LMFDB - Embedded newform 8048.2.a.x.1.1

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			1.1
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)

$f(q)$	$=$	$q-3.42195 q^{3} -3.41581 q^{5} -3.07051 q^{7} +8.70973 q^{9} +O(q^{10})$	$q-3.42195 q^{3} -3.41581 q^{5} -3.07051 q^{7} +8.70973 q^{9} -2.67665 q^{11} -1.00510 q^{13} +11.6887 q^{15} -2.51095 q^{17} -1.80553 q^{19} +10.5071 q^{21} -8.73515 q^{23} +6.66773 q^{25} -19.5384 q^{27} -3.26621 q^{29} +2.15627 q^{31} +9.15936 q^{33} +10.4883 q^{35} -8.81279 q^{37} +3.43941 q^{39} +10.4527 q^{41} +8.99985 q^{43} -29.7507 q^{45} -7.22504 q^{47} +2.42803 q^{49} +8.59232 q^{51} -2.16950 q^{53} +9.14292 q^{55} +6.17843 q^{57} +11.3320 q^{59} -0.407980 q^{61} -26.7433 q^{63} +3.43324 q^{65} -8.93711 q^{67} +29.8912 q^{69} +5.23271 q^{71} -3.72240 q^{73} -22.8166 q^{75} +8.21868 q^{77} -11.2972 q^{79} +40.7302 q^{81} +0.634642 q^{83} +8.57690 q^{85} +11.1768 q^{87} +12.3370 q^{89} +3.08618 q^{91} -7.37865 q^{93} +6.16734 q^{95} +9.05660 q^{97} -23.3129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.42195	−1.97566	−0.987831	−	0.155529i	$-0.950292\pi$
$3$	−3.42195	−1.97566	−0.987831	+	0.155529i	$0.950292\pi$
$4$	0	0
$5$	−3.41581	−1.52759	−0.763797	−	0.645456i	$-0.776667\pi$
$5$	−3.41581	−1.52759	−0.763797	+	0.645456i	$0.776667\pi$
$6$	0	0
$7$	−3.07051	−1.16054	−0.580272	−	0.814423i	$-0.697054\pi$
$7$	−3.07051	−1.16054	−0.580272	+	0.814423i	$0.697054\pi$
$8$	0	0
$9$	8.70973	2.90324
$10$	0	0
$11$	−2.67665	−0.807040	−0.403520	−	0.914971i	$-0.632213\pi$
$11$	−2.67665	−0.807040	−0.403520	+	0.914971i	$0.632213\pi$
$12$	0	0
$13$	−1.00510	−0.278766	−0.139383	−	0.990239i	$-0.544512\pi$
$13$	−1.00510	−0.278766	−0.139383	+	0.990239i	$0.544512\pi$
$14$	0	0
$15$	11.6887	3.01801
$16$	0	0
$17$	−2.51095	−0.608994	−0.304497	−	0.952513i	$-0.598488\pi$
$17$	−2.51095	−0.608994	−0.304497	+	0.952513i	$0.598488\pi$
$18$	0	0
$19$	−1.80553	−0.414217	−0.207108	−	0.978318i	$-0.566405\pi$
$19$	−1.80553	−0.414217	−0.207108	+	0.978318i	$0.566405\pi$
$20$	0	0
$21$	10.5071	2.29284
$22$	0	0
$23$	−8.73515	−1.82141	−0.910703	−	0.413063i	$-0.864459\pi$
$23$	−8.73515	−1.82141	−0.910703	+	0.413063i	$0.864459\pi$
$24$	0	0
$25$	6.66773	1.33355
$26$	0	0
$27$	−19.5384	−3.76016
$28$	0	0
$29$	−3.26621	−0.606519	−0.303260	−	0.952908i	$-0.598075\pi$
$29$	−3.26621	−0.606519	−0.303260	+	0.952908i	$0.598075\pi$
$30$	0	0
$31$	2.15627	0.387278	0.193639	−	0.981073i	$-0.437971\pi$
$31$	2.15627	0.387278	0.193639	+	0.981073i	$0.437971\pi$
$32$	0	0
$33$	9.15936	1.59444
$34$	0	0
$35$	10.4883	1.77284
$36$	0	0
$37$	−8.81279	−1.44881	−0.724407	−	0.689373i	$-0.757886\pi$
$37$	−8.81279	−1.44881	−0.724407	+	0.689373i	$0.757886\pi$
$38$	0	0
$39$	3.43941	0.550747
$40$	0	0
$41$	10.4527	1.63244	0.816219	−	0.577742i	$-0.196066\pi$
$41$	10.4527	1.63244	0.816219	+	0.577742i	$0.196066\pi$
$42$	0	0
$43$	8.99985	1.37246	0.686232	−	0.727383i	$-0.259263\pi$
$43$	8.99985	1.37246	0.686232	+	0.727383i	$0.259263\pi$
$44$	0	0
$45$	−29.7507	−4.43498
$46$	0	0
$47$	−7.22504	−1.05388	−0.526940	−	0.849903i	$-0.676661\pi$
$47$	−7.22504	−1.05388	−0.526940	+	0.849903i	$0.676661\pi$
$48$	0	0
$49$	2.42803	0.346862
$50$	0	0
$51$	8.59232	1.20317
$52$	0	0
$53$	−2.16950	−0.298004	−0.149002	−	0.988837i	$-0.547606\pi$
$53$	−2.16950	−0.298004	−0.149002	+	0.988837i	$0.547606\pi$
$54$	0	0
$55$	9.14292	1.23283
$56$	0	0
$57$	6.17843	0.818353
$58$	0	0
$59$	11.3320	1.47530	0.737652	−	0.675182i	$-0.235935\pi$
$59$	11.3320	1.47530	0.737652	+	0.675182i	$0.235935\pi$
$60$	0	0
$61$	−0.407980	−0.0522365	−0.0261183	−	0.999659i	$-0.508315\pi$
$61$	−0.407980	−0.0522365	−0.0261183	+	0.999659i	$0.508315\pi$
$62$	0	0
$63$	−26.7433	−3.36934
$64$	0	0
$65$	3.43324	0.425841
$66$	0	0
$67$	−8.93711	−1.09184	−0.545921	−	0.837837i	$-0.683820\pi$
$67$	−8.93711	−1.09184	−0.545921	+	0.837837i	$0.683820\pi$
$68$	0	0
$69$	29.8912	3.59848
$70$	0	0
$71$	5.23271	0.621008	0.310504	−	0.950572i	$-0.399502\pi$
$71$	5.23271	0.621008	0.310504	+	0.950572i	$0.399502\pi$
$72$	0	0
$73$	−3.72240	−0.435674	−0.217837	−	0.975985i	$-0.569900\pi$
$73$	−3.72240	−0.435674	−0.217837	+	0.975985i	$0.569900\pi$
$74$	0	0
$75$	−22.8166	−2.63464
$76$	0	0
$77$	8.21868	0.936606
$78$	0	0
$79$	−11.2972	−1.27104	−0.635519	−	0.772085i	$-0.719214\pi$
$79$	−11.2972	−1.27104	−0.635519	+	0.772085i	$0.719214\pi$
$80$	0	0
$81$	40.7302	4.52557
$82$	0	0
