862	−1.64435	−0.822174	+	0.569236i	$0.807239\pi$
$338$	0	0
$339$	−14.3973	−0.781953
$340$	0	0
$341$	−0.240463	−0.0130218
$342$	0	0
$343$	11.0344	0.595802
$344$	0	0
$345$	26.5868	1.43138
$346$	0	0
$347$	−16.2121	−0.870310	−0.435155	−	0.900356i	$-0.643306\pi$
$347$	−16.2121	−0.870310	−0.435155	+	0.900356i	$0.643306\pi$
$348$	0	0
$349$	−20.0437	−1.07292	−0.536458	−	0.843927i	$-0.680238\pi$
$349$	−20.0437	−1.07292	−0.536458	+	0.843927i	$0.680238\pi$
$350$	0	0
$351$	5.83923	0.311675
$352$	0	0
$353$	−31.0479	−1.65251	−0.826256	−	0.563294i	$-0.809534\pi$
$353$	−31.0479	−1.65251	−0.826256	+	0.563294i	$0.809534\pi$
$354$	0	0
$355$	24.4652	1.29848
$356$	0	0
$357$	−6.05532	−0.320481
$358$	0	0
$359$	−1.54542	−0.0815642	−0.0407821	−	0.999168i	$-0.512985\pi$
$359$	−1.54542	−0.0815642	−0.0407821	+	0.999168i	$0.512985\pi$
$360$	0	0
$361$	50.4727	2.65646
$362$	0	0
$363$	22.3466	1.17289
$364$	0	0
$365$	26.4554	1.38474
$366$	0	0
$367$	14.9154	0.778578	0.389289	−	0.921116i	$-0.372721\pi$
$367$	14.9154	0.778578	0.389289	+	0.921116i	$0.372721\pi$
$368$	0	0
$369$	−5.77236	−0.300497
$370$	0	0
$371$	−4.95884	−0.257450
$372$	0	0
$373$	2.04661	0.105969	0.0529847	−	0.998595i	$-0.483127\pi$
$373$	2.04661	0.105969	0.0529847	+	0.998595i	$0.483127\pi$
$374$	0	0
$375$	−89.8178	−4.63817
$376$	0	0
$377$	2.12524	0.109455
$378$	0	0
$379$	−27.3417	−1.40445	−0.702226	−	0.711955i	$-0.747810\pi$
$379$	−27.3417	−1.40445	−0.702226	+	0.711955i	$0.747810\pi$
$380$	0	0
$381$	−15.7029	−0.804484
$382$	0	0
$383$	0.900506	0.0460137	0.0230068	−	0.999735i	$-0.492676\pi$
$383$	0.900506	0.0460137	0.0230068	+	0.999735i	$0.492676\pi$
$384$	0	0
$385$	0.357412	0.0182154
$386$	0	0
$387$	13.6927	0.696041
$388$	0	0
$389$	15.6230	0.792117	0.396059	−	0.918225i	$-0.370378\pi$
$389$	15.6230	0.792117	0.396059	+	0.918225i	$0.370378\pi$
$390$	0	0
$391$	10.5319	0.532619
$392$	0	0
$393$	−21.9601	−1.10774
$394$	0	0
$395$	−57.9220	−2.91437
$396$	0	0
$397$	−12.4493	−0.624811	−0.312406	−	0.949949i	$-0.601135\pi$
$397$	−12.4493	−0.624811	−0.312406	+	0.949949i	$0.601135\pi$
$398$	0	0
$399$	−14.0465	−0.703203
$400$	0	0
$401$	−1.60701	−0.0802502	−0.0401251	−	0.999195i	$-0.512776\pi$
$401$	−1.60701	−0.0802502	−0.0401251	+	0.999195i	$0.512776\pi$
$402$	0	0
$403$	3.82884	0.190728
$404$	0	0
$405$	−49.5908	−2.46418
$406$	0	0
$407$	0.296381	0.0146911
$408$	0	0
$409$	21.8170	1.07878	0.539390	−	0.842056i	$-0.318655\pi$
$409$	21.8170	1.07878	0.539390	+	0.842056i	$0.318655\pi$
$410$	0	0
$411$	35.9386	1.77272
$412$	0	0
$413$	1.29219	0.0635845
$414$	0	0
$415$	−74.7027	−3.66701
$416$	0	0
$417$	−31.1793	−1.52686
$418$	0	0
$419$	−8.73833	−0.426896	−0.213448	−	0.976954i	$-0.568469\pi$
$419$	−8.73833	−0.426896	−0.213448	+	0.976954i	$0.568469\pi$
$420$	0	0
$421$	10.3976	0.506750	0.253375	−	0.967368i	$-0.418459\pi$
