LMFDB - Embedded newform 8048.2.a.u.1.19

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:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+1.16576 q^{3} -3.04539 q^{5} -0.946556 q^{7} -1.64102 q^{9} +O(q^{10})$	$q+1.16576 q^{3} -3.04539 q^{5} -0.946556 q^{7} -1.64102 q^{9} +5.26958 q^{11} +4.98081 q^{13} -3.55018 q^{15} -4.70676 q^{17} -6.27914 q^{19} -1.10345 q^{21} +4.26619 q^{23} +4.27442 q^{25} -5.41029 q^{27} +9.05010 q^{29} -6.05473 q^{31} +6.14304 q^{33} +2.88264 q^{35} +0.0493818 q^{37} +5.80641 q^{39} +4.78649 q^{41} +8.15210 q^{43} +4.99754 q^{45} -12.9257 q^{47} -6.10403 q^{49} -5.48693 q^{51} +3.38346 q^{53} -16.0479 q^{55} -7.31994 q^{57} +6.40570 q^{59} -13.0919 q^{61} +1.55331 q^{63} -15.1685 q^{65} +6.83640 q^{67} +4.97334 q^{69} -0.707071 q^{71} +9.30909 q^{73} +4.98293 q^{75} -4.98795 q^{77} -12.8093 q^{79} -1.38402 q^{81} +6.10655 q^{83} +14.3339 q^{85} +10.5502 q^{87} +7.70505 q^{89} -4.71462 q^{91} -7.05833 q^{93} +19.1225 q^{95} +7.42978 q^{97} -8.64746 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	1.16576	0.673049	0.336524	−	0.941675i	$-0.390749\pi$
$3$	1.16576	0.673049	0.336524	+	0.941675i	$0.390749\pi$
$4$	0	0
$5$	−3.04539	−1.36194	−0.680971	−	0.732311i	$-0.738442\pi$
$5$	−3.04539	−1.36194	−0.680971	+	0.732311i	$0.738442\pi$
$6$	0	0
$7$	−0.946556	−0.357765	−0.178882	−	0.983870i	$-0.557248\pi$
$7$	−0.946556	−0.357765	−0.178882	+	0.983870i	$0.557248\pi$
$8$	0	0
$9$	−1.64102	−0.547005
$10$	0	0
$11$	5.26958	1.58884	0.794419	−	0.607371i	$-0.207776\pi$
$11$	5.26958	1.58884	0.794419	+	0.607371i	$0.207776\pi$
$12$	0	0
$13$	4.98081	1.38143	0.690714	−	0.723128i	$-0.257296\pi$
$13$	4.98081	1.38143	0.690714	+	0.723128i	$0.257296\pi$
$14$	0	0
$15$	−3.55018	−0.916653
$16$	0	0
$17$	−4.70676	−1.14156	−0.570779	−	0.821104i	$-0.693359\pi$
$17$	−4.70676	−1.14156	−0.570779	+	0.821104i	$0.693359\pi$
$18$	0	0
$19$	−6.27914	−1.44053	−0.720267	−	0.693697i	$-0.755981\pi$
$19$	−6.27914	−1.44053	−0.720267	+	0.693697i	$0.755981\pi$
$20$	0	0
$21$	−1.10345	−0.240793
$22$	0	0
$23$	4.26619	0.889563	0.444781	−	0.895639i	$-0.353281\pi$
$23$	4.26619	0.889563	0.444781	+	0.895639i	$0.353281\pi$
$24$	0	0
$25$	4.27442	0.854884
$26$	0	0
$27$	−5.41029	−1.04121
$28$	0	0
$29$	9.05010	1.68056	0.840281	−	0.542152i	$-0.182390\pi$
$29$	9.05010	1.68056	0.840281	+	0.542152i	$0.182390\pi$
$30$	0	0
$31$	−6.05473	−1.08746	−0.543731	−	0.839260i	$-0.682989\pi$
$31$	−6.05473	−1.08746	−0.543731	+	0.839260i	$0.682989\pi$
$32$	0	0
$33$	6.14304	1.06937
$34$	0	0
$35$	2.88264	0.487254
$36$	0	0
$37$	0.0493818	0.00811831	0.00405916	−	0.999992i	$-0.498708\pi$
$37$	0.0493818	0.00811831	0.00405916	+	0.999992i	$0.498708\pi$
$38$	0	0
$39$	5.80641	0.929769
$40$	0	0
$41$	4.78649	0.747523	0.373762	−	0.927525i	$-0.378068\pi$
$41$	4.78649	0.747523	0.373762	+	0.927525i	$0.378068\pi$
$42$	0	0
$43$	8.15210	1.24318	0.621592	−	0.783341i	$-0.286486\pi$
$43$	8.15210	1.24318	0.621592	+	0.783341i	$0.286486\pi$
$44$	0	0
$45$	4.99754	0.744989
$46$	0	0
$47$	−12.9257	−1.88540	−0.942701	−	0.333639i	$-0.891723\pi$
$47$	−12.9257	−1.88540	−0.942701	+	0.333639i	$0.891723\pi$
$48$	0	0
$49$	−6.10403	−0.872004
$50$	0	0
$51$	−5.48693	−0.768324
$52$	0	0
$53$	3.38346	0.464754	0.232377	−	0.972626i	$-0.425350\pi$
$53$	3.38346	0.464754	0.232377	+	0.972626i	$0.425350\pi$
$54$	0	0
$55$	−16.0479	−2.16390
$56$	0	0
$57$	−7.31994	−0.969550
$58$	0	0
$59$	6.40570	0.833951	0.416975	−	0.908918i	$-0.363090\pi$
$59$	6.40570	0.833951	0.416975	+	0.908918i	$0.363090\pi$
$60$	0	0
$61$	−13.0919	−1.67625	−0.838126	−	0.545477i	$-0.816349\pi$
$61$	−13.0919	−1.67625	−0.838126	+	0.545477i	$0.816349\pi$
$62$	0	0
$63$	1.55331	0.195699
$64$	0	0
$65$	−15.1685	−1.88142
$66$	0	0
$67$	6.83640	0.835200	0.417600	−	0.908631i	$-0.362871\pi$
$67$	6.83640	0.835200	0.417600	+	0.908631i	$0.362871\pi$
$68$	0	0
$69$	4.97334	0.598719
$70$	0	0
$71$	−0.707071	−0.0839139	−0.0419570	−	0.999119i	$-0.513359\pi$
$71$	−0.707071	−0.0839139	−0.0419570	+	0.999119i	$0.513359\pi$
$72$	0	0
$73$	9.30909	1.08955	0.544773	−	0.838583i	$-0.316616\pi$
$73$	9.30909	1.08955	0.544773	+	0.838583i	$0.316616\pi$
$74$	0	0
$75$	4.98293	0.575379
$76$	0	0
$77$	−4.98795	−0.568430
$78$	0	0
$79$	−12.8093	−1.44116	−0.720581	−	0.693370i	$-0.756125\pi$
$79$	−12.8093	−1.44116	−0.720581	+	0.693370i	$0.756125\pi$
$80$	0	0
$81$	−1.38402	−0.153780
$82$	0	0
$83$	6.10655	0.670281	0.335140	−	0.942168i	$-0.391216\pi$