$83$	0.634642	0.0696610	0.0348305	−	0.999393i	$-0.488911\pi$
$83$	0.634642	0.0696610	0.0348305	+	0.999393i	$0.488911\pi$
$84$	0	0
$85$	8.57690	0.930296
$86$	0	0
$87$	11.1768	1.19828
$88$	0	0
$89$	12.3370	1.30771	0.653857	−	0.756618i	$-0.273150\pi$
$89$	12.3370	1.30771	0.653857	+	0.756618i	$0.273150\pi$
$90$	0	0
$91$	3.08618	0.323520
$92$	0	0
$93$	−7.37865	−0.765131
$94$	0	0
$95$	6.16734	0.632756
$96$	0	0
$97$	9.05660	0.919558	0.459779	−	0.888033i	$-0.347929\pi$
$97$	9.05660	0.919558	0.459779	+	0.888033i	$0.347929\pi$
$98$	0	0
$99$	−23.3129	−2.34303
$100$	0	0
$101$	7.72547	0.768713	0.384357	−	0.923185i	$-0.374423\pi$
$101$	7.72547	0.768713	0.384357	+	0.923185i	$0.374423\pi$
$102$	0	0
$103$	14.7011	1.44854	0.724269	−	0.689517i	$-0.242177\pi$
$103$	14.7011	1.44854	0.724269	+	0.689517i	$0.242177\pi$
$104$	0	0
$105$	−35.8903	−3.50253
$106$	0	0
$107$	−5.62129	−0.543431	−0.271715	−	0.962378i	$-0.587591\pi$
$107$	−5.62129	−0.543431	−0.271715	+	0.962378i	$0.587591\pi$
$108$	0	0
$109$	−8.68750	−0.832112	−0.416056	−	0.909339i	$-0.636588\pi$
$109$	−8.68750	−0.832112	−0.416056	+	0.909339i	$0.636588\pi$
$110$	0	0
$111$	30.1569	2.86237
$112$	0	0
$113$	18.5238	1.74258	0.871288	−	0.490772i	$-0.163285\pi$
$113$	18.5238	1.74258	0.871288	+	0.490772i	$0.163285\pi$
$114$	0	0
$115$	29.8376	2.78237
$116$	0	0
$117$	−8.75418	−0.809324
$118$	0	0
$119$	7.70988	0.706764
$120$	0	0
$121$	−3.83555	−0.348686
$122$	0	0
$123$	−35.7686	−3.22515
$124$	0	0
$125$	−5.69664	−0.509523
$126$	0	0
$127$	5.90920	0.524357	0.262178	−	0.965019i	$-0.415559\pi$
$127$	5.90920	0.524357	0.262178	+	0.965019i	$0.415559\pi$
$128$	0	0
$129$	−30.7970	−2.71153
$130$	0	0
$131$	14.2078	1.24134	0.620671	−	0.784071i	$-0.286860\pi$
$131$	14.2078	1.24134	0.620671	+	0.784071i	$0.286860\pi$
$132$	0	0
$133$	5.54390	0.480717
$134$	0	0
$135$	66.7393	5.74401
$136$	0	0
$137$	9.03947	0.772294	0.386147	−	0.922437i	$-0.373806\pi$
$137$	9.03947	0.772294	0.386147	+	0.922437i	$0.373806\pi$
$138$	0	0
$139$	−14.0619	−1.19271	−0.596357	−	0.802719i	$-0.703386\pi$
$139$	−14.0619	−1.19271	−0.596357	+	0.802719i	$0.703386\pi$
$140$	0	0
$141$	24.7237	2.08211
$142$	0	0
$143$	2.69031	0.224975
$144$	0	0
$145$	11.1567	0.926516
$146$	0	0
$147$	−8.30860	−0.685282
$148$	0	0
$149$	4.17650	0.342153	0.171076	−	0.985258i	$-0.445276\pi$
$149$	4.17650	0.342153	0.171076	+	0.985258i	$0.445276\pi$
$150$	0	0
$151$	0.896921	0.0729904	0.0364952	−	0.999334i	$-0.488381\pi$
$151$	0.896921	0.0729904	0.0364952	+	0.999334i	$0.488381\pi$
$152$	0	0
$153$	−21.8697	−1.76806
$154$	0	0
$155$	−7.36541	−0.591604
$156$	0	0
$157$	5.38200	0.429531	0.214765	−	0.976666i	$-0.431101\pi$
$157$	5.38200	0.429531	0.214765	+	0.976666i	$0.431101\pi$
$158$	0	0
$159$	7.42392	0.588755
$160$	0	0
$161$	26.8214	2.11382
$162$	0	0
$163$	−6.75137	−0.528808	−0.264404	−	0.964412i	$-0.585175\pi$
$163$	−6.75137	−0.528808	−0.264404	+	0.964412i	$0.585175\pi$
$164$	0	0
$165$	−31.2866	−2.43566
$166$	0	0
$167$	7.38004	0.571084	0.285542	−	0.958366i	$-0.407826\pi$
$167$	7.38004	0.571084	0.285542	+	0.958366i	$0.407826\pi$
$168$	0	0
$169$	−11.9898	−0.922290
$170$	0	0
$171$	−15.7257	−1.20257
$172$	0	0
$173$	−0.804872	−0.0611933	−0.0305966	−	0.999532i	$-0.509741\pi$
$173$	−0.804872	−0.0611933	−0.0305966	+	0.999532i	$0.509741\pi$
$174$	0	0
$175$	−20.4733	−1.54764
$176$	0	0
$177$	−38.7776	−2.91470
$178$	0	0
$179$	4.10674	0.306952	0.153476	−	0.988152i	$-0.450953\pi$
$179$	4.10674	0.306952	0.153476	+	0.988152i	$0.450953\pi$
$180$	0	0
$181$	−13.9327	−1.03561	−0.517804	−	0.855499i	$-0.673250\pi$
$181$	−13.9327	−1.03561	−0.517804	+	0.855499i	$0.673250\pi$
$182$	0	0
$183$	1.39609	0.103202
$184$	0	0
$185$	30.1028	2.21320
$186$	0	0
$187$	6.72092	0.491482
$188$	0	0
$189$	59.9928	4.36384
$190$	0	0
$191$	13.7247	0.993085	0.496543	−	0.868012i	$-0.334603\pi$
$191$	13.7247	0.993085	0.496543	+	0.868012i	$0.334603\pi$
$192$	0	0
$193$	20.1964	1.45377	0.726886	−	0.686758i	$-0.240967\pi$
$193$	20.1964	1.45377	0.726886	+	0.686758i	$0.240967\pi$
$194$	0	0
$195$	−11.7484	−0.841318
$196$	0	0
$197$	13.7715	0.981181	0.490590	−	0.871390i	$-0.336781\pi$
$197$	13.7715	0.981181	0.490590	+	0.871390i	$0.336781\pi$
$198$	0	0
$199$	13.5321	0.959264	0.479632	−	0.877470i	$-0.340770\pi$
$199$	13.5321	0.959264	0.479632	+	0.877470i	$0.340770\pi$
$200$	0	0
$201$	30.5823	2.15711
$202$	0	0
$203$	10.0289	0.703892
$204$	0	0
$205$	−35.7044	−2.49371
$206$	0	0
$207$	−76.0808	−5.28798
$208$	0	0