$421$	10.3976	0.506750	0.253375	+	0.967368i	$0.418459\pi$
$422$	0	0
$423$	4.25399	0.206836
$424$	0	0
$425$	−53.5455	−2.59734
$426$	0	0
$427$	9.66995	0.467962
$428$	0	0
$429$	0.302485	0.0146041
$430$	0	0
$431$	14.8133	0.713533	0.356766	−	0.934194i	$-0.383879\pi$
$431$	14.8133	0.713533	0.356766	+	0.934194i	$0.383879\pi$
$432$	0	0
$433$	21.0096	1.00966	0.504829	−	0.863220i	$-0.331556\pi$
$433$	21.0096	1.00966	0.504829	+	0.863220i	$0.331556\pi$
$434$	0	0
$435$	−12.5250	−0.600526
$436$	0	0
$437$	24.4307	1.16868
$438$	0	0
$439$	34.0623	1.62570	0.812852	−	0.582470i	$-0.197914\pi$
$439$	34.0623	1.62570	0.812852	+	0.582470i	$0.197914\pi$
$440$	0	0
$441$	−7.15929	−0.340918
$442$	0	0
$443$	23.8224	1.13184	0.565919	−	0.824461i	$-0.308522\pi$
$443$	23.8224	1.13184	0.565919	+	0.824461i	$0.308522\pi$
$444$	0	0
$445$	−33.0897	−1.56860
$446$	0	0
$447$	−8.95262	−0.423445
$448$	0	0
$449$	−9.46319	−0.446596	−0.223298	−	0.974750i	$-0.571682\pi$
$449$	−9.46319	−0.446596	−0.223298	+	0.974750i	$0.571682\pi$
$450$	0	0
$451$	0.492005	0.0231676
$452$	0	0
$453$	35.1239	1.65026
$454$	0	0
$455$	−5.69100	−0.266798
$456$	0	0
$457$	33.6524	1.57419	0.787097	−	0.616829i	$-0.211583\pi$
$457$	33.6524	1.57419	0.787097	+	0.616829i	$0.211583\pi$
$458$	0	0
$459$	−13.6322	−0.636294
$460$	0	0
$461$	−11.0943	−0.516713	−0.258357	−	0.966050i	$-0.583181\pi$
$461$	−11.0943	−0.516713	−0.258357	+	0.966050i	$0.583181\pi$
$462$	0	0
$463$	−13.6271	−0.633304	−0.316652	−	0.948542i	$-0.602559\pi$
$463$	−13.6271	−0.633304	−0.316652	+	0.948542i	$0.602559\pi$
$464$	0	0
$465$	−22.5650	−1.04643
$466$	0	0
$467$	−16.8293	−0.778766	−0.389383	−	0.921076i	$-0.627312\pi$
$467$	−16.8293	−0.778766	−0.389383	+	0.921076i	$0.627312\pi$
$468$	0	0
$469$	0.986772	0.0455649
$470$	0	0
$471$	22.3111	1.02804
$472$	0	0
$473$	−1.16710	−0.0536631
$474$	0	0
$475$	−124.209	−5.69910
$476$	0	0
$477$	6.78486	0.310657
$478$	0	0
$479$	23.9636	1.09492	0.547462	−	0.836830i	$-0.315594\pi$
$479$	23.9636	1.09492	0.547462	+	0.836830i	$0.315594\pi$
$480$	0	0
$481$	−4.71922	−0.215178
$482$	0	0
$483$	−4.93956	−0.224758
$484$	0	0
$485$	−66.4109	−3.01556
$486$	0	0
$487$	−6.83145	−0.309563	−0.154781	−	0.987949i	$-0.549467\pi$
$487$	−6.83145	−0.309563	−0.154781	+	0.987949i	$0.549467\pi$
$488$	0	0
$489$	29.6799	1.34217
$490$	0	0
$491$	−17.8246	−0.804413	−0.402206	−	0.915549i	$-0.631757\pi$
$491$	−17.8246	−0.804413	−0.402206	+	0.915549i	$0.631757\pi$
$492$	0	0
$493$	−4.96154	−0.223457
$494$	0	0
$495$	−0.489024	−0.0219800
$496$	0	0
$497$	−4.54539	−0.203889
$498$	0	0
$499$	−1.17945	−0.0527993	−0.0263997	−	0.999651i	$-0.508404\pi$
$499$	−1.17945	−0.0527993	−0.0263997	+	0.999651i	$0.508404\pi$
$500$	0	0
$501$	−38.2311	−1.70804
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−5.70571	−0.253901