$83$	6.10655	0.670281	0.335140	+	0.942168i	$0.391216\pi$
$84$	0	0
$85$	14.3339	1.55473
$86$	0	0
$87$	10.5502	1.13110
$88$	0	0
$89$	7.70505	0.816733	0.408367	−	0.912818i	$-0.366099\pi$
$89$	7.70505	0.816733	0.408367	+	0.912818i	$0.366099\pi$
$90$	0	0
$91$	−4.71462	−0.494226
$92$	0	0
$93$	−7.05833	−0.731915
$94$	0	0
$95$	19.1225	1.96192
$96$	0	0
$97$	7.42978	0.754380	0.377190	−	0.926136i	$-0.376890\pi$
$97$	7.42978	0.754380	0.377190	+	0.926136i	$0.376890\pi$
$98$	0	0
$99$	−8.64746	−0.869102
$100$	0	0
$101$	−2.44307	−0.243095	−0.121547	−	0.992586i	$-0.538786\pi$
$101$	−2.44307	−0.243095	−0.121547	+	0.992586i	$0.538786\pi$
$102$	0	0
$103$	0.861446	0.0848808	0.0424404	−	0.999099i	$-0.486487\pi$
$103$	0.861446	0.0848808	0.0424404	+	0.999099i	$0.486487\pi$
$104$	0	0
$105$	3.36045	0.327946
$106$	0	0
$107$	−9.51324	−0.919680	−0.459840	−	0.888002i	$-0.652093\pi$
$107$	−9.51324	−0.919680	−0.459840	+	0.888002i	$0.652093\pi$
$108$	0	0
$109$	10.5857	1.01393	0.506964	−	0.861967i	$-0.330768\pi$
$109$	10.5857	1.01393	0.506964	+	0.861967i	$0.330768\pi$
$110$	0	0
$111$	0.0575671	0.00546402
$112$	0	0
$113$	−3.44339	−0.323927	−0.161964	−	0.986797i	$-0.551783\pi$
$113$	−3.44339	−0.323927	−0.161964	+	0.986797i	$0.551783\pi$
$114$	0	0
$115$	−12.9922	−1.21153
$116$	0	0
$117$	−8.17359	−0.755648
$118$	0	0
$119$	4.45522	0.408409
$120$	0	0
$121$	16.7684	1.52440
$122$	0	0
$123$	5.57987	0.503120
$124$	0	0
$125$	2.20967	0.197639
$126$	0	0
$127$	−5.32565	−0.472575	−0.236288	−	0.971683i	$-0.575931\pi$
$127$	−5.32565	−0.472575	−0.236288	+	0.971683i	$0.575931\pi$
$128$	0	0
$129$	9.50335	0.836723
$130$	0	0
$131$	6.30936	0.551252	0.275626	−	0.961265i	$-0.411115\pi$
$131$	6.30936	0.551252	0.275626	+	0.961265i	$0.411115\pi$
$132$	0	0
$133$	5.94356	0.515372
$134$	0	0
$135$	16.4765	1.41807
$136$	0	0
$137$	−0.728302	−0.0622231	−0.0311115	−	0.999516i	$-0.509905\pi$
$137$	−0.728302	−0.0622231	−0.0311115	+	0.999516i	$0.509905\pi$
$138$	0	0
$139$	12.8073	1.08630	0.543152	−	0.839634i	$-0.317231\pi$
$139$	12.8073	1.08630	0.543152	+	0.839634i	$0.317231\pi$
$140$	0	0
$141$	−15.0682	−1.26897
$142$	0	0
$143$	26.2468	2.19487
$144$	0	0
$145$	−27.5611	−2.28883
$146$	0	0
$147$	−7.11580	−0.586902
$148$	0	0
$149$	21.2953	1.74458	0.872290	−	0.488989i	$-0.162634\pi$
$149$	21.2953	1.74458	0.872290	+	0.488989i	$0.162634\pi$
$150$	0	0
$151$	18.5640	1.51071	0.755357	−	0.655314i	$-0.227464\pi$
$151$	18.5640	1.51071	0.755357	+	0.655314i	$0.227464\pi$
$152$	0	0
$153$	7.72387	0.624438
$154$	0	0
$155$	18.4390	1.48106
$156$	0	0
$157$	7.38222	0.589166	0.294583	−	0.955626i	$-0.404819\pi$
$157$	7.38222	0.589166	0.294583	+	0.955626i	$0.404819\pi$
$158$	0	0
$159$	3.94429	0.312802
$160$	0	0
$161$	−4.03819	−0.318254
$162$	0	0
$163$	8.57330	0.671513	0.335757	−	0.941949i	$-0.391008\pi$
$163$	8.57330	0.671513	0.335757	+	0.941949i	$0.391008\pi$
$164$	0	0
$165$	−18.7080	−1.45641
$166$	0	0
$167$	−24.6909	−1.91064	−0.955318	−	0.295581i	$-0.904487\pi$
$167$	−24.6909	−1.91064	−0.955318	+	0.295581i	$0.904487\pi$
$168$	0	0
$169$	11.8085	0.908345
$170$	0	0
$171$	10.3042	0.787979
$172$	0	0
$173$	−5.69821	−0.433227	−0.216613	−	0.976257i	$-0.569501\pi$
$173$	−5.69821	−0.433227	−0.216613	+	0.976257i	$0.569501\pi$
$174$	0	0
$175$	−4.04598	−0.305847
$176$	0	0
$177$	7.46747	0.561290
$178$	0	0
$179$	−6.64416	−0.496608	−0.248304	−	0.968682i	$-0.579873\pi$
$179$	−6.64416	−0.496608	−0.248304	+	0.968682i	$0.579873\pi$
$180$	0	0
$181$	−1.23739	−0.0919746	−0.0459873	−	0.998942i	$-0.514643\pi$
$181$	−1.23739	−0.0919746	−0.0459873	+	0.998942i	$0.514643\pi$
$182$	0	0
$183$	−15.2620	−1.12820
$184$	0	0
$185$	−0.150387	−0.0110567
$186$	0	0
$187$	−24.8026	−1.81375
$188$	0	0
$189$	5.12114	0.372508
$190$	0	0
$191$	1.78498	0.129157	0.0645783	−	0.997913i	$-0.479430\pi$
$191$	1.78498	0.129157	0.0645783	+	0.997913i	$0.479430\pi$
$192$	0	0
$193$	21.4873	1.54669	0.773345	−	0.633986i	$-0.218582\pi$
$193$	21.4873	1.54669	0.773345	+	0.633986i	$0.218582\pi$
$194$	0	0
$195$	−17.6828	−1.26629
$196$	0	0
$197$	18.4500	1.31451	0.657253	−	0.753670i	$-0.271718\pi$
$197$	18.4500	1.31451	0.657253	+	0.753670i	$0.271718\pi$
$198$	0	0
$199$	−23.5776	−1.67137	−0.835686	−	0.549207i	$-0.814930\pi$
$199$	−23.5776	−1.67137	−0.835686	+	0.549207i	$0.814930\pi$
$200$	0	0
$201$	7.96957	0.562130
$202$	0	0
$203$	−8.56643	−0.601245
$204$	0	0
$205$	−14.5767	−1.01808
$206$	0	0
$207$	−7.00089	−0.486595
$208$	0	0
$209$	−33.0884	−2.28877
$210$	0	0