$209$	4.83277	0.334290
$210$	0	0
$211$	−4.50520	−0.310151	−0.155076	−	0.987903i	$-0.549562\pi$
$211$	−4.50520	−0.310151	−0.155076	+	0.987903i	$0.549562\pi$
$212$	0	0
$213$	−17.9061	−1.22690
$214$	0	0
$215$	−30.7417	−2.09657
$216$	0	0
$217$	−6.62086	−0.449453
$218$	0	0
$219$	12.7379	0.860745
$220$	0	0
$221$	2.52376	0.169767
$222$	0	0
$223$	20.1321	1.34814	0.674071	−	0.738667i	$-0.264544\pi$
$223$	20.1321	1.34814	0.674071	+	0.738667i	$0.264544\pi$
$224$	0	0
$225$	58.0741	3.87161
$226$	0	0
$227$	8.31877	0.552136	0.276068	−	0.961138i	$-0.410968\pi$
$227$	8.31877	0.552136	0.276068	+	0.961138i	$0.410968\pi$
$228$	0	0
$229$	−15.2358	−1.00681	−0.503407	−	0.864050i	$-0.667920\pi$
$229$	−15.2358	−1.00681	−0.503407	+	0.864050i	$0.667920\pi$
$230$	0	0
$231$	−28.1239	−1.85042
$232$	0	0
$233$	−8.51902	−0.558099	−0.279050	−	0.960277i	$-0.590019\pi$
$233$	−8.51902	−0.558099	−0.279050	+	0.960277i	$0.590019\pi$
$234$	0	0
$235$	24.6793	1.60990
$236$	0	0
$237$	38.6585	2.51114
$238$	0	0
$239$	−15.5453	−1.00554	−0.502772	−	0.864419i	$-0.667686\pi$
$239$	−15.5453	−1.00554	−0.502772	+	0.864419i	$0.667686\pi$
$240$	0	0
$241$	16.2228	1.04500	0.522501	−	0.852639i	$-0.324999\pi$
$241$	16.2228	1.04500	0.522501	+	0.852639i	$0.324999\pi$
$242$	0	0
$243$	−80.7614	−5.18084
$244$	0	0
$245$	−8.29369	−0.529864
$246$	0	0
$247$	1.81474	0.115469
$248$	0	0
$249$	−2.17171	−0.137627
$250$	0	0
$251$	23.3481	1.47372	0.736860	−	0.676045i	$-0.236308\pi$
$251$	23.3481	1.47372	0.736860	+	0.676045i	$0.236308\pi$
$252$	0	0
$253$	23.3809	1.46995
$254$	0	0
$255$	−29.3497	−1.83795
$256$	0	0
$257$	−13.7742	−0.859213	−0.429606	−	0.903016i	$-0.641348\pi$
$257$	−13.7742	−0.859213	−0.429606	+	0.903016i	$0.641348\pi$
$258$	0	0
$259$	27.0598	1.68141
$260$	0	0
$261$	−28.4478	−1.76087
$262$	0	0
$263$	−0.502585	−0.0309907	−0.0154953	−	0.999880i	$-0.504933\pi$
$263$	−0.502585	−0.0309907	−0.0154953	+	0.999880i	$0.504933\pi$
$264$	0	0
$265$	7.41059	0.455229
$266$	0	0
$267$	−42.2164	−2.58360
$268$	0	0
$269$	18.7919	1.14576	0.572880	−	0.819639i	$-0.305826\pi$
$269$	18.7919	1.14576	0.572880	+	0.819639i	$0.305826\pi$
$270$	0	0
$271$	−19.5510	−1.18764	−0.593820	−	0.804598i	$-0.702381\pi$
$271$	−19.5510	−1.18764	−0.593820	+	0.804598i	$0.702381\pi$
$272$	0	0
$273$	−10.5608	−0.639166
$274$	0	0
$275$	−17.8472	−1.07623
$276$	0	0
$277$	−20.3361	−1.22188	−0.610940	−	0.791677i	$-0.709208\pi$
$277$	−20.3361	−1.22188	−0.610940	+	0.791677i	$0.709208\pi$
$278$	0	0
$279$	18.7805	1.12436
$280$	0	0
$281$	−22.1007	−1.31842	−0.659210	−	0.751959i	$-0.729109\pi$
$281$	−22.1007	−1.31842	−0.659210	+	0.751959i	$0.729109\pi$
$282$	0	0
$283$	3.22058	0.191444	0.0957219	−	0.995408i	$-0.469484\pi$
$283$	3.22058	0.191444	0.0957219	+	0.995408i	$0.469484\pi$
$284$	0	0
$285$	−21.1043	−1.25011
$286$	0	0
$287$	−32.0952	−1.89452
$288$	0	0
$289$	−10.6952	−0.629127
$290$	0	0
$291$	−30.9912	−1.81674
$292$	0	0
$293$	−5.21662	−0.304758	−0.152379	−	0.988322i	$-0.548693\pi$
$293$	−5.21662	−0.304758	−0.152379	+	0.988322i	$0.548693\pi$
$294$	0	0
$295$	−38.7080	−2.25367
$296$	0	0
$297$	52.2974	3.03460
$298$	0	0
$299$	8.77974	0.507745
$300$	0	0
$301$	−27.6341	−1.59280
$302$	0	0
$303$	−26.4362	−1.51872
$304$	0	0
$305$	1.39358	0.0797962
$306$	0	0
$307$	−26.9224	−1.53654	−0.768272	−	0.640123i	$-0.778883\pi$
$307$	−26.9224	−1.53654	−0.768272	+	0.640123i	$0.778883\pi$
$308$	0	0
$309$	−50.3063	−2.86182
$310$	0	0
$311$	−23.8829	−1.35427	−0.677136	−	0.735858i	$-0.736779\pi$
$311$	−23.8829	−1.35427	−0.677136	+	0.735858i	$0.736779\pi$
$312$	0	0
$313$	−24.4262	−1.38065	−0.690326	−	0.723499i	$-0.742533\pi$
$313$	−24.4262	−1.38065	−0.690326	+	0.723499i	$0.742533\pi$
$314$	0	0
$315$	91.3499	5.14699
$316$	0	0
$317$	−22.3109	−1.25310	−0.626552	−	0.779379i	$-0.715535\pi$
$317$	−22.3109	−1.25310	−0.626552	+	0.779379i	$0.715535\pi$
$318$	0	0
$319$	8.74249	0.489486
$320$	0	0
$321$	19.2358	1.07364
$322$	0	0
$323$	4.53359	0.252255
$324$	0	0
$325$	−6.70176	−0.371747
$326$	0	0
$327$	29.7282	1.64397
$328$	0	0
$329$	22.1845	1.22307
$330$	0	0
$331$	28.6082	1.57245	0.786225	−	0.617940i	$-0.212033\pi$
$331$	28.6082	1.57245	0.786225	+	0.617940i	$0.212033\pi$
$332$	0	0
$333$	−76.7570	−4.20626
$334$	0	0
$335$	30.5274	1.66789
$336$	0	0
$337$	−31.6428	−1.72369	−0.861847	−	0.507168i	$-0.830692\pi$
$337$	−31.6428	−1.72369	−0.861847	+	0.507168i	$0.830692\pi$
$338$	0	0
$339$	−63.3876	−3.44274
$340$	0	0
$341$	−5.77159	−0.312549
$342$	0	0