$506$	0	0
$507$	21.6157	0.959985
$508$	0	0
$509$	−22.5820	−1.00093	−0.500466	−	0.865756i	$-0.666838\pi$
$509$	−22.5820	−1.00093	−0.500466	+	0.865756i	$0.666838\pi$
$510$	0	0
$511$	−4.91517	−0.217434
$512$	0	0
$513$	−31.6224	−1.39616
$514$	0	0
$515$	−34.5324	−1.52168
$516$	0	0
$517$	−0.362587	−0.0159465
$518$	0	0
$519$	−12.0656	−0.529622
$520$	0	0
$521$	−15.3907	−0.674281	−0.337140	−	0.941454i	$-0.609460\pi$
$521$	−15.3907	−0.674281	−0.337140	+	0.941454i	$0.609460\pi$
$522$	0	0
$523$	−13.0581	−0.570990	−0.285495	−	0.958380i	$-0.592158\pi$
$523$	−13.0581	−0.570990	−0.285495	+	0.958380i	$0.592158\pi$
$524$	0	0
$525$	25.1134	1.09604
$526$	0	0
$527$	−8.93874	−0.389378
$528$	0	0
$529$	−14.4087	−0.626467
$530$	0	0
$531$	−1.76802	−0.0767255
$532$	0	0
$533$	−7.83409	−0.339332
$534$	0	0
$535$	5.09998	0.220491
$536$	0	0
$537$	33.0813	1.42756
$538$	0	0
$539$	0.610219	0.0262840
$540$	0	0
$541$	−4.15271	−0.178539	−0.0892695	−	0.996008i	$-0.528453\pi$
$541$	−4.15271	−0.178539	−0.0892695	+	0.996008i	$0.528453\pi$
$542$	0	0
$543$	−17.8204	−0.764745
$544$	0	0
$545$	55.8068	2.39050
$546$	0	0
$547$	4.91362	0.210091	0.105046	−	0.994467i	$-0.466501\pi$
$547$	4.91362	0.210091	0.105046	+	0.994467i	$0.466501\pi$
$548$	0	0
$549$	−13.2308	−0.564675
$550$	0	0
$551$	−11.5092	−0.490310
$552$	0	0
$553$	10.7614	0.457620
$554$	0	0
$555$	27.8124	1.18057
$556$	0	0
$557$	−4.89601	−0.207451	−0.103725	−	0.994606i	$-0.533076\pi$
$557$	−4.89601	−0.207451	−0.103725	+	0.994606i	$0.533076\pi$
$558$	0	0
$559$	18.5834	0.785995
$560$	0	0
$561$	−0.706177	−0.0298148
$562$	0	0
$563$	−30.4658	−1.28398	−0.641990	−	0.766713i	$-0.721891\pi$
$563$	−30.4658	−1.28398	−0.641990	+	0.766713i	$0.721891\pi$
$564$	0	0
$565$	31.5894	1.32898
$566$	0	0
$567$	9.21349	0.386930
$568$	0	0
$569$	−16.0224	−0.671693	−0.335846	−	0.941917i	$-0.609022\pi$
$569$	−16.0224	−0.671693	−0.335846	+	0.941917i	$0.609022\pi$
$570$	0	0
$571$	−3.92027	−0.164058	−0.0820291	−	0.996630i	$-0.526140\pi$
$571$	−3.92027	−0.164058	−0.0820291	+	0.996630i	$0.526140\pi$
$572$	0	0
$573$	53.0567	2.21647
$574$	0	0
$575$	−43.6792	−1.82155
$576$	0	0
$577$	−18.2327	−0.759037	−0.379519	−	0.925184i	$-0.623910\pi$
$577$	−18.2327	−0.759037	−0.379519	+	0.925184i	$0.623910\pi$
$578$	0	0
$579$	−0.166026	−0.00689980
$580$	0	0
$581$	13.8790	0.575799
$582$	0	0
$583$	−0.578304	−0.0239509
$584$	0	0
$585$	7.78663	0.321937
$586$	0	0
$587$	−19.7471	−0.815048	−0.407524	−	0.913194i	$-0.633608\pi$
$587$	−19.7471	−0.815048	−0.407524	+	0.913194i	$0.633608\pi$
$588$	0	0
$589$	−20.7351	−0.854376
$590$	0	0
$591$	37.0705	1.52488
$592$	0	0
$593$	−26.3107	−1.08045	−0.540225	−	0.841521i	$-0.681661\pi$
$593$	−26.3107	−1.08045	−0.540225	+	0.841521i	$0.681661\pi$
$594$	0	0
$595$	13.2861	0.544678
$596$	0	0
$597$	−18.5689	−0.759976
