LMFDB - Embedded newform 5120.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5120,2,Mod(1,5120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5120, 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("5120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.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:	$40.8834058349$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.18688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 4x + 14 $ x^4 - 10x^2 - 4x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1280)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.04953$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.04953 q^{3} +1.00000 q^{5} +0.463747 q^{7} -1.89848 q^{9} -2.89848 q^{11} +5.58333 q^{13} -1.04953 q^{15} -1.34416 q^{17} -6.99755 q^{19} -0.486719 q^{21} +3.84894 q^{23} +1.00000 q^{25} +5.14112 q^{27} +1.55432 q^{29} -3.17157 q^{31} +3.04205 q^{33} +0.463747 q^{35} +9.75592 q^{37} -5.85990 q^{39} -2.89848 q^{41} -2.29463 q^{43} -1.89848 q^{45} +3.29217 q^{47} -6.78494 q^{49} +1.41074 q^{51} -4.65584 q^{53} -2.89848 q^{55} +7.34416 q^{57} +9.68587 q^{59} -4.86946 q^{61} -0.880415 q^{63} +5.58333 q^{65} -8.39370 q^{67} -4.03960 q^{69} -15.8960 q^{71} +0.556770 q^{73} -1.04953 q^{75} -1.34416 q^{77} +4.23917 q^{79} +0.299657 q^{81} -9.99252 q^{83} -1.34416 q^{85} -1.63131 q^{87} -17.3068 q^{89} +2.58926 q^{91} +3.32867 q^{93} -6.99755 q^{95} +9.04205 q^{97} +5.50270 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 8 q^{7} + 8 q^{9} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 4 q^{19} - 16 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} - 8 q^{29} - 24 q^{31} - 12 q^{33} - 8 q^{35} + 8 q^{37} - 32 q^{39} + 4 q^{41}+ \cdots + 52 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 - 8 * q^7 + 8 * q^9 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 4 * q^19 - 16 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 - 8 * q^29 - 24 * q^31 - 12 * q^33 - 8 * q^35 + 8 * q^37 - 32 * q^39 + 4 * q^41 - 12 * q^43 + 8 * q^45 - 8 * q^47 + 8 * q^49 - 20 * q^53 + 4 * q^55 + 28 * q^57 + 12 * q^59 - 8 * q^61 - 12 * q^63 + 4 * q^65 - 28 * q^67 + 32 * q^69 - 24 * q^71 + 12 * q^73 - 4 * q^77 - 20 * q^83 - 4 * q^85 + 12 * q^87 - 24 * q^89 + 16 * q^91 + 8 * q^93 - 4 * q^95 + 12 * q^97 + 52 * q^99

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
$3$	−1.04953	−0.605949	−0.302974	−	0.952999i	$-0.597980\pi$
$3$	−1.04953	−0.605949	−0.302974	+	0.952999i	$0.597980\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.463747	0.175280	0.0876400	−	0.996152i	$-0.472067\pi$
$7$	0.463747	0.175280	0.0876400	+	0.996152i	$0.472067\pi$
$8$	0	0
$9$	−1.89848	−0.632826
$10$	0	0
$11$	−2.89848	−0.873924	−0.436962	−	0.899480i	$-0.643946\pi$
$11$	−2.89848	−0.873924	−0.436962	+	0.899480i	$0.643946\pi$
$12$	0	0
$13$	5.58333	1.54854	0.774269	−	0.632857i	$-0.218118\pi$
$13$	5.58333	1.54854	0.774269	+	0.632857i	$0.218118\pi$
$14$	0	0
$15$	−1.04953	−0.270988
$16$	0	0
$17$	−1.34416	−0.326007	−0.163004	−	0.986625i	$-0.552118\pi$
$17$	−1.34416	−0.326007	−0.163004	+	0.986625i	$0.552118\pi$
$18$	0	0
$19$	−6.99755	−1.60535	−0.802674	−	0.596419i	$-0.796590\pi$
$19$	−6.99755	−1.60535	−0.802674	+	0.596419i	$0.796590\pi$
$20$	0	0
$21$	−0.486719	−0.106211
$22$	0	0
$23$	3.84894	0.802560	0.401280	−	0.915955i	$-0.368565\pi$
$23$	3.84894	0.802560	0.401280	+	0.915955i	$0.368565\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.14112	0.989409
$28$	0	0
$29$	1.55432	0.288629	0.144315	−	0.989532i	$-0.453902\pi$
$29$	1.55432	0.288629	0.144315	+	0.989532i	$0.453902\pi$
$30$	0	0
$31$	−3.17157	−0.569631	−0.284816	−	0.958582i	$-0.591932\pi$
$31$	−3.17157	−0.569631	−0.284816	+	0.958582i	$0.591932\pi$
$32$	0	0
$33$	3.04205	0.529553
$34$	0	0
$35$	0.463747	0.0783876
$36$	0	0
$37$	9.75592	1.60386	0.801932	−	0.597416i	$-0.203806\pi$
$37$	9.75592	1.60386	0.801932	+	0.597416i	$0.203806\pi$
$38$	0	0
$39$	−5.85990	−0.938334
$40$	0	0
$41$	−2.89848	−0.452666	−0.226333	−	0.974050i	$-0.572674\pi$
$41$	−2.89848	−0.452666	−0.226333	+	0.974050i	$0.572674\pi$
$42$	0	0
$43$	−2.29463	−0.349928	−0.174964	−	0.984575i	$-0.555981\pi$
$43$	−2.29463	−0.349928	−0.174964	+	0.984575i	$0.555981\pi$
$44$	0	0
$45$	−1.89848	−0.283008
$46$	0	0
$47$	3.29217	0.480213	0.240107	−	0.970747i	$-0.422818\pi$
$47$	3.29217	0.480213	0.240107	+	0.970747i	$0.422818\pi$
$48$	0	0
$49$	−6.78494	−0.969277
$50$	0	0
$51$	1.41074	0.197544
$52$	0	0
$53$	−4.65584	−0.639529	−0.319764	−	0.947497i	$-0.603604\pi$
$53$	−4.65584	−0.639529	−0.319764	+	0.947497i	$0.603604\pi$