$343$	14.0383	0.757995
$344$	0	0
$345$	−102.103	−5.49702
$346$	0	0
$347$	−3.19130	−0.171318	−0.0856590	−	0.996325i	$-0.527300\pi$
$347$	−3.19130	−0.171318	−0.0856590	+	0.996325i	$0.527300\pi$
$348$	0	0
$349$	30.9135	1.65476	0.827382	−	0.561640i	$-0.189829\pi$
$349$	30.9135	1.65476	0.827382	+	0.561640i	$0.189829\pi$
$350$	0	0
$351$	19.6381	1.04820
$352$	0	0
$353$	−3.70493	−0.197194	−0.0985968	−	0.995127i	$-0.531435\pi$
$353$	−3.70493	−0.197194	−0.0985968	+	0.995127i	$0.531435\pi$
$354$	0	0
$355$	−17.8739	−0.948649
$356$	0	0
$357$	−26.3828	−1.39633
$358$	0	0
$359$	−16.3500	−0.862921	−0.431460	−	0.902132i	$-0.642002\pi$
$359$	−16.3500	−0.862921	−0.431460	+	0.902132i	$0.642002\pi$
$360$	0	0
$361$	−15.7401	−0.828424
$362$	0	0
$363$	13.1250	0.688886
$364$	0	0
$365$	12.7150	0.665533
$366$	0	0
$367$	28.6885	1.49753	0.748765	−	0.662836i	$-0.230647\pi$
$367$	28.6885	1.49753	0.748765	+	0.662836i	$0.230647\pi$
$368$	0	0
$369$	91.0402	4.73937
$370$	0	0
$371$	6.66148	0.345847
$372$	0	0
$373$	−9.56425	−0.495218	−0.247609	−	0.968860i	$-0.579645\pi$
$373$	−9.56425	−0.495218	−0.247609	+	0.968860i	$0.579645\pi$
$374$	0	0
$375$	19.4936	1.00665
$376$	0	0
$377$	3.28288	0.169077
$378$	0	0
$379$	−10.4967	−0.539180	−0.269590	−	0.962975i	$-0.586888\pi$
$379$	−10.4967	−0.539180	−0.269590	+	0.962975i	$0.586888\pi$
$380$	0	0
$381$	−20.2210	−1.03595
$382$	0	0
$383$	−20.4466	−1.04477	−0.522387	−	0.852708i	$-0.674958\pi$
$383$	−20.4466	−1.04477	−0.522387	+	0.852708i	$0.674958\pi$
$384$	0	0
$385$	−28.0734	−1.43075
$386$	0	0
$387$	78.3862	3.98460
$388$	0	0
$389$	22.7572	1.15384	0.576918	−	0.816802i	$-0.304255\pi$
$389$	22.7572	1.15384	0.576918	+	0.816802i	$0.304255\pi$
$390$	0	0
$391$	21.9335	1.10922
$392$	0	0
$393$	−48.6184	−2.45247
$394$	0	0
$395$	38.5892	1.94163
$396$	0	0
$397$	−22.6605	−1.13730	−0.568648	−	0.822581i	$-0.692533\pi$
$397$	−22.6605	−1.13730	−0.568648	+	0.822581i	$0.692533\pi$
$398$	0	0
$399$	−18.9709	−0.949734
$400$	0	0
$401$	34.2486	1.71030	0.855148	−	0.518384i	$-0.173466\pi$
$401$	34.2486	1.71030	0.855148	+	0.518384i	$0.173466\pi$
$402$	0	0
$403$	−2.16728	−0.107960
$404$	0	0
$405$	−139.126	−6.91324
$406$	0	0
$407$	23.5888	1.16925
$408$	0	0
$409$	−38.2828	−1.89296	−0.946480	−	0.322762i	$-0.895389\pi$
$409$	−38.2828	−1.89296	−0.946480	+	0.322762i	$0.895389\pi$
$410$	0	0
$411$	−30.9326	−1.52579
$412$	0	0
$413$	−34.7951	−1.71215
$414$	0	0
$415$	−2.16781	−0.106414
$416$	0	0
$417$	48.1191	2.35640
$418$	0	0
$419$	7.80948	0.381518	0.190759	−	0.981637i	$-0.438905\pi$
$419$	7.80948	0.381518	0.190759	+	0.981637i	$0.438905\pi$
$420$	0	0
$421$	−22.0631	−1.07529	−0.537645	−	0.843171i	$-0.680686\pi$
$421$	−22.0631	−1.07529	−0.537645	+	0.843171i	$0.680686\pi$
$422$	0	0
$423$	−62.9281	−3.05967
$424$	0	0
$425$	−16.7423	−0.812121
$426$	0	0
$427$	1.25271	0.0606228
$428$	0	0
$429$	−9.20610	−0.444475
$430$	0	0
$431$	30.8865	1.48775	0.743876	−	0.668317i	$-0.232985\pi$
$431$	30.8865	1.48775	0.743876	+	0.668317i	$0.232985\pi$
$432$	0	0
$433$	15.2663	0.733653	0.366827	−	0.930289i	$-0.380444\pi$
$433$	15.2663	0.733653	0.366827	+	0.930289i	$0.380444\pi$
$434$	0	0
$435$	−38.1777	−1.83048
$436$	0	0
$437$	15.7716	0.754457
$438$	0	0
$439$	28.9330	1.38090	0.690449	−	0.723381i	$-0.257413\pi$
$439$	28.9330	1.38090	0.690449	+	0.723381i	$0.257413\pi$
$440$	0	0
$441$	21.1475	1.00702
$442$	0	0
$443$	16.8290	0.799572	0.399786	−	0.916609i	$-0.369084\pi$
$443$	16.8290	0.799572	0.399786	+	0.916609i	$0.369084\pi$
$444$	0	0
$445$	−42.1406	−1.99766
$446$	0	0
$447$	−14.2918	−0.675978
$448$	0	0
$449$	−25.0288	−1.18118	−0.590592	−	0.806970i	$-0.701106\pi$
$449$	−25.0288	−1.18118	−0.590592	+	0.806970i	$0.701106\pi$
$450$	0	0
$451$	−27.9782	−1.31744
$452$	0	0
$453$	−3.06922	−0.144204
$454$	0	0
$455$	−10.5418	−0.494207
$456$	0	0
$457$	26.6606	1.24713	0.623565	−	0.781772i	$-0.285684\pi$
$457$	26.6606	1.24713	0.623565	+	0.781772i	$0.285684\pi$
$458$	0	0
$459$	49.0598	2.28992
$460$	0	0
$461$	1.29306	0.0602239	0.0301119	−	0.999547i	$-0.490414\pi$
$461$	1.29306	0.0602239	0.0301119	+	0.999547i	$0.490414\pi$
$462$	0	0
$463$	−31.6495	−1.47088	−0.735439	−	0.677591i	$-0.763024\pi$
$463$	−31.6495	−1.47088	−0.735439	+	0.677591i	$0.763024\pi$
$464$	0	0
$465$	25.2040	1.16881
$466$	0	0
$467$	35.9574	1.66391	0.831955	−	0.554844i	$-0.187222\pi$
$467$	35.9574	1.66391	0.831955	+	0.554844i	$0.187222\pi$
$468$	0	0
$469$	27.4415	1.26713
$470$	0	0