$598$	0	0
$599$	−3.44856	−0.140904	−0.0704522	−	0.997515i	$-0.522444\pi$
$599$	−3.44856	−0.140904	−0.0704522	+	0.997515i	$0.522444\pi$
$600$	0	0
$601$	−48.1876	−1.96561	−0.982806	−	0.184643i	$-0.940887\pi$
$601$	−48.1876	−1.96561	−0.982806	+	0.184643i	$0.940887\pi$
$602$	0	0
$603$	−1.35014	−0.0549817
$604$	0	0
$605$	−49.0312	−1.99340
$606$	0	0
$607$	20.9551	0.850540	0.425270	−	0.905067i	$-0.360179\pi$
$607$	20.9551	0.850540	0.425270	+	0.905067i	$0.360179\pi$
$608$	0	0
$609$	2.32702	0.0942956
$610$	0	0
$611$	5.77339	0.233566
$612$	0	0
$613$	−11.3527	−0.458530	−0.229265	−	0.973364i	$-0.573632\pi$
$613$	−11.3527	−0.458530	−0.229265	+	0.973364i	$0.573632\pi$
$614$	0	0
$615$	46.1698	1.86175
$616$	0	0
$617$	−33.6693	−1.35548	−0.677738	−	0.735303i	$-0.737040\pi$
$617$	−33.6693	−1.35548	−0.677738	+	0.735303i	$0.737040\pi$
$618$	0	0
$619$	28.8179	1.15829	0.579145	−	0.815225i	$-0.303387\pi$
$619$	28.8179	1.15829	0.579145	+	0.815225i	$0.303387\pi$
$620$	0	0
$621$	−11.1203	−0.446242
$622$	0	0
$623$	6.14775	0.246304
$624$	0	0
$625$	122.561	4.90244
$626$	0	0
$627$	−1.63811	−0.0654199
$628$	0	0
$629$	11.0174	0.439293
$630$	0	0
$631$	13.5612	0.539864	0.269932	−	0.962879i	$-0.412999\pi$
$631$	13.5612	0.539864	0.269932	+	0.962879i	$0.412999\pi$
$632$	0	0
$633$	21.2243	0.843591
$634$	0	0
$635$	34.4541	1.36727
$636$	0	0
$637$	−9.71639	−0.384977
$638$	0	0
$639$	6.21916	0.246026
$640$	0	0
$641$	14.9901	0.592073	0.296036	−	0.955177i	$-0.404335\pi$
$641$	14.9901	0.592073	0.296036	+	0.955177i	$0.404335\pi$
$642$	0	0
$643$	−2.77896	−0.109592	−0.0547958	−	0.998498i	$-0.517451\pi$
$643$	−2.77896	−0.109592	−0.0547958	+	0.998498i	$0.517451\pi$
$644$	0	0
$645$	−109.520	−4.31236
$646$	0	0
$647$	−34.1713	−1.34341	−0.671707	−	0.740817i	$-0.734438\pi$
$647$	−34.1713	−1.34341	−0.671707	+	0.740817i	$0.734438\pi$
$648$	0	0
$649$	0.150696	0.00591535
$650$	0	0
$651$	4.19237	0.164312
$652$	0	0
$653$	−21.1668	−0.828320	−0.414160	−	0.910204i	$-0.635925\pi$
$653$	−21.1668	−0.828320	−0.414160	+	0.910204i	$0.635925\pi$
$654$	0	0
$655$	48.1832	1.88268
$656$	0	0
$657$	6.72510	0.262371
$658$	0	0
$659$	−30.1878	−1.17595	−0.587976	−	0.808879i	$-0.700075\pi$
$659$	−30.1878	−1.17595	−0.587976	+	0.808879i	$0.700075\pi$
$660$	0	0
$661$	−20.1968	−0.785564	−0.392782	−	0.919632i	$-0.628487\pi$
$661$	−20.1968	−0.785564	−0.392782	+	0.919632i	$0.628487\pi$
$662$	0	0
$663$	11.2443	0.436693
$664$	0	0
$665$	30.8197	1.19514
$666$	0	0
$667$	−4.04733	−0.156713
$668$	0	0
$669$	−30.3651	−1.17398
$670$	0	0
$671$	1.12772	0.0435351
$672$	0	0
$673$	22.0029	0.848150	0.424075	−	0.905627i	$-0.360599\pi$
$673$	22.0029	0.848150	0.424075	+	0.905627i	$0.360599\pi$
$674$	0	0
$675$	56.5371	2.17612
$676$	0	0
$677$	16.1953	0.622438	0.311219	−	0.950338i	$-0.399263\pi$