$211$	8.98431	0.618505	0.309253	−	0.950980i	$-0.399921\pi$
$211$	8.98431	0.618505	0.309253	+	0.950980i	$0.399921\pi$
$212$	0	0
$213$	−0.824272	−0.0564782
$214$	0	0
$215$	−24.8263	−1.69314
$216$	0	0
$217$	5.73114	0.389055
$218$	0	0
$219$	10.8521	0.733318
$220$	0	0
$221$	−23.4435	−1.57698
$222$	0	0
$223$	−3.57817	−0.239612	−0.119806	−	0.992797i	$-0.538227\pi$
$223$	−3.57817	−0.239612	−0.119806	+	0.992797i	$0.538227\pi$
$224$	0	0
$225$	−7.01439	−0.467626
$226$	0	0
$227$	−14.3381	−0.951651	−0.475826	−	0.879540i	$-0.657851\pi$
$227$	−14.3381	−0.951651	−0.475826	+	0.879540i	$0.657851\pi$
$228$	0	0
$229$	−18.6635	−1.23332	−0.616660	−	0.787229i	$-0.711515\pi$
$229$	−18.6635	−1.23332	−0.616660	+	0.787229i	$0.711515\pi$
$230$	0	0
$231$	−5.81473	−0.382581
$232$	0	0
$233$	12.7731	0.836792	0.418396	−	0.908265i	$-0.362592\pi$
$233$	12.7731	0.836792	0.418396	+	0.908265i	$0.362592\pi$
$234$	0	0
$235$	39.3637	2.56781
$236$	0	0
$237$	−14.9325	−0.969973
$238$	0	0
$239$	9.81773	0.635056	0.317528	−	0.948249i	$-0.397147\pi$
$239$	9.81773	0.635056	0.317528	+	0.948249i	$0.397147\pi$
$240$	0	0
$241$	27.0449	1.74212	0.871059	−	0.491178i	$-0.163434\pi$
$241$	27.0449	1.74212	0.871059	+	0.491178i	$0.163434\pi$
$242$	0	0
$243$	14.6174	0.937708
$244$	0	0
$245$	18.5892	1.18762
$246$	0	0
$247$	−31.2752	−1.98999
$248$	0	0
$249$	7.11874	0.451132
$250$	0	0
$251$	21.9017	1.38242	0.691211	−	0.722653i	$-0.257078\pi$
$251$	21.9017	1.38242	0.691211	+	0.722653i	$0.257078\pi$
$252$	0	0
$253$	22.4810	1.41337
$254$	0	0
$255$	16.7099	1.04641
$256$	0	0
$257$	1.42333	0.0887852	0.0443926	−	0.999014i	$-0.485865\pi$
$257$	1.42333	0.0887852	0.0443926	+	0.999014i	$0.485865\pi$
$258$	0	0
$259$	−0.0467426	−0.00290445
$260$	0	0
$261$	−14.8514	−0.919276
$262$	0	0
$263$	−21.4215	−1.32091	−0.660453	−	0.750867i	$-0.729636\pi$
$263$	−21.4215	−1.32091	−0.660453	+	0.750867i	$0.729636\pi$
$264$	0	0
$265$	−10.3040	−0.632968
$266$	0	0
$267$	8.98220	0.549701
$268$	0	0
$269$	28.3745	1.73002	0.865011	−	0.501753i	$-0.167311\pi$
$269$	28.3745	1.73002	0.865011	+	0.501753i	$0.167311\pi$
$270$	0	0
$271$	11.9277	0.724557	0.362278	−	0.932070i	$-0.381999\pi$
$271$	11.9277	0.724557	0.362278	+	0.932070i	$0.381999\pi$
$272$	0	0
$273$	−5.49609	−0.332639
$274$	0	0
$275$	22.5244	1.35827
$276$	0	0
$277$	4.48698	0.269597	0.134798	−	0.990873i	$-0.456961\pi$
$277$	4.48698	0.269597	0.134798	+	0.990873i	$0.456961\pi$
$278$	0	0
$279$	9.93590	0.594847
$280$	0	0
$281$	14.2557	0.850424	0.425212	−	0.905094i	$-0.360200\pi$
$281$	14.2557	0.850424	0.425212	+	0.905094i	$0.360200\pi$
$282$	0	0
$283$	4.40751	0.261999	0.131000	−	0.991382i	$-0.458181\pi$
$283$	4.40751	0.261999	0.131000	+	0.991382i	$0.458181\pi$
$284$	0	0
$285$	22.2921	1.32047
$286$	0	0
$287$	−4.53068	−0.267438
$288$	0	0
$289$	5.15361	0.303153
$290$	0	0
$291$	8.66130	0.507735
$292$	0	0
$293$	−24.6104	−1.43776	−0.718879	−	0.695136i	$-0.755344\pi$
$293$	−24.6104	−1.43776	−0.718879	+	0.695136i	$0.755344\pi$
$294$	0	0
$295$	−19.5079	−1.13579
$296$	0	0
$297$	−28.5099	−1.65431
$298$	0	0
$299$	21.2491	1.22887
$300$	0	0
$301$	−7.71642	−0.444767
$302$	0	0
$303$	−2.84802	−0.163615
$304$	0	0
$305$	39.8701	2.28296
$306$	0	0
$307$	20.7200	1.18255	0.591275	−	0.806470i	$-0.298625\pi$
$307$	20.7200	1.18255	0.591275	+	0.806470i	$0.298625\pi$
$308$	0	0
$309$	1.00424	0.0571290
$310$	0	0
$311$	19.1818	1.08770	0.543849	−	0.839183i	$-0.316967\pi$
$311$	19.1818	1.08770	0.543849	+	0.839183i	$0.316967\pi$
$312$	0	0
$313$	17.8254	1.00755	0.503775	−	0.863835i	$-0.331944\pi$
$313$	17.8254	1.00755	0.503775	+	0.863835i	$0.331944\pi$
$314$	0	0
$315$	−4.73045	−0.266531
$316$	0	0
$317$	17.9832	1.01004	0.505020	−	0.863108i	$-0.331485\pi$
$317$	17.9832	1.01004	0.505020	+	0.863108i	$0.331485\pi$
$318$	0	0
$319$	47.6902	2.67014
$320$	0	0
$321$	−11.0901	−0.618990
$322$	0	0
$323$	29.5544	1.64445
$324$	0	0
$325$	21.2901	1.18096
$326$	0	0
$327$	12.3404	0.682424
$328$	0	0
$329$	12.2349	0.674530
$330$	0	0
$331$	14.0001	0.769516	0.384758	−	0.923018i	$-0.374285\pi$
$331$	14.0001	0.769516	0.384758	+	0.923018i	$0.374285\pi$
$332$	0	0
$333$	−0.0810363	−0.00444076
$334$	0	0
$335$	−20.8195	−1.13749
$336$	0	0
$337$	7.61057	0.414574	0.207287	−	0.978280i	$-0.433537\pi$
$337$	7.61057	0.414574	0.207287	+	0.978280i	$0.433537\pi$
$338$	0	0
$339$	−4.01415	−0.218019
$340$	0	0
$341$	−31.9059	−1.72780
$342$	0	0
$343$	12.4037	0.669737
$344$	0	0
$345$	−15.1458	−0.815420
$346$	0	0