$471$	−18.4169	−0.848608
$472$	0	0
$473$	−24.0894	−1.10763
$474$	0	0
$475$	−12.0388	−0.552377
$476$	0	0
$477$	−18.8958	−0.865178
$478$	0	0
$479$	−1.37045	−0.0626176	−0.0313088	−	0.999510i	$-0.509968\pi$
$479$	−1.37045	−0.0626176	−0.0313088	+	0.999510i	$0.509968\pi$
$480$	0	0
$481$	8.85777	0.403879
$482$	0	0
$483$	−91.7814	−4.17620
$484$	0	0
$485$	−30.9356	−1.40471
$486$	0	0
$487$	−10.5188	−0.476653	−0.238327	−	0.971185i	$-0.576599\pi$
$487$	−10.5188	−0.476653	−0.238327	+	0.971185i	$0.576599\pi$
$488$	0	0
$489$	23.1028	1.04475
$490$	0	0
$491$	−28.2028	−1.27278	−0.636388	−	0.771369i	$-0.719572\pi$
$491$	−28.2028	−1.27278	−0.636388	+	0.771369i	$0.719572\pi$
$492$	0	0
$493$	8.20127	0.369366
$494$	0	0
$495$	79.6323	3.57921
$496$	0	0
$497$	−16.0671	−0.720707
$498$	0	0
$499$	−7.37872	−0.330317	−0.165159	−	0.986267i	$-0.552814\pi$
$499$	−7.37872	−0.330317	−0.165159	+	0.986267i	$0.552814\pi$
$500$	0	0
$501$	−25.2541	−1.12827
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−26.3887	−1.17428
$506$	0	0
$507$	41.0284	1.82213
$508$	0	0
$509$	13.4012	0.593998	0.296999	−	0.954878i	$-0.404014\pi$
$509$	13.4012	0.593998	0.296999	+	0.954878i	$0.404014\pi$
$510$	0	0
$511$	11.4297	0.505619
$512$	0	0
$513$	35.2771	1.55752
$514$	0	0
$515$	−50.2160	−2.21278
$516$	0	0
$517$	19.3389	0.850523
$518$	0	0
$519$	2.75423	0.120897
$520$	0	0
$521$	−3.00933	−0.131841	−0.0659206	−	0.997825i	$-0.520998\pi$
$521$	−3.00933	−0.131841	−0.0659206	+	0.997825i	$0.520998\pi$
$522$	0	0
$523$	−4.48747	−0.196223	−0.0981116	−	0.995175i	$-0.531280\pi$
$523$	−4.48747	−0.196223	−0.0981116	+	0.995175i	$0.531280\pi$
$524$	0	0
$525$	70.0587	3.05761
$526$	0	0
$527$	−5.41428	−0.235850
$528$	0	0
$529$	53.3029	2.31752
$530$	0	0
$531$	98.6988	4.28316
$532$	0	0
$533$	−10.5061	−0.455068
$534$	0	0
$535$	19.2012	0.830142
$536$	0	0
$537$	−14.0531	−0.606434
$538$	0	0
$539$	−6.49900	−0.279932
$540$	0	0
$541$	33.1475	1.42512	0.712560	−	0.701611i	$-0.247536\pi$
$541$	33.1475	1.42512	0.712560	+	0.701611i	$0.247536\pi$
$542$	0	0
$543$	47.6769	2.04601
$544$	0	0
$545$	29.6748	1.27113
$546$	0	0
$547$	36.5726	1.56373	0.781866	−	0.623446i	$-0.214268\pi$
$547$	36.5726	1.56373	0.781866	+	0.623446i	$0.214268\pi$
$548$	0	0
$549$	−3.55340	−0.151655
$550$	0	0
$551$	5.89723	0.251231
$552$	0	0
$553$	34.6883	1.47510
$554$	0	0
$555$	−103.010	−4.37254
$556$	0	0
$557$	−2.82286	−0.119608	−0.0598041	−	0.998210i	$-0.519048\pi$
$557$	−2.82286	−0.119608	−0.0598041	+	0.998210i	$0.519048\pi$
$558$	0	0
$559$	−9.04578	−0.382596
$560$	0	0
$561$	−22.9986	−0.971004
$562$	0	0
$563$	37.9443	1.59916	0.799580	−	0.600560i	$-0.205055\pi$
$563$	37.9443	1.59916	0.799580	+	0.600560i	$0.205055\pi$
$564$	0	0
$565$	−63.2738	−2.66195
$566$	0	0
$567$	−125.062	−5.25213
$568$	0	0
$569$	7.79256	0.326681	0.163340	−	0.986570i	$-0.447773\pi$
$569$	7.79256	0.326681	0.163340	+	0.986570i	$0.447773\pi$
$570$	0	0
$571$	10.5833	0.442896	0.221448	−	0.975172i	$-0.428922\pi$
$571$	10.5833	0.442896	0.221448	+	0.975172i	$0.428922\pi$
$572$	0	0
$573$	−46.9652	−1.96200
$574$	0	0
$575$	−58.2436	−2.42893
$576$	0	0
$577$	24.2222	1.00838	0.504191	−	0.863592i	$-0.331791\pi$
$577$	24.2222	1.00838	0.504191	+	0.863592i	$0.331791\pi$
$578$	0	0
$579$	−69.1112	−2.87216
$580$	0	0
$581$	−1.94867	−0.0808446
$582$	0	0
$583$	5.80699	0.240501
$584$	0	0
$585$	29.9026	1.23632
$586$	0	0
$587$	−30.9609	−1.27789	−0.638946	−	0.769252i	$-0.720629\pi$
$587$	−30.9609	−1.27789	−0.638946	+	0.769252i	$0.720629\pi$
$588$	0	0
$589$	−3.89321	−0.160417
$590$	0	0
$591$	−47.1255	−1.93848
$592$	0	0
$593$	−46.7246	−1.91875	−0.959375	−	0.282135i	$-0.908957\pi$
$593$	−46.7246	−1.91875	−0.959375	+	0.282135i	$0.908957\pi$
$594$	0	0
$595$	−26.3355	−1.07965
$596$	0	0
$597$	−46.3061	−1.89518
$598$	0	0
$599$	5.06501	0.206951	0.103475	−	0.994632i	$-0.467004\pi$
$599$	5.06501	0.206951	0.103475	+	0.994632i	$0.467004\pi$
$600$	0	0
$601$	−38.2501	−1.56026	−0.780128	−	0.625621i	$-0.784846\pi$
$601$	−38.2501	−1.56026	−0.780128	+	0.625621i	$0.784846\pi$
$602$	0	0
$603$	−77.8398	−3.16988
$604$	0	0
$605$	13.1015	0.532651
$606$	0	0
$607$	26.8222	1.08868	0.544340	−	0.838865i	$-0.316780\pi$
$607$	26.8222	1.08868	0.544340	+	0.838865i	$0.316780\pi$
$608$	0	0
$609$	−34.3184	−1.39065
$610$	0	0
$611$	7.26191	0.293785
$612$	0	0
$613$	32.9451	1.33064	0.665320	−	0.746558i	$-0.268295\pi$
$613$	32.9451	1.33064	0.665320	+	0.746558i	$0.268295\pi$
$614$	0	0
$615$	122.179	4.92672