$54$	0	0
$55$	−2.89848	−0.390831
$56$	0	0
$57$	7.34416	0.972758
$58$	0	0
$59$	9.68587	1.26099	0.630496	−	0.776192i	$-0.282851\pi$
$59$	9.68587	1.26099	0.630496	+	0.776192i	$0.282851\pi$
$60$	0	0
$61$	−4.86946	−0.623471	−0.311735	−	0.950169i	$-0.600910\pi$
$61$	−4.86946	−0.623471	−0.311735	+	0.950169i	$0.600910\pi$
$62$	0	0
$63$	−0.880415	−0.110922
$64$	0	0
$65$	5.58333	0.692527
$66$	0	0
$67$	−8.39370	−1.02545	−0.512727	−	0.858552i	$-0.671365\pi$
$67$	−8.39370	−1.02545	−0.512727	+	0.858552i	$0.671365\pi$
$68$	0	0
$69$	−4.03960	−0.486310
$70$	0	0
$71$	−15.8960	−1.88651	−0.943256	−	0.332068i	$-0.892254\pi$
$71$	−15.8960	−1.88651	−0.943256	+	0.332068i	$0.892254\pi$
$72$	0	0
$73$	0.556770	0.0651650	0.0325825	−	0.999469i	$-0.489627\pi$
$73$	0.556770	0.0651650	0.0325825	+	0.999469i	$0.489627\pi$
$74$	0	0
$75$	−1.04953	−0.121190
$76$	0	0
$77$	−1.34416	−0.153181
$78$	0	0
$79$	4.23917	0.476944	0.238472	−	0.971149i	$-0.423353\pi$
$79$	4.23917	0.476944	0.238472	+	0.971149i	$0.423353\pi$
$80$	0	0
$81$	0.299657	0.0332952
$82$	0	0
$83$	−9.99252	−1.09682	−0.548411	−	0.836209i	$-0.684767\pi$
$83$	−9.99252	−1.09682	−0.548411	+	0.836209i	$0.684767\pi$
$84$	0	0
$85$	−1.34416	−0.145795
$86$	0	0
$87$	−1.63131	−0.174895
$88$	0	0
$89$	−17.3068	−1.83451	−0.917257	−	0.398296i	$-0.869602\pi$
$89$	−17.3068	−1.83451	−0.917257	+	0.398296i	$0.869602\pi$
$90$	0	0
$91$	2.58926	0.271428
$92$	0	0
$93$	3.32867	0.345167
$94$	0	0
$95$	−6.99755	−0.717933
$96$	0	0
$97$	9.04205	0.918081	0.459041	−	0.888415i	$-0.348193\pi$
$97$	9.04205	0.918081	0.459041	+	0.888415i	$0.348193\pi$
$98$	0	0
$99$	5.50270	0.553042
$100$	0	0
$101$	9.75245	0.970405	0.485203	−	0.874402i	$-0.338746\pi$
$101$	9.75245	0.970405	0.485203	+	0.874402i	$0.338746\pi$
$102$	0	0
$103$	−15.0156	−1.47953	−0.739766	−	0.672864i	$-0.765064\pi$
$103$	−15.0156	−1.47953	−0.739766	+	0.672864i	$0.765064\pi$
$104$	0	0
$105$	−0.486719	−0.0474989
$106$	0	0
$107$	16.6434	1.60898	0.804491	−	0.593964i	$-0.202438\pi$
$107$	16.6434	1.60898	0.804491	+	0.593964i	$0.202438\pi$
$108$	0	0
$109$	5.85990	0.561276	0.280638	−	0.959814i	$-0.409454\pi$
$109$	5.85990	0.561276	0.280638	+	0.959814i	$0.409454\pi$
$110$	0	0
$111$	−10.2392	−0.971859
$112$	0	0
$113$	7.99509	0.752115	0.376058	−	0.926596i	$-0.377280\pi$
$113$	7.99509	0.752115	0.376058	+	0.926596i	$0.377280\pi$
$114$	0	0
$115$	3.84894	0.358916
$116$	0	0
$117$	−10.5998	−0.979955
$118$	0	0
$119$	−0.623352	−0.0571426
$120$	0	0
$121$	−2.59882	−0.236257
$122$	0	0
$123$	3.04205	0.274293
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.9795	−0.974272	−0.487136	−	0.873326i	$-0.661958\pi$
$127$	−10.9795	−0.974272	−0.487136	+	0.873326i	$0.661958\pi$
$128$	0	0
$129$	2.40829	0.212038
$130$	0	0
$131$	6.21015	0.542584	0.271292	−	0.962497i	$-0.412549\pi$
$131$	6.21015	0.542584	0.271292	+	0.962497i	$0.412549\pi$
$132$	0	0
$133$	−3.24509	−0.281385
$134$	0	0
$135$	5.14112	0.442477
$136$	0	0
$137$	−21.8960	−1.87070	−0.935352	−	0.353719i	$-0.884917\pi$
$137$	−21.8960	−1.87070	−0.935352	+	0.353719i	$0.884917\pi$
$138$	0	0
$139$	−6.30922	−0.535141	−0.267571	−	0.963538i	$-0.586221\pi$
$139$	−6.30922	−0.535141	−0.267571	+	0.963538i	$0.586221\pi$
$140$	0	0
$141$	−3.45525	−0.290984
$142$	0	0
$143$	−16.1832	−1.35330
$144$	0	0
$145$	1.55432	0.129079
$146$	0	0
$147$	7.12102	0.587332
$148$	0	0
$149$	10.3862	0.850872	0.425436	−	0.904989i	$-0.360121\pi$
$149$	10.3862	0.850872	0.425436	+	0.904989i	$0.360121\pi$
$150$	0	0
$151$	−17.0266	−1.38560	−0.692801	−	0.721129i	$-0.743624\pi$
$151$	−17.0266	−1.38560	−0.692801	+	0.721129i	$0.743624\pi$
$152$	0	0
$153$	2.55186	0.206306
$154$	0	0
$155$	−3.17157	−0.254747
$156$	0	0
$157$	5.45381	0.435262	0.217631	−	0.976031i	$-0.430167\pi$
$157$	5.45381	0.435262	0.217631	+	0.976031i	$0.430167\pi$
$158$	0	0
$159$	4.88646	0.387521
$160$	0	0
$161$	1.78494	0.140673
$162$	0	0
$163$	−17.4308	−1.36529	−0.682644	−	0.730751i	$-0.739170\pi$
$163$	−17.4308	−1.36529	−0.682644	+	0.730751i	$0.739170\pi$
$164$	0	0
$165$	3.04205	0.236823
$166$	0	0
$167$	18.6314	1.44174	0.720872	−	0.693069i	$-0.243742\pi$
$167$	18.6314	1.44174	0.720872	+	0.693069i	$0.243742\pi$
$168$	0	0
$169$	18.1736	1.39797
$170$	0	0
$171$	13.2847	1.01591
$172$	0	0
$173$	6.38131	0.485162	0.242581	−	0.970131i	$-0.422006\pi$
$173$	6.38131	0.485162	0.242581	+	0.970131i	$0.422006\pi$
$174$	0	0
$175$	0.463747	0.0350560
$176$	0	0
$177$	−10.1656	−0.764097
$178$	0	0
$179$	−21.4828	−1.60570	−0.802851	−	0.596180i	$-0.796685\pi$
$179$	−21.4828	−1.60570	−0.802851	+	0.596180i	$0.796685\pi$