$347$	−9.89632	−0.531262	−0.265631	−	0.964075i	$-0.585580\pi$
$347$	−9.89632	−0.531262	−0.265631	+	0.964075i	$0.585580\pi$
$348$	0	0
$349$	26.2356	1.40436	0.702179	−	0.712001i	$-0.252211\pi$
$349$	26.2356	1.40436	0.702179	+	0.712001i	$0.252211\pi$
$350$	0	0
$351$	−26.9476	−1.43836
$352$	0	0
$353$	13.4861	0.717791	0.358896	−	0.933378i	$-0.383153\pi$
$353$	13.4861	0.717791	0.358896	+	0.933378i	$0.383153\pi$
$354$	0	0
$355$	2.15331	0.114286
$356$	0	0
$357$	5.19369	0.274879
$358$	0	0
$359$	−20.7781	−1.09663	−0.548314	−	0.836272i	$-0.684730\pi$
$359$	−20.7781	−1.09663	−0.548314	+	0.836272i	$0.684730\pi$
$360$	0	0
$361$	20.4276	1.07514
$362$	0	0
$363$	19.5479	1.02600
$364$	0	0
$365$	−28.3498	−1.48390
$366$	0	0
$367$	10.0667	0.525477	0.262738	−	0.964867i	$-0.415374\pi$
$367$	10.0667	0.525477	0.262738	+	0.964867i	$0.415374\pi$
$368$	0	0
$369$	−7.85470	−0.408899
$370$	0	0
$371$	−3.20264	−0.166273
$372$	0	0
$373$	14.8220	0.767455	0.383727	−	0.923446i	$-0.374640\pi$
$373$	14.8220	0.767455	0.383727	+	0.923446i	$0.374640\pi$
$374$	0	0
$375$	2.57594	0.133021
$376$	0	0
$377$	45.0768	2.32158
$378$	0	0
$379$	31.3165	1.60862	0.804310	−	0.594209i	$-0.202535\pi$
$379$	31.3165	1.60862	0.804310	+	0.594209i	$0.202535\pi$
$380$	0	0
$381$	−6.20841	−0.318066
$382$	0	0
$383$	22.7814	1.16407	0.582037	−	0.813162i	$-0.302256\pi$
$383$	22.7814	1.16407	0.582037	+	0.813162i	$0.302256\pi$
$384$	0	0
$385$	15.1903	0.774168
$386$	0	0
$387$	−13.3777	−0.680028
$388$	0	0
$389$	−12.6937	−0.643596	−0.321798	−	0.946808i	$-0.604287\pi$
$389$	−12.6937	−0.643596	−0.321798	+	0.946808i	$0.604287\pi$
$390$	0	0
$391$	−20.0800	−1.01549
$392$	0	0
$393$	7.35517	0.371019
$394$	0	0
$395$	39.0095	1.96278
$396$	0	0
$397$	−14.5079	−0.728131	−0.364065	−	0.931373i	$-0.618612\pi$
$397$	−14.5079	−0.728131	−0.364065	+	0.931373i	$0.618612\pi$
$398$	0	0
$399$	6.92873	0.346871
$400$	0	0
$401$	−30.1900	−1.50762	−0.753808	−	0.657095i	$-0.771785\pi$
$401$	−30.1900	−1.50762	−0.753808	+	0.657095i	$0.771785\pi$
$402$	0	0
$403$	−30.1575	−1.50225
$404$	0	0
$405$	4.21490	0.209440
$406$	0	0
$407$	0.260221	0.0128987
$408$	0	0
$409$	13.4655	0.665824	0.332912	−	0.942958i	$-0.391969\pi$
$409$	13.4655	0.665824	0.332912	+	0.942958i	$0.391969\pi$
$410$	0	0
$411$	−0.849022	−0.0418792
$412$	0	0
$413$	−6.06335	−0.298358
$414$	0	0
$415$	−18.5968	−0.912883
$416$	0	0
$417$	14.9302	0.731136
$418$	0	0
$419$	−11.3193	−0.552983	−0.276491	−	0.961016i	$-0.589172\pi$
$419$	−11.3193	−0.552983	−0.276491	+	0.961016i	$0.589172\pi$
$420$	0	0
$421$	−1.51985	−0.0740728	−0.0370364	−	0.999314i	$-0.511792\pi$
$421$	−1.51985	−0.0740728	−0.0370364	+	0.999314i	$0.511792\pi$
$422$	0	0
$423$	21.2112	1.03132
$424$	0	0
$425$	−20.1187	−0.975899
$426$	0	0
$427$	12.3923	0.599704
$428$	0	0
$429$	30.5973	1.47725
$430$	0	0
$431$	8.01315	0.385980	0.192990	−	0.981201i	$-0.438182\pi$
$431$	8.01315	0.385980	0.192990	+	0.981201i	$0.438182\pi$
$432$	0	0
$433$	−0.923286	−0.0443703	−0.0221852	−	0.999754i	$-0.507062\pi$
$433$	−0.923286	−0.0443703	−0.0221852	+	0.999754i	$0.507062\pi$
$434$	0	0
$435$	−32.1295	−1.54049
$436$	0	0
$437$	−26.7880	−1.28144
$438$	0	0
$439$	16.5904	0.791819	0.395909	−	0.918290i	$-0.370429\pi$
$439$	16.5904	0.791819	0.395909	+	0.918290i	$0.370429\pi$
$440$	0	0
$441$	10.0168	0.476991
$442$	0	0
$443$	−14.0669	−0.668338	−0.334169	−	0.942513i	$-0.608456\pi$
$443$	−14.0669	−0.668338	−0.334169	+	0.942513i	$0.608456\pi$
$444$	0	0
$445$	−23.4649	−1.11234
$446$	0	0
$447$	24.8251	1.17419
$448$	0	0
$449$	7.99294	0.377210	0.188605	−	0.982053i	$-0.439603\pi$
$449$	7.99294	0.377210	0.188605	+	0.982053i	$0.439603\pi$
$450$	0	0
$451$	25.2228	1.18769
$452$	0	0
$453$	21.6410	1.01678
$454$	0	0
$455$	14.3579	0.673107
$456$	0	0
$457$	8.58940	0.401795	0.200898	−	0.979612i	$-0.435614\pi$
$457$	8.58940	0.401795	0.200898	+	0.979612i	$0.435614\pi$
$458$	0	0
$459$	25.4649	1.18860
$460$	0	0
$461$	−26.6903	−1.24309	−0.621547	−	0.783377i	$-0.713495\pi$
$461$	−26.6903	−1.24309	−0.621547	+	0.783377i	$0.713495\pi$
$462$	0	0
$463$	32.9401	1.53085	0.765427	−	0.643522i	$-0.222528\pi$
$463$	32.9401	1.53085	0.765427	+	0.643522i	$0.222528\pi$
$464$	0	0
$465$	21.4954	0.996825
$466$	0	0
$467$	23.4770	1.08638	0.543192	−	0.839608i	$-0.317216\pi$
$467$	23.4770	1.08638	0.543192	+	0.839608i	$0.317216\pi$
$468$	0	0
$469$	−6.47104	−0.298805
$470$	0	0
$471$	8.60586	0.396537
$472$	0	0
$473$	42.9581	1.97522
$474$	0	0
$475$	−26.8397	−1.23149