$677$	16.1953	0.622438	0.311219	+	0.950338i	$0.399263\pi$
$678$	0	0
$679$	12.3385	0.473509
$680$	0	0
$681$	19.5329	0.748501
$682$	0	0
$683$	4.04086	0.154619	0.0773096	−	0.997007i	$-0.475367\pi$
$683$	4.04086	0.154619	0.0773096	+	0.997007i	$0.475367\pi$
$684$	0	0
$685$	−78.8537	−3.01284
$686$	0	0
$687$	20.3757	0.777381
$688$	0	0
$689$	9.20822	0.350805
$690$	0	0
$691$	9.75760	0.371197	0.185598	−	0.982626i	$-0.440578\pi$
$691$	9.75760	0.371197	0.185598	+	0.982626i	$0.440578\pi$
$692$	0	0
$693$	0.0908560	0.00345133
$694$	0	0
$695$	68.4112	2.59499
$696$	0	0
$697$	18.2893	0.692758
$698$	0	0
$699$	9.51147	0.359757
$700$	0	0
$701$	19.5876	0.739814	0.369907	−	0.929069i	$-0.379390\pi$
$701$	19.5876	0.739814	0.369907	+	0.929069i	$0.379390\pi$
$702$	0	0
$703$	25.5570	0.963900
$704$	0	0
$705$	−34.0252	−1.28146
$706$	0	0
$707$	1.06007	0.0398679
$708$	0	0
$709$	5.31630	0.199658	0.0998288	−	0.995005i	$-0.468170\pi$
$709$	5.31630	0.199658	0.0998288	+	0.995005i	$0.468170\pi$
$710$	0	0
$711$	−14.7241	−0.552195
$712$	0	0
$713$	−7.29169	−0.273076
$714$	0	0
$715$	−0.663690	−0.0248206
$716$	0	0
$717$	42.7109	1.59507
$718$	0	0
$719$	−26.8375	−1.00087	−0.500434	−	0.865775i	$-0.666826\pi$
$719$	−26.8375	−1.00087	−0.500434	+	0.865775i	$0.666826\pi$
$720$	0	0
$721$	6.41579	0.238937
$722$	0	0
$723$	−57.6771	−2.14503
$724$	0	0
$725$	20.5772	0.764217
$726$	0	0
$727$	23.0650	0.855434	0.427717	−	0.903913i	$-0.359318\pi$
$727$	23.0650	0.855434	0.427717	+	0.903913i	$0.359318\pi$
$728$	0	0
$729$	10.5356	0.390207
$730$	0	0
$731$	−43.3845	−1.60463
$732$	0	0
$733$	42.7383	1.57857	0.789287	−	0.614025i	$-0.210451\pi$
$733$	42.7383	1.57857	0.789287	+	0.614025i	$0.210451\pi$
$734$	0	0
$735$	57.2630	2.11218
$736$	0	0
$737$	0.115078	0.00423896
$738$	0	0
$739$	0.649472	0.0238912	0.0119456	−	0.999929i	$-0.496198\pi$
$739$	0.649472	0.0238912	0.0119456	+	0.999929i	$0.496198\pi$
$740$	0	0
$741$	26.0833	0.958195
$742$	0	0
$743$	−17.6669	−0.648137	−0.324068	−	0.946034i	$-0.605051\pi$
$743$	−17.6669	−0.648137	−0.324068	+	0.946034i	$0.605051\pi$
$744$	0	0
$745$	19.6432	0.719670
$746$	0	0
$747$	−18.9898	−0.694799
$748$	0	0
$749$	−0.947527	−0.0346219
$750$	0	0
$751$	7.53204	0.274848	0.137424	−	0.990512i	$-0.456118\pi$
$751$	7.53204	0.274848	0.137424	+	0.990512i	$0.456118\pi$
$752$	0	0
$753$	−5.89661	−0.214885
$754$	0	0
$755$	−77.0661	−2.80472
$756$	0	0
$757$	3.73106	0.135608	0.0678039	−	0.997699i	$-0.478401\pi$
$757$	3.73106	0.135608	0.0678039	+	0.997699i	$0.478401\pi$
$758$	0	0
$759$	−0.576056	−0.0209095
$760$	0	0
$761$	−2.45314	−0.0889264	−0.0444632	−	0.999011i	$-0.514158\pi$
$761$	−2.45314	−0.0889264	−0.0444632	+	0.999011i	$0.514158\pi$
$762$	0	0
$763$	−10.3684	−0.375360
$764$	0	0
$765$	−18.1785	−0.657246
$766$	0	0
$767$	−2.39951	−0.0866412
$768$	0	0
$769$	11.2733	0.406526	0.203263	−	0.979124i	$-0.434845\pi$