$180$	0	0
$181$	16.5228	1.22813	0.614067	−	0.789254i	$-0.289533\pi$
$181$	16.5228	1.22813	0.614067	+	0.789254i	$0.289533\pi$
$182$	0	0
$183$	5.11067	0.377791
$184$	0	0
$185$	9.75592	0.717270
$186$	0	0
$187$	3.89603	0.284906
$188$	0	0
$189$	2.38418	0.173424
$190$	0	0
$191$	−3.89603	−0.281906	−0.140953	−	0.990016i	$-0.545017\pi$
$191$	−3.89603	−0.281906	−0.140953	+	0.990016i	$0.545017\pi$
$192$	0	0
$193$	−19.0371	−1.37032	−0.685162	−	0.728391i	$-0.740269\pi$
$193$	−19.0371	−1.37032	−0.685162	+	0.728391i	$0.740269\pi$
$194$	0	0
$195$	−5.85990	−0.419636
$196$	0	0
$197$	−0.0665816	−0.00474374	−0.00237187	−	0.999997i	$-0.500755\pi$
$197$	−0.0665816	−0.00474374	−0.00237187	+	0.999997i	$0.500755\pi$
$198$	0	0
$199$	−11.3137	−0.802008	−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137	−0.802008	−0.401004	+	0.916076i	$0.631339\pi$
$200$	0	0
$201$	8.80947	0.621372
$202$	0	0
$203$	0.720810	0.0505910
$204$	0	0
$205$	−2.89848	−0.202439
$206$	0	0
$207$	−7.30714	−0.507881
$208$	0	0
$209$	20.2822	1.40295
$210$	0	0
$211$	0.478170	0.0329186	0.0164593	−	0.999865i	$-0.494761\pi$
$211$	0.478170	0.0329186	0.0164593	+	0.999865i	$0.494761\pi$
$212$	0	0
$213$	16.6834	1.14313
$214$	0	0
$215$	−2.29463	−0.156492
$216$	0	0
$217$	−1.47081	−0.0998450
$218$	0	0
$219$	−0.584349	−0.0394867
$220$	0	0
$221$	−7.50490	−0.504834
$222$	0	0
$223$	−19.6049	−1.31284	−0.656419	−	0.754396i	$-0.727930\pi$
$223$	−19.6049	−1.31284	−0.656419	+	0.754396i	$0.727930\pi$
$224$	0	0
$225$	−1.89848	−0.126565
$226$	0	0
$227$	−22.8465	−1.51637	−0.758187	−	0.652037i	$-0.773915\pi$
$227$	−22.8465	−1.51637	−0.758187	+	0.652037i	$0.773915\pi$
$228$	0	0
$229$	−26.5829	−1.75665	−0.878324	−	0.478066i	$-0.841338\pi$
$229$	−26.5829	−1.75665	−0.878324	+	0.478066i	$0.841338\pi$
$230$	0	0
$231$	1.41074	0.0928201
$232$	0	0
$233$	17.2402	1.12944	0.564721	−	0.825282i	$-0.308984\pi$
$233$	17.2402	1.12944	0.564721	+	0.825282i	$0.308984\pi$
$234$	0	0
$235$	3.29217	0.214758
$236$	0	0
$237$	−4.44915	−0.289004
$238$	0	0
$239$	−27.6930	−1.79131	−0.895655	−	0.444749i	$-0.853293\pi$
$239$	−27.6930	−1.79131	−0.895655	+	0.444749i	$0.853293\pi$
$240$	0	0
$241$	9.09661	0.585964	0.292982	−	0.956118i	$-0.405352\pi$
$241$	9.09661	0.585964	0.292982	+	0.956118i	$0.405352\pi$
$242$	0	0
$243$	−15.7379	−1.00958
$244$	0	0
$245$	−6.78494	−0.433474
$246$	0	0
$247$	−39.0696	−2.48594
$248$	0	0
$249$	10.4875	0.664617
$250$	0	0
$251$	12.2102	0.770698	0.385349	−	0.922771i	$-0.374081\pi$
$251$	12.2102	0.770698	0.385349	+	0.922771i	$0.374081\pi$
$252$	0	0
$253$	−11.1561	−0.701377
$254$	0	0
$255$	1.41074	0.0883442
$256$	0	0
$257$	−23.0647	−1.43874	−0.719369	−	0.694628i	$-0.755569\pi$
$257$	−23.0647	−1.43874	−0.719369	+	0.694628i	$0.755569\pi$
$258$	0	0
$259$	4.52428	0.281125
$260$	0	0
$261$	−2.95084	−0.182652
$262$	0	0
$263$	21.1833	1.30622	0.653109	−	0.757264i	$-0.273464\pi$
$263$	21.1833	1.30622	0.653109	+	0.757264i	$0.273464\pi$
$264$	0	0
$265$	−4.65584	−0.286006
$266$	0	0
$267$	18.1640	1.11162
$268$	0	0
$269$	−8.98553	−0.547857	−0.273929	−	0.961750i	$-0.588323\pi$
$269$	−8.98553	−0.547857	−0.273929	+	0.961750i	$0.588323\pi$
$270$	0	0
$271$	10.8235	0.657482	0.328741	−	0.944420i	$-0.393376\pi$
$271$	10.8235	0.657482	0.328741	+	0.944420i	$0.393376\pi$
$272$	0	0
$273$	−2.71751	−0.164471
$274$	0	0
$275$	−2.89848	−0.174785
$276$	0	0
$277$	−23.3449	−1.40266	−0.701330	−	0.712836i	$-0.747410\pi$
$277$	−23.3449	−1.40266	−0.701330	+	0.712836i	$0.747410\pi$
$278$	0	0
$279$	6.02116	0.360478
$280$	0	0
$281$	10.6064	0.632726	0.316363	−	0.948638i	$-0.397538\pi$
$281$	10.6064	0.632726	0.316363	+	0.948638i	$0.397538\pi$
$282$	0	0
$283$	−3.96003	−0.235399	−0.117700	−	0.993049i	$-0.537552\pi$
$283$	−3.96003	−0.235399	−0.117700	+	0.993049i	$0.537552\pi$
$284$	0	0
$285$	7.34416	0.435031
$286$	0	0
$287$	−1.34416	−0.0793434
$288$	0	0
$289$	−15.1932	−0.893719
$290$	0	0
$291$	−9.48994	−0.556310
$292$	0	0
$293$	−19.9951	−1.16813	−0.584063	−	0.811708i	$-0.698538\pi$
$293$	−19.9951	−1.16813	−0.584063	+	0.811708i	$0.698538\pi$
$294$	0	0
$295$	9.68587	0.563933
$296$	0	0
$297$	−14.9014	−0.864668
$298$	0	0
$299$	21.4899	1.24280
$300$	0	0
$301$	−1.06413	−0.0613353
$302$	0	0
$303$	−10.2355	−0.588016
$304$	0	0
$305$	−4.86946	−0.278825
$306$	0	0
$307$	30.6759	1.75077	0.875384	−	0.483428i	$-0.160609\pi$
$307$	30.6759	1.75077	0.875384	+	0.483428i	$0.160609\pi$
$308$	0	0
$309$	15.7594	0.896520
$310$	0	0
$311$	−0.490189	−0.0277960	−0.0138980	−	0.999903i	$-0.504424\pi$
$311$	−0.490189	−0.0277960	−0.0138980	+	0.999903i	$0.504424\pi$