$616$	0	0
$617$	−31.6363	−1.27363	−0.636815	−	0.771016i	$-0.719749\pi$
$617$	−31.6363	−1.27363	−0.636815	+	0.771016i	$0.719749\pi$
$618$	0	0
$619$	7.42947	0.298616	0.149308	−	0.988791i	$-0.452295\pi$
$619$	7.42947	0.298616	0.149308	+	0.988791i	$0.452295\pi$
$620$	0	0
$621$	170.671	6.84878
$622$	0	0
$623$	−37.8807	−1.51766
$624$	0	0
$625$	−13.8800	−0.555201
$626$	0	0
$627$	−16.5375	−0.660444
$628$	0	0
$629$	22.1284	0.882318
$630$	0	0
$631$	34.7077	1.38169	0.690846	−	0.723002i	$-0.257238\pi$
$631$	34.7077	1.38169	0.690846	+	0.723002i	$0.257238\pi$
$632$	0	0
$633$	15.4166	0.612754
$634$	0	0
$635$	−20.1847	−0.801005
$636$	0	0
$637$	−2.44043	−0.0966932
$638$	0	0
$639$	45.5755	1.80294
$640$	0	0
$641$	0.314210	0.0124105	0.00620527	−	0.999981i	$-0.498025\pi$
$641$	0.314210	0.0124105	0.00620527	+	0.999981i	$0.498025\pi$
$642$	0	0
$643$	15.7558	0.621347	0.310674	−	0.950517i	$-0.399445\pi$
$643$	15.7558	0.621347	0.310674	+	0.950517i	$0.399445\pi$
$644$	0	0
$645$	105.197	4.14211
$646$	0	0
$647$	−13.6189	−0.535416	−0.267708	−	0.963500i	$-0.586266\pi$
$647$	−13.6189	−0.535416	−0.267708	+	0.963500i	$0.586266\pi$
$648$	0	0
$649$	−30.3318	−1.19063
$650$	0	0
$651$	22.6562	0.887968
$652$	0	0
$653$	9.65881	0.377978	0.188989	−	0.981979i	$-0.439479\pi$
$653$	9.65881	0.377978	0.188989	+	0.981979i	$0.439479\pi$
$654$	0	0
$655$	−48.5311	−1.89627
$656$	0	0
$657$	−32.4211	−1.26487
$658$	0	0
$659$	−22.6152	−0.880963	−0.440481	−	0.897762i	$-0.645192\pi$
$659$	−22.6152	−0.880963	−0.440481	+	0.897762i	$0.645192\pi$
$660$	0	0
$661$	−43.7607	−1.70209	−0.851047	−	0.525090i	$-0.824032\pi$
$661$	−43.7607	−1.70209	−0.851047	+	0.525090i	$0.824032\pi$
$662$	0	0
$663$	−8.63618	−0.335401
$664$	0	0
$665$	−18.9369	−0.734341
$666$	0	0
$667$	28.5308	1.10472
$668$	0	0
$669$	−68.8908	−2.66347
$670$	0	0
$671$	1.09202	0.0421570
$672$	0	0
$673$	−5.02933	−0.193866	−0.0969332	−	0.995291i	$-0.530903\pi$
$673$	−5.02933	−0.193866	−0.0969332	+	0.995291i	$0.530903\pi$
$674$	0	0
$675$	−130.277	−5.01435
$676$	0	0
$677$	29.8500	1.14723	0.573614	−	0.819126i	$-0.305541\pi$
$677$	29.8500	1.14723	0.573614	+	0.819126i	$0.305541\pi$
$678$	0	0
$679$	−27.8084	−1.06719
$680$	0	0
$681$	−28.4664	−1.09083
$682$	0	0
$683$	−25.8581	−0.989431	−0.494716	−	0.869055i	$-0.664728\pi$
$683$	−25.8581	−0.989431	−0.494716	+	0.869055i	$0.664728\pi$
$684$	0	0
$685$	−30.8771	−1.17975
$686$	0	0
$687$	52.1363	1.98912
$688$	0	0
$689$	2.18057	0.0830733
$690$	0	0
$691$	−13.9671	−0.531334	−0.265667	−	0.964065i	$-0.585592\pi$
$691$	−13.9671	−0.531334	−0.265667	+	0.964065i	$0.585592\pi$
$692$	0	0
$693$	71.5825	2.71919
$694$	0	0
$695$	48.0327	1.82199
$696$	0	0
$697$	−26.2462	−0.994145
$698$	0	0
$699$	29.1516	1.10262
$700$	0	0
$701$	20.4759	0.773364	0.386682	−	0.922213i	$-0.373621\pi$
$701$	20.4759	0.773364	0.386682	+	0.922213i	$0.373621\pi$
$702$	0	0
$703$	15.9118	0.600123
$704$	0	0
$705$	−84.4513	−3.18062
$706$	0	0
$707$	−23.7211	−0.892125
$708$	0	0
$709$	−10.6962	−0.401703	−0.200852	−	0.979622i	$-0.564371\pi$
$709$	−10.6962	−0.401703	−0.200852	+	0.979622i	$0.564371\pi$
$710$	0	0
$711$	−98.3958	−3.69013
$712$	0	0
$713$	−18.8354	−0.705390
$714$	0	0
$715$	−9.18958	−0.343671
$716$	0	0
$717$	53.1953	1.98662
$718$	0	0
$719$	−40.9965	−1.52891	−0.764455	−	0.644677i	$-0.776992\pi$
$719$	−40.9965	−1.52891	−0.764455	+	0.644677i	$0.776992\pi$
$720$	0	0
$721$	−45.1398	−1.68109
$722$	0	0
$723$	−55.5136	−2.06457
$724$	0	0
$725$	−21.7782	−0.808821
$726$	0	0
$727$	16.5206	0.612715	0.306357	−	0.951917i	$-0.400890\pi$
$727$	16.5206	0.612715	0.306357	+	0.951917i	$0.400890\pi$
$728$	0	0
$729$	154.171	5.71002
$730$	0	0
$731$	−22.5981	−0.835822
$732$	0	0
$733$	11.8427	0.437420	0.218710	−	0.975790i	$-0.429815\pi$
$733$	11.8427	0.437420	0.218710	+	0.975790i	$0.429815\pi$
$734$	0	0
$735$	28.3806	1.04683
$736$	0	0
$737$	23.9215	0.881161
$738$	0	0
$739$	11.6046	0.426884	0.213442	−	0.976956i	$-0.431533\pi$
$739$	11.6046	0.426884	0.213442	+	0.976956i	$0.431533\pi$
$740$	0	0
$741$	−6.20996	−0.228129
$742$	0	0
$743$	44.0794	1.61712	0.808558	−	0.588417i	$-0.200249\pi$
$743$	44.0794	1.61712	0.808558	+	0.588417i	$0.200249\pi$
$744$	0	0
$745$	−14.2661	−0.522670
$746$	0	0
$747$	5.52756	0.202243
$748$	0	0
$749$	17.2602	0.630675
$750$	0	0
$751$	−39.2159	−1.43101	−0.715505	−	0.698608i	$-0.753803\pi$
$751$	−39.2159	−1.43101	−0.715505	+	0.698608i	$0.753803\pi$
$752$	0	0
$753$	−79.8960	−2.91157
$754$	0	0