$769$	11.2733	0.406526	0.203263	+	0.979124i	$0.434845\pi$
$770$	0	0
$771$	31.6018	1.13811
$772$	0	0
$773$	−43.0607	−1.54879	−0.774393	−	0.632705i	$-0.781945\pi$
$773$	−43.0607	−1.54879	−0.774393	+	0.632705i	$0.781945\pi$
$774$	0	0
$775$	37.0720	1.33166
$776$	0	0
$777$	−5.16729	−0.185375
$778$	0	0
$779$	42.4256	1.52006
$780$	0	0
$781$	−0.530088	−0.0189680
$782$	0	0
$783$	5.23875	0.187218
$784$	0	0
$785$	−48.9532	−1.74722
$786$	0	0
$787$	−36.4349	−1.29876	−0.649382	−	0.760462i	$-0.724972\pi$
$787$	−36.4349	−1.29876	−0.649382	+	0.760462i	$0.724972\pi$
$788$	0	0
$789$	6.94665	0.247307
$790$	0	0
$791$	−5.86901	−0.208678
$792$	0	0
$793$	−17.9564	−0.637651
$794$	0	0
$795$	−54.2681	−1.92469
$796$	0	0
$797$	31.1724	1.10418	0.552092	−	0.833783i	$-0.313830\pi$
$797$	31.1724	1.10418	0.552092	+	0.833783i	$0.313830\pi$
$798$	0	0
$799$	−13.4785	−0.476834
$800$	0	0
$801$	−8.41156	−0.297208
$802$	0	0
$803$	−0.573211	−0.0202282
$804$	0	0
$805$	10.8380	0.381990
$806$	0	0
$807$	−14.6456	−0.515549
$808$	0	0
$809$	26.8016	0.942294	0.471147	−	0.882055i	$-0.343840\pi$
$809$	26.8016	0.942294	0.471147	+	0.882055i	$0.343840\pi$
$810$	0	0
$811$	26.5851	0.933527	0.466764	−	0.884382i	$-0.345420\pi$
$811$	26.5851	0.933527	0.466764	+	0.884382i	$0.345420\pi$
$812$	0	0
$813$	−65.5385	−2.29854
$814$	0	0
$815$	−65.1214	−2.28110
$816$	0	0
$817$	−100.639	−3.52090
$818$	0	0
$819$	−1.44668	−0.0505511
$820$	0	0
$821$	54.2548	1.89351	0.946754	−	0.321959i	$-0.104341\pi$
$821$	54.2548	1.89351	0.946754	+	0.321959i	$0.104341\pi$
$822$	0	0
$823$	21.7378	0.757732	0.378866	−	0.925451i	$-0.376314\pi$
$823$	21.7378	0.757732	0.378866	+	0.925451i	$0.376314\pi$
$824$	0	0
$825$	2.92875	0.101966
$826$	0	0
$827$	−23.2671	−0.809077	−0.404538	−	0.914521i	$-0.632568\pi$
$827$	−23.2671	−0.809077	−0.404538	+	0.914521i	$0.632568\pi$
$828$	0	0
$829$	−51.9647	−1.80481	−0.902404	−	0.430891i	$-0.858199\pi$
$829$	−51.9647	−1.80481	−0.902404	+	0.430891i	$0.858199\pi$
$830$	0	0
$831$	−42.4940	−1.47410
$832$	0	0
$833$	22.6837	0.785944
$834$	0	0
$835$	83.8837	2.90292
$836$	0	0
$837$	9.43815	0.326230
$838$	0	0
$839$	6.20131	0.214093	0.107047	−	0.994254i	$-0.465861\pi$
$839$	6.20131	0.214093	0.107047	+	0.994254i	$0.465861\pi$
$840$	0	0
$841$	−27.0933	−0.934252
$842$	0	0
$843$	−24.2031	−0.833598
$844$	0	0
$845$	−47.4274	−1.63155
$846$	0	0
$847$	9.10953	0.313007
$848$	0	0
$849$	6.41724	0.220239
$850$	0	0
$851$	8.98733	0.308082
$852$	0	0
$853$	−9.53933	−0.326620	−0.163310	−	0.986575i	$-0.552217\pi$
$853$	−9.53933	−0.326620	−0.163310	+	0.986575i	$0.552217\pi$
$854$	0	0
$855$	−42.1686	−1.44213
$856$	0	0
$857$	4.07636	0.139246	0.0696230	−	0.997573i	$-0.477820\pi$
$857$	4.07636	0.139246	0.0696230	+	0.997573i	$0.477820\pi$
$858$	0	0