$312$	0	0
$313$	−22.5794	−1.27627	−0.638133	−	0.769926i	$-0.720293\pi$
$313$	−22.5794	−1.27627	−0.638133	+	0.769926i	$0.720293\pi$
$314$	0	0
$315$	−0.880415	−0.0496057
$316$	0	0
$317$	26.3488	1.47990	0.739949	−	0.672663i	$-0.234850\pi$
$317$	26.3488	1.47990	0.739949	+	0.672663i	$0.234850\pi$
$318$	0	0
$319$	−4.50515	−0.252240
$320$	0	0
$321$	−17.4679	−0.974961
$322$	0	0
$323$	9.40584	0.523355
$324$	0	0
$325$	5.58333	0.309708
$326$	0	0
$327$	−6.15016	−0.340105
$328$	0	0
$329$	1.52674	0.0841718
$330$	0	0
$331$	25.0966	1.37943	0.689717	−	0.724079i	$-0.257735\pi$
$331$	25.0966	1.37943	0.689717	+	0.724079i	$0.257735\pi$
$332$	0	0
$333$	−18.5214	−1.01497
$334$	0	0
$335$	−8.39370	−0.458597
$336$	0	0
$337$	−12.2972	−0.669871	−0.334936	−	0.942241i	$-0.608715\pi$
$337$	−12.2972	−0.669871	−0.334936	+	0.942241i	$0.608715\pi$
$338$	0	0
$339$	−8.39112	−0.455743
$340$	0	0
$341$	9.19274	0.497815
$342$	0	0
$343$	−6.39273	−0.345175
$344$	0	0
$345$	−4.03960	−0.217485
$346$	0	0
$347$	−6.19065	−0.332332	−0.166166	−	0.986098i	$-0.553139\pi$
$347$	−6.19065	−0.332332	−0.166166	+	0.986098i	$0.553139\pi$
$348$	0	0
$349$	−26.8631	−1.43795	−0.718975	−	0.695036i	$-0.755388\pi$
$349$	−26.8631	−1.43795	−0.718975	+	0.695036i	$0.755388\pi$
$350$	0	0
$351$	28.7046	1.53214
$352$	0	0
$353$	24.6785	1.31350	0.656752	−	0.754106i	$-0.271930\pi$
$353$	24.6785	1.31350	0.656752	+	0.754106i	$0.271930\pi$
$354$	0	0
$355$	−15.8960	−0.843673
$356$	0	0
$357$	0.654229	0.0346255
$358$	0	0
$359$	−28.5745	−1.50811	−0.754053	−	0.656813i	$-0.771904\pi$
$359$	−28.5745	−1.50811	−0.754053	+	0.656813i	$0.771904\pi$
$360$	0	0
$361$	29.9657	1.57714
$362$	0	0
$363$	2.72755	0.143159
$364$	0	0
$365$	0.556770	0.0291427
$366$	0	0
$367$	9.86594	0.514998	0.257499	−	0.966279i	$-0.417102\pi$
$367$	9.86594	0.514998	0.257499	+	0.966279i	$0.417102\pi$
$368$	0	0
$369$	5.50270	0.286459
$370$	0	0
$371$	−2.15913	−0.112097
$372$	0	0
$373$	11.1566	0.577667	0.288834	−	0.957379i	$-0.406733\pi$
$373$	11.1566	0.577667	0.288834	+	0.957379i	$0.406733\pi$
$374$	0	0
$375$	−1.04953	−0.0541977
$376$	0	0
$377$	8.67827	0.446953
$378$	0	0
$379$	4.79941	0.246529	0.123265	−	0.992374i	$-0.460664\pi$
$379$	4.79941	0.246529	0.123265	+	0.992374i	$0.460664\pi$
$380$	0	0
$381$	11.5233	0.590359
$382$	0	0
$383$	6.25012	0.319366	0.159683	−	0.987168i	$-0.448953\pi$
$383$	6.25012	0.319366	0.159683	+	0.987168i	$0.448953\pi$
$384$	0	0
$385$	−1.34416	−0.0685048
$386$	0	0
$387$	4.35630	0.221443
$388$	0	0
$389$	−4.30211	−0.218126	−0.109063	−	0.994035i	$-0.534785\pi$
$389$	−4.30211	−0.218126	−0.109063	+	0.994035i	$0.534785\pi$
$390$	0	0
$391$	−5.17361	−0.261640
$392$	0	0
$393$	−6.51777	−0.328778
$394$	0	0
$395$	4.23917	0.213296
$396$	0	0
$397$	9.81245	0.492473	0.246236	−	0.969210i	$-0.420806\pi$
$397$	9.81245	0.492473	0.246236	+	0.969210i	$0.420806\pi$
$398$	0	0
$399$	3.40584	0.170505
$400$	0	0
$401$	11.1736	0.557983	0.278992	−	0.960294i	$-0.410000\pi$
$401$	11.1736	0.557983	0.278992	+	0.960294i	$0.410000\pi$
$402$	0	0
$403$	−17.7079	−0.882096
$404$	0	0
$405$	0.299657	0.0148901
$406$	0	0
$407$	−28.2773	−1.40166
$408$	0	0
$409$	−39.6611	−1.96111	−0.980557	−	0.196233i	$-0.937129\pi$
$409$	−39.6611	−1.96111	−0.980557	+	0.196233i	$0.937129\pi$
$410$	0	0
$411$	22.9806	1.13355
$412$	0	0
$413$	4.49180	0.221027
$414$	0	0
$415$	−9.99252	−0.490513
$416$	0	0
$417$	6.62174	0.324268
$418$	0	0
$419$	−5.39382	−0.263505	−0.131753	−	0.991283i	$-0.542060\pi$
$419$	−5.39382	−0.263505	−0.131753	+	0.991283i	$0.542060\pi$
$420$	0	0
$421$	−11.9541	−0.582605	−0.291303	−	0.956631i	$-0.594089\pi$
$421$	−11.9541	−0.582605	−0.291303	+	0.956631i	$0.594089\pi$
$422$	0	0
$423$	−6.25012	−0.303891
$424$	0	0
$425$	−1.34416	−0.0652014
$426$	0	0
$427$	−2.25820	−0.109282
$428$	0	0
$429$	16.9848	0.820033
$430$	0	0
$431$	23.3909	1.12670	0.563349	−	0.826219i	$-0.309513\pi$
$431$	23.3909	1.12670	0.563349	+	0.826219i	$0.309513\pi$
$432$	0	0
$433$	−3.33925	−0.160474	−0.0802372	−	0.996776i	$-0.525568\pi$
$433$	−3.33925	−0.160474	−0.0802372	+	0.996776i	$0.525568\pi$
$434$	0	0
$435$	−1.63131	−0.0782152
$436$	0	0
$437$	−26.9332	−1.28839
$438$	0	0
$439$	1.27064	0.0606444	0.0303222	−	0.999540i	$-0.490347\pi$
$439$	1.27064	0.0606444	0.0303222	+	0.999540i	$0.490347\pi$
$440$	0	0
$441$	12.8811	0.613384
$442$	0	0
$443$	−24.7474	−1.17579	−0.587893	−	0.808939i	$-0.700042\pi$
$443$	−24.7474	−1.17579	−0.587893	+	0.808939i	$0.700042\pi$
$444$	0	0
$445$	−17.3068	−0.820420
$446$	0	0
$447$	−10.9007	−0.515585
$448$	0	0
$449$	−15.9831	−0.754288	−0.377144	−	0.926155i	$-0.623094\pi$