$755$	−3.06371	−0.111500
$756$	0	0
$757$	−27.5973	−1.00304	−0.501520	−	0.865146i	$-0.667226\pi$
$757$	−27.5973	−1.00304	−0.501520	+	0.865146i	$0.667226\pi$
$758$	0	0
$759$	−80.0084	−2.90412
$760$	0	0
$761$	−48.6368	−1.76308	−0.881541	−	0.472107i	$-0.843494\pi$
$761$	−48.6368	−1.76308	−0.881541	+	0.472107i	$0.843494\pi$
$762$	0	0
$763$	26.6751	0.965702
$764$	0	0
$765$	74.7025	2.70087
$766$	0	0
$767$	−11.3899	−0.411264
$768$	0	0
$769$	27.2804	0.983757	0.491878	−	0.870664i	$-0.336310\pi$
$769$	27.2804	0.983757	0.491878	+	0.870664i	$0.336310\pi$
$770$	0	0
$771$	47.1347	1.69751
$772$	0	0
$773$	−7.57966	−0.272621	−0.136311	−	0.990666i	$-0.543525\pi$
$773$	−7.57966	−0.272621	−0.136311	+	0.990666i	$0.543525\pi$
$774$	0	0
$775$	14.3774	0.516453
$776$	0	0
$777$	−92.5971	−3.32190
$778$	0	0
$779$	−18.8727	−0.676184
$780$	0	0
$781$	−14.0061	−0.501179
$782$	0	0
$783$	63.8164	2.28061
$784$	0	0
$785$	−18.3839	−0.656149
$786$	0	0
$787$	42.3110	1.50822	0.754112	−	0.656746i	$-0.228068\pi$
$787$	42.3110	1.50822	0.754112	+	0.656746i	$0.228068\pi$
$788$	0	0
$789$	1.71982	0.0612272
$790$	0	0
$791$	−56.8776	−2.02234
$792$	0	0
$793$	0.410063	0.0145617
$794$	0	0
$795$	−25.3587	−0.899379
$796$	0	0
$797$	17.7535	0.628861	0.314431	−	0.949280i	$-0.398186\pi$
$797$	17.7535	0.628861	0.314431	+	0.949280i	$0.398186\pi$
$798$	0	0
$799$	18.1417	0.641806
$800$	0	0
$801$	107.451	3.79661
$802$	0	0
$803$	9.96356	0.351607
$804$	0	0
$805$	−91.6166	−3.22906
$806$	0	0
$807$	−64.3048	−2.26364
$808$	0	0
$809$	32.4810	1.14197	0.570985	−	0.820961i	$-0.306562\pi$
$809$	32.4810	1.14197	0.570985	+	0.820961i	$0.306562\pi$
$810$	0	0
$811$	28.0195	0.983899	0.491950	−	0.870624i	$-0.336284\pi$
$811$	28.0195	0.983899	0.491950	+	0.870624i	$0.336284\pi$
$812$	0	0
$813$	66.9026	2.34638
$814$	0	0
$815$	23.0614	0.807804
$816$	0	0
$817$	−16.2495	−0.568498
$818$	0	0
$819$	26.8798	0.939256
$820$	0	0
$821$	−37.5822	−1.31163	−0.655814	−	0.754922i	$-0.727675\pi$
$821$	−37.5822	−1.31163	−0.655814	+	0.754922i	$0.727675\pi$
$822$	0	0
$823$	33.3227	1.16156	0.580778	−	0.814062i	$-0.302748\pi$
$823$	33.3227	1.16156	0.580778	+	0.814062i	$0.302748\pi$
$824$	0	0
$825$	61.0721	2.12626
$826$	0	0
$827$	24.8491	0.864087	0.432044	−	0.901853i	$-0.357793\pi$
$827$	24.8491	0.864087	0.432044	+	0.901853i	$0.357793\pi$
$828$	0	0
$829$	6.40969	0.222618	0.111309	−	0.993786i	$-0.464496\pi$
$829$	6.40969	0.222618	0.111309	+	0.993786i	$0.464496\pi$
$830$	0	0
$831$	69.5892	2.41402
$832$	0	0
$833$	−6.09666	−0.211237
$834$	0	0
$835$	−25.2088	−0.872385
$836$	0	0
$837$	−42.1301	−1.45623
$838$	0	0
$839$	5.85295	0.202066	0.101033	−	0.994883i	$-0.467785\pi$
$839$	5.85295	0.202066	0.101033	+	0.994883i	$0.467785\pi$
$840$	0	0
$841$	−18.3319	−0.632134
$842$	0	0
$843$	75.6276	2.60475
$844$	0	0
$845$	40.9547	1.40888
$846$	0	0
$847$	11.7771	0.404665
$848$	0	0
$849$	−11.0207	−0.378228
$850$	0	0
$851$	76.9811	2.63888
$852$	0	0
$853$	44.0895	1.50960	0.754798	−	0.655957i	$-0.227735\pi$
$853$	44.0895	1.50960	0.754798	+	0.655957i	$0.227735\pi$
$854$	0	0
$855$	53.7158	1.83704
$856$	0	0
$857$	1.88120	0.0642605	0.0321302	−	0.999484i	$-0.489771\pi$
$857$	1.88120	0.0642605	0.0321302	+	0.999484i	$0.489771\pi$
$858$	0	0
$859$	−12.5933	−0.429679	−0.214840	−	0.976649i	$-0.568923\pi$
$859$	−12.5933	−0.429679	−0.214840	+	0.976649i	$0.568923\pi$
$860$	0	0
$861$	109.828	3.74293
$862$	0	0
$863$	49.7641	1.69399	0.846995	−	0.531601i	$-0.178409\pi$
$863$	49.7641	1.69399	0.846995	+	0.531601i	$0.178409\pi$
$864$	0	0
$865$	2.74928	0.0934785
$866$	0	0
$867$	36.5983	1.24294
$868$	0	0
$869$	30.2387	1.02578
$870$	0	0
$871$	8.98273	0.304368
$872$	0	0
$873$	78.8805	2.66970
$874$	0	0
$875$	17.4916	0.591324
$876$	0	0
$877$	10.1237	0.341853	0.170926	−	0.985284i	$-0.445324\pi$
$877$	10.1237	0.341853	0.170926	+	0.985284i	$0.445324\pi$
$878$	0	0
$879$	17.8510	0.602099
$880$	0	0
$881$	−54.1145	−1.82316	−0.911582	−	0.411119i	$-0.865138\pi$
$881$	−54.1145	−1.82316	−0.911582	+	0.411119i	$0.865138\pi$
$882$	0	0
$883$	−2.73874	−0.0921658	−0.0460829	−	0.998938i	$-0.514674\pi$
$883$	−2.73874	−0.0921658	−0.0460829	+	0.998938i	$0.514674\pi$
$884$	0	0
$885$	132.457	4.45248
$886$	0	0
$887$	6.77920	0.227623	0.113812	−	0.993502i	$-0.463694\pi$
$887$	6.77920	0.227623	0.113812	+	0.993502i	$0.463694\pi$
$888$	0	0
$889$	−18.1443	−0.608539
$890$	0	0
$891$	−109.020	−3.65232
$892$	0	0
$893$	13.0450	0.436535
$894$	0	0
$895$	−14.0278	−0.468899