$476$	0	0
$477$	−5.55231	−0.254223
$478$	0	0
$479$	−11.0771	−0.506128	−0.253064	−	0.967450i	$-0.581438\pi$
$479$	−11.0771	−0.506128	−0.253064	+	0.967450i	$0.581438\pi$
$480$	0	0
$481$	0.245961	0.0112149
$482$	0	0
$483$	−4.70754	−0.214201
$484$	0	0
$485$	−22.6266	−1.02742
$486$	0	0
$487$	−14.3579	−0.650620	−0.325310	−	0.945607i	$-0.605469\pi$
$487$	−14.3579	−0.650620	−0.325310	+	0.945607i	$0.605469\pi$
$488$	0	0
$489$	9.99437	0.451961
$490$	0	0
$491$	−24.8206	−1.12014	−0.560068	−	0.828447i	$-0.689225\pi$
$491$	−24.8206	−1.12014	−0.560068	+	0.828447i	$0.689225\pi$
$492$	0	0
$493$	−42.5967	−1.91846
$494$	0	0
$495$	26.3349	1.18367
$496$	0	0
$497$	0.669283	0.0300214
$498$	0	0
$499$	−8.94277	−0.400333	−0.200167	−	0.979762i	$-0.564148\pi$
$499$	−8.94277	−0.400333	−0.200167	+	0.979762i	$0.564148\pi$
$500$	0	0
$501$	−28.7835	−1.28595
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	7.44011	0.331081
$506$	0	0
$507$	13.7658	0.611360
$508$	0	0
$509$	−29.0226	−1.28641	−0.643203	−	0.765696i	$-0.722395\pi$
$509$	−29.0226	−1.28641	−0.643203	+	0.765696i	$0.722395\pi$
$510$	0	0
$511$	−8.81158	−0.389801
$512$	0	0
$513$	33.9719	1.49990
$514$	0	0
$515$	−2.62344	−0.115603
$516$	0	0
$517$	−68.1128	−2.99560
$518$	0	0
$519$	−6.64272	−0.291583
$520$	0	0
$521$	−12.7528	−0.558710	−0.279355	−	0.960188i	$-0.590121\pi$
$521$	−12.7528	−0.558710	−0.279355	+	0.960188i	$0.590121\pi$
$522$	0	0
$523$	−33.4251	−1.46158	−0.730789	−	0.682603i	$-0.760848\pi$
$523$	−33.4251	−1.46158	−0.730789	+	0.682603i	$0.760848\pi$
$524$	0	0
$525$	−4.71662	−0.205850
$526$	0	0
$527$	28.4982	1.24140
$528$	0	0
$529$	−4.79960	−0.208678
$530$	0	0
$531$	−10.5118	−0.456175
$532$	0	0
$533$	23.8406	1.03265
$534$	0	0
$535$	28.9716	1.25255
$536$	0	0
$537$	−7.74546	−0.334241
$538$	0	0
$539$	−32.1657	−1.38547
$540$	0	0
$541$	40.1089	1.72442	0.862208	−	0.506554i	$-0.169081\pi$
$541$	40.1089	1.72442	0.862208	+	0.506554i	$0.169081\pi$
$542$	0	0
$543$	−1.44250	−0.0619034
$544$	0	0
$545$	−32.2377	−1.38091
$546$	0	0
$547$	10.6304	0.454525	0.227262	−	0.973834i	$-0.427022\pi$
$547$	10.6304	0.454525	0.227262	+	0.973834i	$0.427022\pi$
$548$	0	0
$549$	21.4841	0.916918
$550$	0	0
$551$	−56.8268	−2.42090
$552$	0	0
$553$	12.1248	0.515597
$554$	0	0
$555$	−0.175314	−0.00744168
$556$	0	0
$557$	−1.94138	−0.0822588	−0.0411294	−	0.999154i	$-0.513096\pi$
$557$	−1.94138	−0.0822588	−0.0411294	+	0.999154i	$0.513096\pi$
$558$	0	0
$559$	40.6041	1.71737
$560$	0	0
$561$	−28.9138	−1.22074
$562$	0	0
$563$	9.66422	0.407298	0.203649	−	0.979044i	$-0.434720\pi$
$563$	9.66422	0.407298	0.203649	+	0.979044i	$0.434720\pi$
$564$	0	0
$565$	10.4865	0.441170
$566$	0	0
$567$	1.31006	0.0550172
$568$	0	0
$569$	−3.24217	−0.135919	−0.0679595	−	0.997688i	$-0.521649\pi$
$569$	−3.24217	−0.135919	−0.0679595	+	0.997688i	$0.521649\pi$
$570$	0	0
$571$	20.2048	0.845546	0.422773	−	0.906236i	$-0.361057\pi$
$571$	20.2048	0.845546	0.422773	+	0.906236i	$0.361057\pi$
$572$	0	0
$573$	2.08085	0.0869287
$574$	0	0
$575$	18.2355	0.760473
$576$	0	0
$577$	18.9102	0.787241	0.393621	−	0.919273i	$-0.371222\pi$
$577$	18.9102	0.787241	0.393621	+	0.919273i	$0.371222\pi$
$578$	0	0
$579$	25.0489	1.04100
$580$	0	0
$581$	−5.78019	−0.239803
$582$	0	0
$583$	17.8294	0.738419
$584$	0	0
$585$	24.8918	1.02915
$586$	0	0
$587$	0.0156371	0.000645414 0	0.000322707	−	1.00000i	$-0.499897\pi$
$587$	0.0156371	0.000645414 0	0.000322707		1.00000i	$0.499897\pi$
$588$	0	0
$589$	38.0185	1.56652
$590$	0	0
$591$	21.5082	0.884727
$592$	0	0
$593$	−24.0728	−0.988551	−0.494275	−	0.869305i	$-0.664567\pi$
$593$	−24.0728	−0.988551	−0.494275	+	0.869305i	$0.664567\pi$
$594$	0	0
$595$	−13.5679	−0.556229
$596$	0	0
$597$	−27.4857	−1.12492
$598$	0	0
$599$	−14.1490	−0.578112	−0.289056	−	0.957312i	$-0.593341\pi$
$599$	−14.1490	−0.578112	−0.289056	+	0.957312i	$0.593341\pi$
$600$	0	0
$601$	14.9260	0.608845	0.304422	−	0.952537i	$-0.401537\pi$
$601$	14.9260	0.608845	0.304422	+	0.952537i	$0.401537\pi$
$602$	0	0
$603$	−11.2186	−0.456858
$604$	0	0
$605$	−51.0665	−2.07615
$606$	0	0
$607$	13.3949	0.543681	0.271840	−	0.962342i	$-0.412368\pi$
$607$	13.3949	0.543681	0.271840	+	0.962342i	$0.412368\pi$
$608$	0	0
$609$	−9.98636	−0.404668
$610$	0	0
$611$	−64.3803	−2.60455
$612$	0	0
$613$	20.2642	0.818463	0.409232	−	0.912431i	$-0.365797\pi$
$613$	20.2642	0.818463	0.409232	+	0.912431i	$0.365797\pi$
$614$	0	0
$615$	−16.9929	−0.685220
$616$	0	0
$617$	−25.2260	−1.01556	−0.507781	−	0.861486i	$-0.669534\pi$