$449$	−15.9831	−0.754288	−0.377144	+	0.926155i	$0.623094\pi$
$450$	0	0
$451$	8.40118	0.395596
$452$	0	0
$453$	17.8700	0.839604
$454$	0	0
$455$	2.58926	0.121386
$456$	0	0
$457$	−6.78739	−0.317501	−0.158750	−	0.987319i	$-0.550747\pi$
$457$	−6.78739	−0.317501	−0.158750	+	0.987319i	$0.550747\pi$
$458$	0	0
$459$	−6.91050	−0.322554
$460$	0	0
$461$	−29.8316	−1.38940	−0.694699	−	0.719300i	$-0.744463\pi$
$461$	−29.8316	−1.38940	−0.694699	+	0.719300i	$0.744463\pi$
$462$	0	0
$463$	−21.3234	−0.990982	−0.495491	−	0.868613i	$-0.665012\pi$
$463$	−21.3234	−0.990982	−0.495491	+	0.868613i	$0.665012\pi$
$464$	0	0
$465$	3.32867	0.154364
$466$	0	0
$467$	6.66332	0.308342	0.154171	−	0.988044i	$-0.450729\pi$
$467$	6.66332	0.308342	0.154171	+	0.988044i	$0.450729\pi$
$468$	0	0
$469$	−3.89256	−0.179741
$470$	0	0
$471$	−5.72396	−0.263746
$472$	0	0
$473$	6.65093	0.305810
$474$	0	0
$475$	−6.99755	−0.321069
$476$	0	0
$477$	8.83901	0.404710
$478$	0	0
$479$	−10.1932	−0.465740	−0.232870	−	0.972508i	$-0.574812\pi$
$479$	−10.1932	−0.465740	−0.232870	+	0.972508i	$0.574812\pi$
$480$	0	0
$481$	54.4706	2.48364
$482$	0	0
$483$	−1.87335	−0.0852405
$484$	0	0
$485$	9.04205	0.410578
$486$	0	0
$487$	−5.87940	−0.266421	−0.133210	−	0.991088i	$-0.542529\pi$
$487$	−5.87940	−0.266421	−0.133210	+	0.991088i	$0.542529\pi$
$488$	0	0
$489$	18.2943	0.827295
$490$	0	0
$491$	4.27022	0.192712	0.0963562	−	0.995347i	$-0.469281\pi$
$491$	4.27022	0.192712	0.0963562	+	0.995347i	$0.469281\pi$
$492$	0	0
$493$	−2.08925	−0.0940952
$494$	0	0
$495$	5.50270	0.247328
$496$	0	0
$497$	−7.37174	−0.330668
$498$	0	0
$499$	29.9583	1.34112	0.670559	−	0.741856i	$-0.266054\pi$
$499$	29.9583	1.34112	0.670559	+	0.741856i	$0.266054\pi$
$500$	0	0
$501$	−19.5543	−0.873622
$502$	0	0
$503$	31.7450	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$503$	31.7450	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$504$	0	0
$505$	9.75245	0.433978
$506$	0	0
$507$	−19.0738	−0.847098
$508$	0	0
$509$	−2.38071	−0.105523	−0.0527616	−	0.998607i	$-0.516802\pi$
$509$	−2.38071	−0.105523	−0.0527616	+	0.998607i	$0.516802\pi$
$510$	0	0
$511$	0.258201	0.0114221
$512$	0	0
$513$	−35.9752	−1.58834
$514$	0	0
$515$	−15.0156	−0.661667
$516$	0	0
$517$	−9.54230	−0.419670
$518$	0	0
$519$	−6.69740	−0.293983
$520$	0	0
$521$	−23.1736	−1.01525	−0.507627	−	0.861577i	$-0.669477\pi$
$521$	−23.1736	−1.01525	−0.507627	+	0.861577i	$0.669477\pi$
$522$	0	0
$523$	−41.7180	−1.82420	−0.912100	−	0.409968i	$-0.865540\pi$
$523$	−41.7180	−1.82420	−0.912100	+	0.409968i	$0.865540\pi$
$524$	0	0
$525$	−0.486719	−0.0212421
$526$	0	0
$527$	4.26311	0.185704
$528$	0	0
$529$	−8.18562	−0.355897
$530$	0	0
$531$	−18.3884	−0.797989
$532$	0	0
$533$	−16.1832	−0.700971
$534$	0	0
$535$	16.6434	0.719559
$536$	0	0
$537$	22.5470	0.972973
$538$	0	0
$539$	19.6660	0.847075
$540$	0	0
$541$	−18.9240	−0.813607	−0.406804	−	0.913516i	$-0.633357\pi$
$541$	−18.9240	−0.813607	−0.406804	+	0.913516i	$0.633357\pi$
$542$	0	0
$543$	−17.3413	−0.744186
$544$	0	0
$545$	5.85990	0.251010
$546$	0	0
$547$	6.66332	0.284903	0.142452	−	0.989802i	$-0.454502\pi$
$547$	6.66332	0.284903	0.142452	+	0.989802i	$0.454502\pi$
$548$	0	0
$549$	9.24457	0.394549
$550$	0	0
$551$	−10.8764	−0.463350
$552$	0	0
$553$	1.96590	0.0835988
$554$	0	0
$555$	−10.2392	−0.434629
$556$	0	0
$557$	−14.7724	−0.625928	−0.312964	−	0.949765i	$-0.601322\pi$
$557$	−14.7724	−0.625928	−0.312964	+	0.949765i	$0.601322\pi$
$558$	0	0
$559$	−12.8117	−0.541876
$560$	0	0
$561$	−4.08901	−0.172638
$562$	0	0
$563$	−29.0495	−1.22429	−0.612146	−	0.790745i	$-0.709694\pi$
$563$	−29.0495	−1.22429	−0.612146	+	0.790745i	$0.709694\pi$
$564$	0	0
$565$	7.99509	0.336356
$566$	0	0
$567$	0.138965	0.00583599
$568$	0	0
$569$	19.8690	0.832954	0.416477	−	0.909146i	$-0.363265\pi$
$569$	19.8690	0.832954	0.416477	+	0.909146i	$0.363265\pi$
$570$	0	0
$571$	22.2943	0.932986	0.466493	−	0.884525i	$-0.345517\pi$
$571$	22.2943	0.932986	0.466493	+	0.884525i	$0.345517\pi$
$572$	0	0
$573$	4.08901	0.170821
$574$	0	0
$575$	3.84894	0.160512
$576$	0	0
$577$	15.1927	0.632482	0.316241	−	0.948679i	$-0.397579\pi$
$577$	15.1927	0.632482	0.316241	+	0.948679i	$0.397579\pi$
$578$	0	0
$579$	19.9801	0.830346
$580$	0	0
$581$	−4.63400	−0.192251
$582$	0	0
$583$	13.4948	0.558899
$584$	0	0
$585$	−10.5998	−0.438249
$586$	0	0
$587$	−30.7051	−1.26734	−0.633668	−	0.773605i	$-0.718451\pi$
$587$	−30.7051	−1.26734	−0.633668	+	0.773605i	$0.718451\pi$
$588$	0	0
$589$	22.1932	0.914456
$590$	0	0
$591$	0.0698796	0.00287446
$592$	0	0
$593$	−21.4757	−0.881902	−0.440951	−	0.897531i	$-0.645359\pi$