$859$	10.6251	0.362525	0.181263	−	0.983435i	$-0.441982\pi$
$859$	10.6251	0.362525	0.181263	+	0.983435i	$0.441982\pi$
$860$	0	0
$861$	−8.57790	−0.292334
$862$	0	0
$863$	18.7328	0.637674	0.318837	−	0.947810i	$-0.396708\pi$
$863$	18.7328	0.637674	0.318837	+	0.947810i	$0.396708\pi$
$864$	0	0
$865$	26.4735	0.900125
$866$	0	0
$867$	8.31428	0.282368
$868$	0	0
$869$	1.25500	0.0425729
$870$	0	0
$871$	−1.83237	−0.0620874
$872$	0	0
$873$	−16.8820	−0.571368
$874$	0	0
$875$	−36.6139	−1.23778
$876$	0	0
$877$	−45.1218	−1.52365	−0.761827	−	0.647780i	$-0.775698\pi$
$877$	−45.1218	−1.52365	−0.761827	+	0.647780i	$0.775698\pi$
$878$	0	0
$879$	42.2153	1.42389
$880$	0	0
$881$	8.17898	0.275557	0.137778	−	0.990463i	$-0.456004\pi$
$881$	8.17898	0.275557	0.137778	+	0.990463i	$0.456004\pi$
$882$	0	0
$883$	−37.2673	−1.25415	−0.627073	−	0.778961i	$-0.715747\pi$
$883$	−37.2673	−1.25415	−0.627073	+	0.778961i	$0.715747\pi$
$884$	0	0
$885$	14.1414	0.475357
$886$	0	0
$887$	−3.11112	−0.104461	−0.0522305	−	0.998635i	$-0.516633\pi$
$887$	−3.11112	−0.104461	−0.0522305	+	0.998635i	$0.516633\pi$
$888$	0	0
$889$	−6.40124	−0.214691
$890$	0	0
$891$	1.07448	0.0359966
$892$	0	0
$893$	−31.2659	−1.04627
$894$	0	0
$895$	−72.5844	−2.42623
$896$	0	0
$897$	9.17243	0.306258
$898$	0	0
$899$	3.43510	0.114567
$900$	0	0
$901$	−21.4973	−0.716180
$902$	0	0
$903$	20.3478	0.677133
$904$	0	0
$905$	39.1001	1.29973
$906$	0	0
$907$	−58.6324	−1.94686	−0.973428	−	0.228994i	$-0.926456\pi$
$907$	−58.6324	−1.94686	−0.973428	+	0.228994i	$0.926456\pi$
$908$	0	0
$909$	−1.45042	−0.0481073
$910$	0	0
$911$	−8.36194	−0.277044	−0.138522	−	0.990359i	$-0.544235\pi$
$911$	−8.36194	−0.277044	−0.138522	+	0.990359i	$0.544235\pi$
$912$	0	0
$913$	1.61859	0.0535674
$914$	0	0
$915$	105.825	3.49847
$916$	0	0
$917$	−8.95198	−0.295621
$918$	0	0
$919$	−14.5589	−0.480253	−0.240127	−	0.970742i	$-0.577189\pi$
$919$	−14.5589	−0.480253	−0.240127	+	0.970742i	$0.577189\pi$
$920$	0	0
$921$	51.1565	1.68567
$922$	0	0
$923$	8.44047	0.277822
$924$	0	0
$925$	−45.6929	−1.50237
$926$	0	0
$927$	−8.77831	−0.288318
$928$	0	0
$929$	2.37351	0.0778723	0.0389361	−	0.999242i	$-0.487603\pi$
$929$	2.37351	0.0778723	0.0389361	+	0.999242i	$0.487603\pi$
$930$	0	0
$931$	52.6192	1.72453
$932$	0	0
$933$	−48.9512	−1.60259
$934$	0	0
$935$	1.54944	0.0506721
$936$	0	0
$937$	32.0019	1.04546	0.522728	−	0.852500i	$-0.324914\pi$
$937$	32.0019	1.04546	0.522728	+	0.852500i	$0.324914\pi$
$938$	0	0
$939$	38.2531	1.24834
$940$	0	0
$941$	14.9262	0.486580	0.243290	−	0.969954i	$-0.421773\pi$
$941$	14.9262	0.486580	0.243290	+	0.969954i	$0.421773\pi$
$942$	0	0
$943$	14.9193	0.485841
$944$	0	0
$945$	−14.0284	−0.456345
$946$	0	0
$947$	−24.6714	−0.801714	−0.400857	−	0.916141i	$-0.631288\pi$
$947$	−24.6714	−0.801714	−0.400857	+	0.916141i	$0.631288\pi$
$948$	0	0