$617$	−25.2260	−1.01556	−0.507781	+	0.861486i	$0.669534\pi$
$618$	0	0
$619$	14.4142	0.579354	0.289677	−	0.957124i	$-0.406452\pi$
$619$	14.4142	0.579354	0.289677	+	0.957124i	$0.406452\pi$
$620$	0	0
$621$	−23.0813	−0.926222
$622$	0	0
$623$	−7.29326	−0.292198
$624$	0	0
$625$	−28.1014	−1.12406
$626$	0	0
$627$	−38.5730	−1.54046
$628$	0	0
$629$	−0.232428	−0.00926752
$630$	0	0
$631$	−1.30592	−0.0519877	−0.0259938	−	0.999662i	$-0.508275\pi$
$631$	−1.30592	−0.0519877	−0.0259938	+	0.999662i	$0.508275\pi$
$632$	0	0
$633$	10.4735	0.416284
$634$	0	0
$635$	16.2187	0.643620
$636$	0	0
$637$	−30.4030	−1.20461
$638$	0	0
$639$	1.16031	0.0459013
$640$	0	0
$641$	20.5617	0.812139	0.406070	−	0.913842i	$-0.366899\pi$
$641$	20.5617	0.812139	0.406070	+	0.913842i	$0.366899\pi$
$642$	0	0
$643$	−27.9532	−1.10237	−0.551183	−	0.834385i	$-0.685823\pi$
$643$	−27.9532	−1.10237	−0.551183	+	0.834385i	$0.685823\pi$
$644$	0	0
$645$	−28.9414	−1.13957
$646$	0	0
$647$	20.3933	0.801743	0.400872	−	0.916134i	$-0.368707\pi$
$647$	20.3933	0.801743	0.400872	+	0.916134i	$0.368707\pi$
$648$	0	0
$649$	33.7553	1.32501
$650$	0	0
$651$	6.68111	0.261853
$652$	0	0
$653$	−19.0975	−0.747341	−0.373671	−	0.927561i	$-0.621901\pi$
$653$	−19.0975	−0.747341	−0.373671	+	0.927561i	$0.621901\pi$
$654$	0	0
$655$	−19.2145	−0.750772
$656$	0	0
$657$	−15.2764	−0.595988
$658$	0	0
$659$	45.5316	1.77366	0.886829	−	0.462098i	$-0.152903\pi$
$659$	45.5316	1.77366	0.886829	+	0.462098i	$0.152903\pi$
$660$	0	0
$661$	−34.7182	−1.35038	−0.675191	−	0.737643i	$-0.735939\pi$
$661$	−34.7182	−1.35038	−0.675191	+	0.737643i	$0.735939\pi$
$662$	0	0
$663$	−27.3294	−1.06138
$664$	0	0
$665$	−18.1005	−0.701906
$666$	0	0
$667$	38.6095	1.49496
$668$	0	0
$669$	−4.17127	−0.161271
$670$	0	0
$671$	−68.9890	−2.66329
$672$	0	0
$673$	43.3056	1.66931	0.834653	−	0.550775i	$-0.185668\pi$
$673$	43.3056	1.66931	0.834653	+	0.550775i	$0.185668\pi$
$674$	0	0
$675$	−23.1258	−0.890114
$676$	0	0
$677$	32.8850	1.26387	0.631936	−	0.775020i	$-0.282260\pi$
$677$	32.8850	1.26387	0.631936	+	0.775020i	$0.282260\pi$
$678$	0	0
$679$	−7.03270	−0.269890
$680$	0	0
$681$	−16.7147	−0.640508
$682$	0	0
$683$	−23.0225	−0.880932	−0.440466	−	0.897769i	$-0.645187\pi$
$683$	−23.0225	−0.880932	−0.440466	+	0.897769i	$0.645187\pi$
$684$	0	0
$685$	2.21797	0.0847442
$686$	0	0
$687$	−21.7571	−0.830085
$688$	0	0
$689$	16.8524	0.642025
$690$	0	0
$691$	−21.7543	−0.827571	−0.413785	−	0.910374i	$-0.635794\pi$
$691$	−21.7543	−0.827571	−0.413785	+	0.910374i	$0.635794\pi$
$692$	0	0
$693$	8.18531	0.310934
$694$	0	0
$695$	−39.0034	−1.47948
$696$	0	0
$697$	−22.5288	−0.853341
$698$	0	0
$699$	14.8903	0.563202
$700$	0	0
$701$	30.6194	1.15648	0.578240	−	0.815867i	$-0.303740\pi$
$701$	30.6194	1.15648	0.578240	+	0.815867i	$0.303740\pi$
$702$	0	0
$703$	−0.310075	−0.0116947
$704$	0	0
$705$	45.8885	1.72826
$706$	0	0
$707$	2.31251	0.0869707
$708$	0	0
$709$	−38.1556	−1.43296	−0.716482	−	0.697605i	$-0.754249\pi$
$709$	−38.1556	−1.43296	−0.716482	+	0.697605i	$0.754249\pi$
$710$	0	0
$711$	21.0203	0.788323
$712$	0	0
$713$	−25.8306	−0.967365
$714$	0	0
$715$	−79.9317	−2.98928
$716$	0	0
$717$	11.4451	0.427424
$718$	0	0
$719$	12.4956	0.466006	0.233003	−	0.972476i	$-0.425145\pi$
$719$	12.4956	0.466006	0.233003	+	0.972476i	$0.425145\pi$
$720$	0	0
$721$	−0.815407	−0.0303674
$722$	0	0
$723$	31.5278	1.17253
$724$	0	0
$725$	38.6839	1.43669
$726$	0	0
$727$	−35.5447	−1.31828	−0.659140	−	0.752020i	$-0.729080\pi$
$727$	−35.5447	−1.31828	−0.659140	+	0.752020i	$0.729080\pi$
$728$	0	0
$729$	21.1924	0.784904
$730$	0	0
$731$	−38.3700	−1.41917
$732$	0	0
$733$	−35.0084	−1.29306	−0.646531	−	0.762887i	$-0.723781\pi$
$733$	−35.0084	−1.29306	−0.646531	+	0.762887i	$0.723781\pi$
$734$	0	0
$735$	21.6704	0.799326
$736$	0	0
$737$	36.0250	1.32700
$738$	0	0
$739$	−50.7286	−1.86608	−0.933041	−	0.359770i	$-0.882855\pi$
$739$	−50.7286	−1.86608	−0.933041	+	0.359770i	$0.882855\pi$
$740$	0	0
$741$	−36.4592	−1.33936
$742$	0	0
$743$	−28.4073	−1.04216	−0.521082	−	0.853507i	$-0.674471\pi$
$743$	−28.4073	−1.04216	−0.521082	+	0.853507i	$0.674471\pi$
$744$	0	0
$745$	−64.8526	−2.37602
$746$	0	0
$747$	−10.0209	−0.366647
$748$	0	0
$749$	9.00482	0.329029
$750$	0	0
$751$	22.4979	0.820960	0.410480	−	0.911870i	$-0.365361\pi$
$751$	22.4979	0.820960	0.410480	+	0.911870i	$0.365361\pi$
$752$	0	0
$753$	25.5320	0.930437
$754$	0	0
$755$	−56.5346	−2.05750
$756$	0	0