$593$	−21.4757	−0.881902	−0.440951	+	0.897531i	$0.645359\pi$
$594$	0	0
$595$	−0.623352	−0.0255549
$596$	0	0
$597$	11.8741	0.485975
$598$	0	0
$599$	26.7025	1.09104	0.545518	−	0.838099i	$-0.316333\pi$
$599$	26.7025	1.09104	0.545518	+	0.838099i	$0.316333\pi$
$600$	0	0
$601$	−7.19053	−0.293308	−0.146654	−	0.989188i	$-0.546850\pi$
$601$	−7.19053	−0.293308	−0.146654	+	0.989188i	$0.546850\pi$
$602$	0	0
$603$	15.9353	0.648934
$604$	0	0
$605$	−2.59882	−0.105657
$606$	0	0
$607$	−5.73930	−0.232951	−0.116475	−	0.993194i	$-0.537160\pi$
$607$	−5.73930	−0.232951	−0.116475	+	0.993194i	$0.537160\pi$
$608$	0	0
$609$	−0.756515	−0.0306555
$610$	0	0
$611$	18.3813	0.743628
$612$	0	0
$613$	−10.7670	−0.434875	−0.217437	−	0.976074i	$-0.569770\pi$
$613$	−10.7670	−0.434875	−0.217437	+	0.976074i	$0.569770\pi$
$614$	0	0
$615$	3.04205	0.122667
$616$	0	0
$617$	5.32012	0.214180	0.107090	−	0.994249i	$-0.465847\pi$
$617$	5.32012	0.214180	0.107090	+	0.994249i	$0.465847\pi$
$618$	0	0
$619$	−30.6564	−1.23219	−0.616093	−	0.787673i	$-0.711285\pi$
$619$	−30.6564	−1.23219	−0.616093	+	0.787673i	$0.711285\pi$
$620$	0	0
$621$	19.7879	0.794060
$622$	0	0
$623$	−8.02597	−0.321554
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−21.2869	−0.850117
$628$	0	0
$629$	−13.1135	−0.522871
$630$	0	0
$631$	−7.43468	−0.295970	−0.147985	−	0.988990i	$-0.547279\pi$
$631$	−7.43468	−0.295970	−0.147985	+	0.988990i	$0.547279\pi$
$632$	0	0
$633$	−0.501855	−0.0199470
$634$	0	0
$635$	−10.9795	−0.435707
$636$	0	0
$637$	−37.8826	−1.50096
$638$	0	0
$639$	30.1783	1.19383
$640$	0	0
$641$	−18.9034	−0.746639	−0.373319	−	0.927703i	$-0.621780\pi$
$641$	−18.9034	−0.746639	−0.373319	+	0.927703i	$0.621780\pi$
$642$	0	0
$643$	−27.5023	−1.08459	−0.542293	−	0.840189i	$-0.682444\pi$
$643$	−27.5023	−1.08459	−0.542293	+	0.840189i	$0.682444\pi$
$644$	0	0
$645$	2.40829	0.0948263
$646$	0	0
$647$	−31.9756	−1.25709	−0.628545	−	0.777773i	$-0.716349\pi$
$647$	−31.9756	−1.25709	−0.628545	+	0.777773i	$0.716349\pi$
$648$	0	0
$649$	−28.0743	−1.10201
$650$	0	0
$651$	1.54366	0.0605010
$652$	0	0
$653$	15.4234	0.603563	0.301781	−	0.953377i	$-0.402419\pi$
$653$	15.4234	0.603563	0.301781	+	0.953377i	$0.402419\pi$
$654$	0	0
$655$	6.21015	0.242651
$656$	0	0
$657$	−1.05702	−0.0412381
$658$	0	0
$659$	−5.96811	−0.232485	−0.116242	−	0.993221i	$-0.537085\pi$
$659$	−5.96811	−0.232485	−0.116242	+	0.993221i	$0.537085\pi$
$660$	0	0
$661$	−16.7464	−0.651358	−0.325679	−	0.945480i	$-0.605593\pi$
$661$	−16.7464	−0.651358	−0.325679	+	0.945480i	$0.605593\pi$
$662$	0	0
$663$	7.87665	0.305904
$664$	0	0
$665$	−3.24509	−0.125839
$666$	0	0
$667$	5.98248	0.231642
$668$	0	0
$669$	20.5760	0.795513
$670$	0	0
$671$	14.1140	0.544866
$672$	0	0
$673$	30.4089	1.17218	0.586088	−	0.810247i	$-0.300667\pi$
$673$	30.4089	1.17218	0.586088	+	0.810247i	$0.300667\pi$
$674$	0	0
$675$	5.14112	0.197882
$676$	0	0
$677$	46.1409	1.77334	0.886669	−	0.462405i	$-0.153013\pi$
$677$	46.1409	1.77334	0.886669	+	0.462405i	$0.153013\pi$
$678$	0	0
$679$	4.19323	0.160921
$680$	0	0
$681$	23.9782	0.918845
$682$	0	0
$683$	−25.1563	−0.962579	−0.481290	−	0.876562i	$-0.659831\pi$
$683$	−25.1563	−0.962579	−0.481290	+	0.876562i	$0.659831\pi$
$684$	0	0
$685$	−21.8960	−0.836604
$686$	0	0
$687$	27.8997	1.06444
$688$	0	0
$689$	−25.9951	−0.990334
$690$	0	0
$691$	44.0863	1.67712	0.838561	−	0.544808i	$-0.183397\pi$
$691$	44.0863	1.67712	0.838561	+	0.544808i	$0.183397\pi$
$692$	0	0
$693$	2.55186	0.0969373
$694$	0	0
$695$	−6.30922	−0.239322
$696$	0	0
$697$	3.89603	0.147572
$698$	0	0
$699$	−18.0942	−0.684384
$700$	0	0
$701$	0.896274	0.0338518	0.0169259	−	0.999857i	$-0.494612\pi$
$701$	0.896274	0.0338518	0.0169259	+	0.999857i	$0.494612\pi$
$702$	0	0
$703$	−68.2675	−2.57476
$704$	0	0
$705$	−3.45525	−0.130132
$706$	0	0
$707$	4.52267	0.170093
$708$	0	0
$709$	15.9836	0.600277	0.300138	−	0.953896i	$-0.402967\pi$
$709$	15.9836	0.600277	0.300138	+	0.953896i	$0.402967\pi$
$710$	0	0
$711$	−8.04798	−0.301823
$712$	0	0
$713$	−12.2072	−0.457164
$714$	0	0
$715$	−16.1832	−0.605216
$716$	0	0
$717$	29.0647	1.08544
$718$	0	0
$719$	19.7820	0.737744	0.368872	−	0.929480i	$-0.379744\pi$
$719$	19.7820	0.737744	0.368872	+	0.929480i	$0.379744\pi$
$720$	0	0
$721$	−6.96345	−0.259332
$722$	0	0
$723$	−9.54720	−0.355064
$724$	0	0
$725$	1.55432	0.0577259
$726$	0	0
$727$	5.00706	0.185702	0.0928508	−	0.995680i	$-0.470402\pi$
$727$	5.00706	0.185702	0.0928508	+	0.995680i	$0.470402\pi$
$728$	0	0
$729$	15.6184	0.578461
$730$	0	0
$731$	3.08435	0.114079
$732$	0	0