$896$	0	0
$897$	−30.0438	−1.00313
$898$	0	0
$899$	−7.04283	−0.234892
$900$	0	0
$901$	5.44750	0.181483
$902$	0	0
$903$	94.5626	3.14685
$904$	0	0
$905$	47.5913	1.58199
$906$	0	0
$907$	11.1216	0.369287	0.184643	−	0.982806i	$-0.440887\pi$
$907$	11.1216	0.369287	0.184643	+	0.982806i	$0.440887\pi$
$908$	0	0
$909$	67.2868	2.23176
$910$	0	0
$911$	−23.1864	−0.768199	−0.384099	−	0.923292i	$-0.625488\pi$
$911$	−23.1864	−0.768199	−0.384099	+	0.923292i	$0.625488\pi$
$912$	0	0
$913$	−1.69871	−0.0562192
$914$	0	0
$915$	−4.76876	−0.157650
$916$	0	0
$917$	−43.6252	−1.44063
$918$	0	0
$919$	−28.5966	−0.943316	−0.471658	−	0.881782i	$-0.656344\pi$
$919$	−28.5966	−0.943316	−0.471658	+	0.881782i	$0.656344\pi$
$920$	0	0
$921$	92.1271	3.03569
$922$	0	0
$923$	−5.25941	−0.173116
$924$	0	0
$925$	−58.7613	−1.93206
$926$	0	0
$927$	128.042	4.20546
$928$	0	0
$929$	−6.84094	−0.224444	−0.112222	−	0.993683i	$-0.535797\pi$
$929$	−6.84094	−0.224444	−0.112222	+	0.993683i	$0.535797\pi$
$930$	0	0
$931$	−4.38389	−0.143676
$932$	0	0
$933$	81.7259	2.67559
$934$	0	0
$935$	−22.9574	−0.750786
$936$	0	0
$937$	−36.7062	−1.19914	−0.599570	−	0.800323i	$-0.704662\pi$
$937$	−36.7062	−1.19914	−0.599570	+	0.800323i	$0.704662\pi$
$938$	0	0
$939$	83.5852	2.72770
$940$	0	0
$941$	47.3641	1.54403	0.772013	−	0.635607i	$-0.219250\pi$
$941$	47.3641	1.54403	0.772013	+	0.635607i	$0.219250\pi$
$942$	0	0
$943$	−91.3060	−2.97333
$944$	0	0
$945$	−204.924	−6.66617
$946$	0	0
$947$	23.4451	0.761862	0.380931	−	0.924604i	$-0.375604\pi$
$947$	23.4451	0.761862	0.380931	+	0.924604i	$0.375604\pi$
$948$	0	0
$949$	3.74140	0.121451
$950$	0	0
$951$	76.3467	2.47571
$952$	0	0
$953$	32.7862	1.06205	0.531024	−	0.847357i	$-0.321807\pi$
$953$	32.7862	1.06205	0.531024	+	0.847357i	$0.321807\pi$
$954$	0	0
$955$	−46.8809	−1.51703
$956$	0	0
$957$	−29.9163	−0.967058
$958$	0	0
$959$	−27.7558	−0.896281
$960$	0	0
$961$	−26.3505	−0.850016
$962$	0	0
$963$	−48.9599	−1.57771
$964$	0	0
$965$	−68.9871	−2.22077
$966$	0	0
$967$	−48.1036	−1.54691	−0.773453	−	0.633854i	$-0.781472\pi$
$967$	−48.1036	−1.54691	−0.773453	+	0.633854i	$0.781472\pi$
$968$	0	0
$969$	−15.5137	−0.498372
$970$	0	0
$971$	47.2710	1.51700	0.758499	−	0.651674i	$-0.225933\pi$
$971$	47.2710	1.51700	0.758499	+	0.651674i	$0.225933\pi$
$972$	0	0
$973$	43.1772	1.38420
$974$	0	0
$975$	22.9331	0.734446
$976$	0	0
$977$	−16.5578	−0.529730	−0.264865	−	0.964286i	$-0.585327\pi$
$977$	−16.5578	−0.529730	−0.264865	+	0.964286i	$0.585327\pi$
$978$	0	0
$979$	−33.0217	−1.05538
$980$	0	0
$981$	−75.6658	−2.41582
$982$	0	0
$983$	29.9633	0.955680	0.477840	−	0.878447i	$-0.341420\pi$
$983$	29.9633	0.955680	0.477840	+	0.878447i	$0.341420\pi$
$984$	0	0
$985$	−47.0409	−1.49885
$986$	0	0
$987$	−75.9144	−2.41638
$988$	0	0
$989$	−78.6151	−2.49981
$990$	0	0
$991$	20.3105	0.645184	0.322592	−	0.946538i	$-0.395446\pi$
$991$	20.3105	0.645184	0.322592	+	0.946538i	$0.395446\pi$
$992$	0	0
$993$	−97.8959	−3.10663
$994$	0	0
$995$	−46.2230	−1.46537
$996$	0	0
$997$	39.5285	1.25188	0.625940	−	0.779871i	$-0.284716\pi$
$997$	39.5285	1.25188	0.625940	+	0.779871i	$0.284716\pi$
$998$	0	0
$999$	172.188	5.44778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.1		33
4.3	odd	2		4024.2.a.g.1.33	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.33	✓	33	4.3	odd	2
8048.2.a.x.1.1		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)

\(f(q)\)	\(=\)	\(q-3.42195 q^{3} -3.41581 q^{5} -3.07051 q^{7} +8.70973 q^{9} +O(q^{10})\)	\(q-3.42195 q^{3} -3.41581 q^{5} -3.07051 q^{7} +8.70973 q^{9} -2.67665 q^{11} -1.00510 q^{13} +11.6887 q^{15} -2.51095 q^{17} -1.80553 q^{19} +10.5071 q^{21} -8.73515 q^{23} +6.66773 q^{25} -19.5384 q^{27} -3.26621 q^{29} +2.15627 q^{31} +9.15936 q^{33} +10.4883 q^{35} -8.81279 q^{37} +3.43941 q^{39} +10.4527 q^{41} +8.99985 q^{43} -29.7507 q^{45} -7.22504 q^{47} +2.42803 q^{49} +8.59232 q^{51} -2.16950 q^{53} +9.14292 q^{55} +6.17843 q^{57} +11.3320 q^{59} -0.407980 q^{61} -26.7433 q^{63} +3.43324 q^{65} -8.93711 q^{67} +29.8912 q^{69} +5.23271 q^{71} -3.72240 q^{73} -22.8166 q^{75} +8.21868 q^{77} -11.2972 q^{79} +40.7302 q^{81} +0.634642 q^{83} +8.57690 q^{85} +11.1768 q^{87} +12.3370 q^{89} +3.08618 q^{91} -7.37865 q^{93} +6.16734 q^{95} +9.05660 q^{97} -23.3129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 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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