$757$	−9.04087	−0.328596	−0.164298	−	0.986411i	$-0.552536\pi$
$757$	−9.04087	−0.328596	−0.164298	+	0.986411i	$0.552536\pi$
$758$	0	0
$759$	26.2074	0.951268
$760$	0	0
$761$	−13.3889	−0.485347	−0.242674	−	0.970108i	$-0.578024\pi$
$761$	−13.3889	−0.485347	−0.242674	+	0.970108i	$0.578024\pi$
$762$	0	0
$763$	−10.0200	−0.362748
$764$	0	0
$765$	−23.5222	−0.850448
$766$	0	0
$767$	31.9056	1.15204
$768$	0	0
$769$	24.1417	0.870572	0.435286	−	0.900292i	$-0.356647\pi$
$769$	24.1417	0.870572	0.435286	+	0.900292i	$0.356647\pi$
$770$	0	0
$771$	1.65926	0.0597568
$772$	0	0
$773$	0.580377	0.0208747	0.0104374	−	0.999946i	$-0.496678\pi$
$773$	0.580377	0.0208747	0.0104374	+	0.999946i	$0.496678\pi$
$774$	0	0
$775$	−25.8805	−0.929654
$776$	0	0
$777$	−0.0544905	−0.00195483
$778$	0	0
$779$	−30.0550	−1.07683
$780$	0	0
$781$	−3.72597	−0.133326
$782$	0	0
$783$	−48.9636	−1.74982
$784$	0	0
$785$	−22.4818	−0.802409
$786$	0	0
$787$	−41.3871	−1.47529	−0.737645	−	0.675189i	$-0.764062\pi$
$787$	−41.3871	−1.47529	−0.737645	+	0.675189i	$0.764062\pi$
$788$	0	0
$789$	−24.9722	−0.889035
$790$	0	0
$791$	3.25937	0.115890
$792$	0	0
$793$	−65.2085	−2.31562
$794$	0	0
$795$	−12.0119	−0.426018
$796$	0	0
$797$	53.2368	1.88575	0.942873	−	0.333154i	$-0.108113\pi$
$797$	53.2368	1.88575	0.942873	+	0.333154i	$0.108113\pi$
$798$	0	0
$799$	60.8380	2.15229
$800$	0	0
$801$	−12.6441	−0.446757
$802$	0	0
$803$	49.0550	1.73111
$804$	0	0
$805$	12.2979	0.433443
$806$	0	0
$807$	33.0777	1.16439
$808$	0	0
$809$	−24.9831	−0.878358	−0.439179	−	0.898400i	$-0.644731\pi$
$809$	−24.9831	−0.878358	−0.439179	+	0.898400i	$0.644731\pi$
$810$	0	0
$811$	−9.60801	−0.337383	−0.168691	−	0.985669i	$-0.553954\pi$
$811$	−9.60801	−0.337383	−0.168691	+	0.985669i	$0.553954\pi$
$812$	0	0
$813$	13.9048	0.487662
$814$	0	0
$815$	−26.1091	−0.914561
$816$	0	0
$817$	−51.1882	−1.79085
$818$	0	0
$819$	7.73676	0.270344
$820$	0	0
$821$	30.3206	1.05820	0.529098	−	0.848561i	$-0.322531\pi$
$821$	30.3206	1.05820	0.529098	+	0.848561i	$0.322531\pi$
$822$	0	0
$823$	−19.7390	−0.688060	−0.344030	−	0.938959i	$-0.611792\pi$
$823$	−19.7390	−0.688060	−0.344030	+	0.938959i	$0.611792\pi$
$824$	0	0
$825$	26.2579	0.914183
$826$	0	0
$827$	−23.3164	−0.810792	−0.405396	−	0.914141i	$-0.632866\pi$
$827$	−23.3164	−0.810792	−0.405396	+	0.914141i	$0.632866\pi$
$828$	0	0
$829$	−6.98350	−0.242547	−0.121274	−	0.992619i	$-0.538698\pi$
$829$	−6.98350	−0.242547	−0.121274	+	0.992619i	$0.538698\pi$
$830$	0	0
$831$	5.23072	0.181452
$832$	0	0
$833$	28.7302	0.995443
$834$	0	0
$835$	75.1934	2.60217
$836$	0	0
$837$	32.7578	1.13228
$838$	0	0
$839$	−26.3200	−0.908666	−0.454333	−	0.890832i	$-0.650122\pi$
$839$	−26.3200	−0.908666	−0.454333	+	0.890832i	$0.650122\pi$
$840$	0	0
$841$	52.9043	1.82429
$842$	0	0
$843$	16.6187	0.572377
$844$	0	0
$845$	−35.9615	−1.23711
$846$	0	0
$847$	−15.8723	−0.545378
$848$	0	0
$849$	5.13808	0.176338
$850$	0	0
$851$	0.210672	0.00722175
$852$	0	0
$853$	3.58233	0.122657	0.0613283	−	0.998118i	$-0.480466\pi$
$853$	3.58233	0.122657	0.0613283	+	0.998118i	$0.480466\pi$
$854$	0	0
$855$	−31.3802	−1.07318
$856$	0	0
$857$	−4.74136	−0.161962	−0.0809810	−	0.996716i	$-0.525805\pi$
$857$	−4.74136	−0.161962	−0.0809810	+	0.996716i	$0.525805\pi$
$858$	0	0
$859$	−19.9131	−0.679425	−0.339713	−	0.940529i	$-0.610330\pi$
$859$	−19.9131	−0.679425	−0.339713	+	0.940529i	$0.610330\pi$
$860$	0	0
$861$	−5.28166	−0.179999
$862$	0	0
$863$	−26.7053	−0.909059	−0.454529	−	0.890732i	$-0.650193\pi$
$863$	−26.7053	−0.909059	−0.454529	+	0.890732i	$0.650193\pi$
$864$	0	0
$865$	17.3533	0.590030
$866$	0	0
$867$	6.00784	0.204037
$868$	0	0
$869$	−67.4998	−2.28977
$870$	0	0
$871$	34.0508	1.15377
$872$	0	0
$873$	−12.1924	−0.412650
$874$	0	0
$875$	−2.09158	−0.0707084
$876$	0	0
$877$	51.9905	1.75559	0.877796	−	0.479034i	$-0.159013\pi$
$877$	51.9905	1.75559	0.877796	+	0.479034i	$0.159013\pi$
$878$	0	0
$879$	−28.6897	−0.967681
$880$	0	0
$881$	22.5537	0.759855	0.379927	−	0.925016i	$-0.375949\pi$
$881$	22.5537	0.759855	0.379927	+	0.925016i	$0.375949\pi$
$882$	0	0
$883$	19.0973	0.642674	0.321337	−	0.946965i	$-0.395868\pi$
$883$	19.0973	0.642674	0.321337	+	0.946965i	$0.395868\pi$
$884$	0	0
$885$	−22.7414	−0.764444
$886$	0	0
$887$	21.4067	0.718768	0.359384	−	0.933190i	$-0.382987\pi$
$887$	21.4067	0.718768	0.359384	+	0.933190i	$0.382987\pi$
$888$	0	0
$889$	5.04103	0.169071
$890$	0	0
$891$	−7.29322	−0.244332
$892$	0	0
$893$	81.1620	2.71598
$894$	0	0
$895$	20.2341	0.676351
$896$	0	0