$733$	2.15219	0.0794931	0.0397465	−	0.999210i	$-0.487345\pi$
$733$	2.15219	0.0794931	0.0397465	+	0.999210i	$0.487345\pi$
$734$	0	0
$735$	7.12102	0.262663
$736$	0	0
$737$	24.3289	0.896168
$738$	0	0
$739$	21.2749	0.782609	0.391305	−	0.920261i	$-0.372024\pi$
$739$	21.2749	0.782609	0.391305	+	0.920261i	$0.372024\pi$
$740$	0	0
$741$	41.0049	1.50635
$742$	0	0
$743$	40.7480	1.49490	0.747450	−	0.664318i	$-0.231278\pi$
$743$	40.7480	1.49490	0.747450	+	0.664318i	$0.231278\pi$
$744$	0	0
$745$	10.3862	0.380521
$746$	0	0
$747$	18.9706	0.694097
$748$	0	0
$749$	7.71836	0.282023
$750$	0	0
$751$	20.8596	0.761179	0.380590	−	0.924744i	$-0.375721\pi$
$751$	20.8596	0.761179	0.380590	+	0.924744i	$0.375721\pi$
$752$	0	0
$753$	−12.8150	−0.467004
$754$	0	0
$755$	−17.0266	−0.619660
$756$	0	0
$757$	19.9492	0.725064	0.362532	−	0.931971i	$-0.381912\pi$
$757$	19.9492	0.725064	0.362532	+	0.931971i	$0.381912\pi$
$758$	0	0
$759$	11.7087	0.424998
$760$	0	0
$761$	13.7778	0.499446	0.249723	−	0.968317i	$-0.419660\pi$
$761$	13.7778	0.499446	0.249723	+	0.968317i	$0.419660\pi$
$762$	0	0
$763$	2.71751	0.0983806
$764$	0	0
$765$	2.55186	0.0922628
$766$	0	0
$767$	54.0794	1.95270
$768$	0	0
$769$	−6.89137	−0.248509	−0.124255	−	0.992250i	$-0.539654\pi$
$769$	−6.89137	−0.248509	−0.124255	+	0.992250i	$0.539654\pi$
$770$	0	0
$771$	24.2072	0.871801
$772$	0	0
$773$	16.2355	0.583951	0.291976	−	0.956426i	$-0.405687\pi$
$773$	16.2355	0.583951	0.291976	+	0.956426i	$0.405687\pi$
$774$	0	0
$775$	−3.17157	−0.113926
$776$	0	0
$777$	−4.74839	−0.170348
$778$	0	0
$779$	20.2822	0.726687
$780$	0	0
$781$	46.0743	1.64867
$782$	0	0
$783$	7.99093	0.285572
$784$	0	0
$785$	5.45381	0.194655
$786$	0	0
$787$	21.2993	0.759238	0.379619	−	0.925143i	$-0.376055\pi$
$787$	21.2993	0.759238	0.379619	+	0.925143i	$0.376055\pi$
$788$	0	0
$789$	−22.2326	−0.791501
$790$	0	0
$791$	3.70770	0.131831
$792$	0	0
$793$	−27.1878	−0.965468
$794$	0	0
$795$	4.88646	0.173305
$796$	0	0
$797$	14.7159	0.521264	0.260632	−	0.965438i	$-0.416069\pi$
$797$	14.7159	0.521264	0.260632	+	0.965438i	$0.416069\pi$
$798$	0	0
$799$	−4.42522	−0.156553
$800$	0	0
$801$	32.8565	1.16093
$802$	0	0
$803$	−1.61379	−0.0569493
$804$	0	0
$805$	1.78494	0.0629108
$806$	0	0
$807$	9.43062	0.331973
$808$	0	0
$809$	−17.7970	−0.625708	−0.312854	−	0.949801i	$-0.601285\pi$
$809$	−17.7970	−0.625708	−0.312854	+	0.949801i	$0.601285\pi$
$810$	0	0
$811$	3.01742	0.105956	0.0529779	−	0.998596i	$-0.483129\pi$
$811$	3.01742	0.105956	0.0529779	+	0.998596i	$0.483129\pi$
$812$	0	0
$813$	−11.3597	−0.398400
$814$	0	0
$815$	−17.4308	−0.610576
$816$	0	0
$817$	16.0568	0.561755
$818$	0	0
$819$	−4.91565	−0.171767
$820$	0	0
$821$	38.6184	1.34779	0.673897	−	0.738826i	$-0.264619\pi$
$821$	38.6184	1.34779	0.673897	+	0.738826i	$0.264619\pi$
$822$	0	0
$823$	41.1720	1.43516	0.717582	−	0.696474i	$-0.245249\pi$
$823$	41.1720	1.43516	0.717582	+	0.696474i	$0.245249\pi$
$824$	0	0
$825$	3.04205	0.105911
$826$	0	0
$827$	51.3645	1.78612	0.893059	−	0.449939i	$-0.148554\pi$
$827$	51.3645	1.78612	0.893059	+	0.449939i	$0.148554\pi$
$828$	0	0
$829$	33.7412	1.17188	0.585940	−	0.810354i	$-0.300725\pi$
$829$	33.7412	1.17188	0.585940	+	0.810354i	$0.300725\pi$
$830$	0	0
$831$	24.5013	0.849940
$832$	0	0
$833$	9.12006	0.315991
$834$	0	0
$835$	18.6314	0.644767
$836$	0	0
$837$	−16.3054	−0.563598
$838$	0	0
$839$	17.7869	0.614072	0.307036	−	0.951698i	$-0.400663\pi$
$839$	17.7869	0.614072	0.307036	+	0.951698i	$0.400663\pi$
$840$	0	0
$841$	−26.5841	−0.916693
$842$	0	0
$843$	−11.1318	−0.383400
$844$	0	0
$845$	18.1736	0.625191
$846$	0	0
$847$	−1.20520	−0.0414111
$848$	0	0
$849$	4.15619	0.142640
$850$	0	0
$851$	37.5500	1.28720
$852$	0	0
$853$	6.12952	0.209871	0.104935	−	0.994479i	$-0.466536\pi$
$853$	6.12952	0.209871	0.104935	+	0.994479i	$0.466536\pi$
$854$	0	0
$855$	13.2847	0.454327
$856$	0	0
$857$	−39.7072	−1.35637	−0.678186	−	0.734890i	$-0.737234\pi$
$857$	−39.7072	−1.35637	−0.678186	+	0.734890i	$0.737234\pi$
$858$	0	0
$859$	−4.88351	−0.166623	−0.0833117	−	0.996524i	$-0.526550\pi$
$859$	−4.88351	−0.166623	−0.0833117	+	0.996524i	$0.526550\pi$
$860$	0	0
$861$	1.41074	0.0480780
$862$	0	0
$863$	45.1975	1.53854	0.769271	−	0.638923i	$-0.220620\pi$
$863$	45.1975	1.53854	0.769271	+	0.638923i	$0.220620\pi$
$864$	0	0
$865$	6.38131	0.216971
$866$	0	0
$867$	15.9458	0.541548
$868$	0	0
$869$	−12.2871	−0.416813
$870$	0	0
$871$	−46.8648	−1.58795
$872$	0	0
$873$	−17.1661	−0.580986
$874$	0	0
$875$	0.463747	0.0156775
$876$	0	0
$877$	−20.3305	−0.686511	−0.343255	−	0.939242i	$-0.611530\pi$
$877$	−20.3305	−0.686511	−0.343255	+	0.939242i	$0.611530\pi$