$949$	9.12712	0.296279
$950$	0	0
$951$	−41.1621	−1.33477
$952$	0	0
$953$	1.01906	0.0330106	0.0165053	−	0.999864i	$-0.494746\pi$
$953$	1.01906	0.0330106	0.0165053	+	0.999864i	$0.494746\pi$
$954$	0	0
$955$	−116.413	−3.76703
$956$	0	0
$957$	0.271379	0.00877244
$958$	0	0
$959$	14.6503	0.473081
$960$	0	0
$961$	−24.8113	−0.800365
$962$	0	0
$963$	1.29644	0.0417772
$964$	0	0
$965$	0.364281	0.0117266
$966$	0	0
$967$	0.525505	0.0168991	0.00844954	−	0.999964i	$-0.497310\pi$
$967$	0.525505	0.0168991	0.00844954	+	0.999964i	$0.497310\pi$
$968$	0	0
$969$	−60.8936	−1.95619
$970$	0	0
$971$	36.8504	1.18258	0.591292	−	0.806457i	$-0.298618\pi$
$971$	36.8504	1.18258	0.591292	+	0.806457i	$0.298618\pi$
$972$	0	0
$973$	−12.7102	−0.407469
$974$	0	0
$975$	−46.6339	−1.49348
$976$	0	0
$977$	−16.3528	−0.523171	−0.261586	−	0.965180i	$-0.584245\pi$
$977$	−16.3528	−0.523171	−0.261586	+	0.965180i	$0.584245\pi$
$978$	0	0
$979$	0.716956	0.0229140
$980$	0	0
$981$	14.1864	0.452936
$982$	0	0
$983$	−12.1552	−0.387691	−0.193846	−	0.981032i	$-0.562096\pi$
$983$	−12.1552	−0.387691	−0.193846	+	0.981032i	$0.562096\pi$
$984$	0	0
$985$	−81.3372	−2.59162
$986$	0	0
$987$	6.32155	0.201217
$988$	0	0
$989$	−35.3905	−1.12535
$990$	0	0
$991$	13.7096	0.435500	0.217750	−	0.976005i	$-0.430128\pi$
$991$	13.7096	0.435500	0.217750	+	0.976005i	$0.430128\pi$
$992$	0	0
$993$	61.0472	1.93728
$994$	0	0
$995$	40.7425	1.29163
$996$	0	0
$997$	−38.5298	−1.22025	−0.610125	−	0.792305i	$-0.708881\pi$
$997$	−38.5298	−1.22025	−0.610125	+	0.792305i	$0.708881\pi$
$998$	0	0
$999$	−11.6330	−0.368050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.9		33
4.3	odd	2		4024.2.a.g.1.25	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.25	✓	33	4.3	odd	2
8048.2.a.x.1.9		33	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.03324 q^{3} +4.46117 q^{5} -0.828843 q^{7} +1.13405 q^{9} +O(q^{10})\)	\(q-2.03324 q^{3} +4.46117 q^{5} -0.828843 q^{7} +1.13405 q^{9} -0.0966604 q^{11} +1.53910 q^{13} -9.07062 q^{15} -3.59316 q^{17} -8.33503 q^{19} +1.68523 q^{21} -2.93109 q^{23} +14.9021 q^{25} +3.79392 q^{27} +1.38083 q^{29} +2.48771 q^{31} +0.196533 q^{33} -3.69761 q^{35} -3.06621 q^{37} -3.12936 q^{39} -5.09004 q^{41} +12.0742 q^{43} +5.05920 q^{45} +3.75114 q^{47} -6.31302 q^{49} +7.30575 q^{51} +5.98285 q^{53} -0.431219 q^{55} +16.9471 q^{57} -1.55903 q^{59} -11.6668 q^{61} -0.939950 q^{63} +6.86620 q^{65} -1.19054 q^{67} +5.95959 q^{69} +5.48402 q^{71} +5.93015 q^{73} -30.2994 q^{75} +0.0801163 q^{77} -12.9836 q^{79} -11.1161 q^{81} -16.7451 q^{83} -16.0297 q^{85} -2.80755 q^{87} -7.41727 q^{89} -1.27567 q^{91} -5.05810 q^{93} -37.1840 q^{95} -14.8864 q^{97} -0.109618 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * q^99

Introduction

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