$897$	24.7712	0.827088
$898$	0	0
$899$	−54.7959	−1.82755
$900$	0	0
$901$	−15.9251	−0.530543
$902$	0	0
$903$	−8.99546	−0.299350
$904$	0	0
$905$	3.76834	0.125264
$906$	0	0
$907$	−6.26775	−0.208117	−0.104059	−	0.994571i	$-0.533183\pi$
$907$	−6.26775	−0.208117	−0.104059	+	0.994571i	$0.533183\pi$
$908$	0	0
$909$	4.00912	0.132974
$910$	0	0
$911$	8.04040	0.266390	0.133195	−	0.991090i	$-0.457476\pi$
$911$	8.04040	0.266390	0.133195	+	0.991090i	$0.457476\pi$
$912$	0	0
$913$	32.1789	1.06497
$914$	0	0
$915$	46.4788	1.53654
$916$	0	0
$917$	−5.97217	−0.197218
$918$	0	0
$919$	30.5624	1.00816	0.504080	−	0.863657i	$-0.331832\pi$
$919$	30.5624	1.00816	0.504080	+	0.863657i	$0.331832\pi$
$920$	0	0
$921$	24.1544	0.795914
$922$	0	0
$923$	−3.52179	−0.115921
$924$	0	0
$925$	0.211078	0.00694022
$926$	0	0
$927$	−1.41365	−0.0464302
$928$	0	0
$929$	−14.6661	−0.481180	−0.240590	−	0.970627i	$-0.577341\pi$
$929$	−14.6661	−0.481180	−0.240590	+	0.970627i	$0.577341\pi$
$930$	0	0
$931$	38.3281	1.25615
$932$	0	0
$933$	22.3612	0.732074
$934$	0	0
$935$	75.5338	2.47022
$936$	0	0
$937$	39.3587	1.28579	0.642897	−	0.765953i	$-0.277732\pi$
$937$	39.3587	1.28579	0.642897	+	0.765953i	$0.277732\pi$
$938$	0	0
$939$	20.7800	0.678131
$940$	0	0
$941$	−35.4002	−1.15401	−0.577006	−	0.816740i	$-0.695779\pi$
$941$	−35.4002	−1.15401	−0.577006	+	0.816740i	$0.695779\pi$
$942$	0	0
$943$	20.4201	0.664969
$944$	0	0
$945$	−15.5959	−0.507334
$946$	0	0
$947$	−37.1198	−1.20623	−0.603116	−	0.797654i	$-0.706074\pi$
$947$	−37.1198	−1.20623	−0.603116	+	0.797654i	$0.706074\pi$
$948$	0	0
$949$	46.3668	1.50513
$950$	0	0
$951$	20.9641	0.679806
$952$	0	0
$953$	17.0538	0.552425	0.276213	−	0.961097i	$-0.410921\pi$
$953$	17.0538	0.552425	0.276213	+	0.961097i	$0.410921\pi$
$954$	0	0
$955$	−5.43596	−0.175904
$956$	0	0
$957$	55.5951	1.79713
$958$	0	0
$959$	0.689379	0.0222612
$960$	0	0
$961$	5.65976	0.182573
$962$	0	0
$963$	15.6114	0.503070
$964$	0	0
$965$	−65.4373	−2.10650
$966$	0	0
$967$	28.0236	0.901178	0.450589	−	0.892732i	$-0.351214\pi$
$967$	28.0236	0.901178	0.450589	+	0.892732i	$0.351214\pi$
$968$	0	0
$969$	34.4532	1.10680
$970$	0	0
$971$	24.9706	0.801345	0.400672	−	0.916221i	$-0.368777\pi$
$971$	24.9706	0.801345	0.400672	+	0.916221i	$0.368777\pi$
$972$	0	0
$973$	−12.1229	−0.388641
$974$	0	0
$975$	24.8190	0.794845
$976$	0	0
$977$	−43.3293	−1.38623	−0.693114	−	0.720828i	$-0.743762\pi$
$977$	−43.3293	−1.38623	−0.693114	+	0.720828i	$0.743762\pi$
$978$	0	0
$979$	40.6023	1.29766
$980$	0	0
$981$	−17.3713	−0.554624
$982$	0	0
$983$	−1.82326	−0.0581530	−0.0290765	−	0.999577i	$-0.509257\pi$
$983$	−1.82326	−0.0581530	−0.0290765	+	0.999577i	$0.509257\pi$
$984$	0	0
$985$	−56.1874	−1.79028
$986$	0	0
$987$	14.2629	0.453992
$988$	0	0
$989$	34.7784	1.10589
$990$	0	0
$991$	−4.34705	−0.138089	−0.0690443	−	0.997614i	$-0.521995\pi$
$991$	−4.34705	−0.138089	−0.0690443	+	0.997614i	$0.521995\pi$
$992$	0	0
$993$	16.3207	0.517922
$994$	0	0
$995$	71.8031	2.27631
$996$	0	0
$997$	−7.66217	−0.242663	−0.121332	−	0.992612i	$-0.538716\pi$
$997$	−7.66217	−0.242663	−0.121332	+	0.992612i	$0.538716\pi$
$998$	0	0
$999$	−0.267170	−0.00845287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.19		26
4.3	odd	2		503.2.a.f.1.11	✓	26
12.11	even	2		4527.2.a.o.1.16		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.11	✓	26	4.3	odd	2
4527.2.a.o.1.16		26	12.11	even	2
8048.2.a.u.1.19		26	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)

\(f(q)\)	\(=\)	\(q+1.16576 q^{3} -3.04539 q^{5} -0.946556 q^{7} -1.64102 q^{9} +O(q^{10})\)	\(q+1.16576 q^{3} -3.04539 q^{5} -0.946556 q^{7} -1.64102 q^{9} +5.26958 q^{11} +4.98081 q^{13} -3.55018 q^{15} -4.70676 q^{17} -6.27914 q^{19} -1.10345 q^{21} +4.26619 q^{23} +4.27442 q^{25} -5.41029 q^{27} +9.05010 q^{29} -6.05473 q^{31} +6.14304 q^{33} +2.88264 q^{35} +0.0493818 q^{37} +5.80641 q^{39} +4.78649 q^{41} +8.15210 q^{43} +4.99754 q^{45} -12.9257 q^{47} -6.10403 q^{49} -5.48693 q^{51} +3.38346 q^{53} -16.0479 q^{55} -7.31994 q^{57} +6.40570 q^{59} -13.0919 q^{61} +1.55331 q^{63} -15.1685 q^{65} +6.83640 q^{67} +4.97334 q^{69} -0.707071 q^{71} +9.30909 q^{73} +4.98293 q^{75} -4.98795 q^{77} -12.8093 q^{79} -1.38402 q^{81} +6.10655 q^{83} +14.3339 q^{85} +10.5502 q^{87} +7.70505 q^{89} -4.71462 q^{91} -7.05833 q^{93} +19.1225 q^{95} +7.42978 q^{97} -8.64746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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