$878$	0	0
$879$	20.9855	0.707824
$880$	0	0
$881$	12.2102	0.411371	0.205685	−	0.978618i	$-0.434058\pi$
$881$	12.2102	0.411371	0.205685	+	0.978618i	$0.434058\pi$
$882$	0	0
$883$	26.4164	0.888982	0.444491	−	0.895783i	$-0.353385\pi$
$883$	26.4164	0.888982	0.444491	+	0.895783i	$0.353385\pi$
$884$	0	0
$885$	−10.1656	−0.341715
$886$	0	0
$887$	42.4950	1.42684	0.713421	−	0.700736i	$-0.247145\pi$
$887$	42.4950	1.42684	0.713421	+	0.700736i	$0.247145\pi$
$888$	0	0
$889$	−5.09171	−0.170770
$890$	0	0
$891$	−0.868549	−0.0290975
$892$	0	0
$893$	−23.0371	−0.770909
$894$	0	0
$895$	−21.4828	−0.718092
$896$	0	0
$897$	−22.5544	−0.753070
$898$	0	0
$899$	−4.92963	−0.164412
$900$	0	0
$901$	6.25820	0.208491
$902$	0	0
$903$	1.11684	0.0371661
$904$	0	0
$905$	16.5228	0.549238
$906$	0	0
$907$	5.97009	0.198234	0.0991168	−	0.995076i	$-0.468398\pi$
$907$	5.97009	0.198234	0.0991168	+	0.995076i	$0.468398\pi$
$908$	0	0
$909$	−18.5148	−0.614098
$910$	0	0
$911$	33.0306	1.09435	0.547177	−	0.837017i	$-0.315703\pi$
$911$	33.0306	1.09435	0.547177	+	0.837017i	$0.315703\pi$
$912$	0	0
$913$	28.9631	0.958539
$914$	0	0
$915$	5.11067	0.168953
$916$	0	0
$917$	2.87994	0.0951041
$918$	0	0
$919$	−18.3501	−0.605313	−0.302657	−	0.953100i	$-0.597874\pi$
$919$	−18.3501	−0.605313	−0.302657	+	0.953100i	$0.597874\pi$
$920$	0	0
$921$	−32.1954	−1.06088
$922$	0	0
$923$	−88.7528	−2.92133
$924$	0	0
$925$	9.75592	0.320773
$926$	0	0
$927$	28.5068	0.936287
$928$	0	0
$929$	50.7648	1.66554	0.832770	−	0.553619i	$-0.186754\pi$
$929$	50.7648	1.66554	0.832770	+	0.553619i	$0.186754\pi$
$930$	0	0
$931$	47.4779	1.55603
$932$	0	0
$933$	0.514470	0.0168430
$934$	0	0
$935$	3.89603	0.127414
$936$	0	0
$937$	57.2804	1.87127	0.935634	−	0.352971i	$-0.114829\pi$
$937$	57.2804	1.87127	0.935634	+	0.352971i	$0.114829\pi$
$938$	0	0
$939$	23.6979	0.773351
$940$	0	0
$941$	−59.3824	−1.93581	−0.967905	−	0.251315i	$-0.919137\pi$
$941$	−59.3824	−1.93581	−0.967905	+	0.251315i	$0.919137\pi$
$942$	0	0
$943$	−11.1561	−0.363292
$944$	0	0
$945$	2.38418	0.0775574
$946$	0	0
$947$	−15.2152	−0.494427	−0.247214	−	0.968961i	$-0.579515\pi$
$947$	−15.2152	−0.494427	−0.247214	+	0.968961i	$0.579515\pi$
$948$	0	0
$949$	3.10863	0.100911
$950$	0	0
$951$	−27.6540	−0.896742
$952$	0	0
$953$	33.6640	1.09049	0.545243	−	0.838278i	$-0.316438\pi$
$953$	33.6640	1.09049	0.545243	+	0.838278i	$0.316438\pi$
$954$	0	0
$955$	−3.89603	−0.126072
$956$	0	0
$957$	4.72831	0.152845
$958$	0	0
$959$	−10.1542	−0.327897
$960$	0	0
$961$	−20.9411	−0.675520
$962$	0	0
$963$	−31.5972	−1.01821
$964$	0	0
$965$	−19.0371	−0.612827
$966$	0	0
$967$	12.6075	0.405430	0.202715	−	0.979238i	$-0.435024\pi$
$967$	12.6075	0.405430	0.202715	+	0.979238i	$0.435024\pi$
$968$	0	0
$969$	−9.87174	−0.317126
$970$	0	0
$971$	−0.567180	−0.0182017	−0.00910084	−	0.999959i	$-0.502897\pi$
$971$	−0.567180	−0.0182017	−0.00910084	+	0.999959i	$0.502897\pi$
$972$	0	0
$973$	−2.92589	−0.0937996
$974$	0	0
$975$	−5.85990	−0.187667
$976$	0	0
$977$	21.1220	0.675752	0.337876	−	0.941191i	$-0.390292\pi$
$977$	21.1220	0.675752	0.337876	+	0.941191i	$0.390292\pi$
$978$	0	0
$979$	50.1633	1.60323
$980$	0	0
$981$	−11.1249	−0.355190
$982$	0	0
$983$	35.2583	1.12457	0.562283	−	0.826945i	$-0.309923\pi$
$983$	35.2583	1.12457	0.562283	+	0.826945i	$0.309923\pi$
$984$	0	0
$985$	−0.0665816	−0.00212147
$986$	0	0
$987$	−1.60236	−0.0510038
$988$	0	0
$989$	−8.83190	−0.280838
$990$	0	0
$991$	−11.9471	−0.379513	−0.189756	−	0.981831i	$-0.560770\pi$
$991$	−11.9471	−0.379513	−0.189756	+	0.981831i	$0.560770\pi$
$992$	0	0
$993$	−26.3397	−0.835867
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	−10.5929	−0.335480	−0.167740	−	0.985831i	$-0.553647\pi$
$997$	−10.5929	−0.335480	−0.167740	+	0.985831i	$0.553647\pi$
$998$	0	0
$999$	50.1564	1.58688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.j.1.2		4
4.3	odd	2		5120.2.a.p.1.3		4
8.3	odd	2		5120.2.a.i.1.2		4
8.5	even	2		5120.2.a.c.1.3		4
32.3	odd	8		1280.2.l.c.321.3	yes	8
32.5	even	8		1280.2.l.b.961.3	yes	8
32.11	odd	8		1280.2.l.c.961.3	yes	8
32.13	even	8		1280.2.l.b.321.3	✓	8
32.19	odd	8		1280.2.l.e.321.2	yes	8
32.21	even	8		1280.2.l.h.961.2	yes	8
32.27	odd	8		1280.2.l.e.961.2	yes	8
32.29	even	8		1280.2.l.h.321.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1280.2.l.b.321.3	✓	8	32.13	even	8
1280.2.l.b.961.3	yes	8	32.5	even	8
1280.2.l.c.321.3	yes	8	32.3	odd	8
1280.2.l.c.961.3	yes	8	32.11	odd	8
1280.2.l.e.321.2	yes	8	32.19	odd	8
1280.2.l.e.961.2	yes	8	32.27	odd	8
1280.2.l.h.321.2	yes	8	32.29	even	8
1280.2.l.h.961.2	yes	8	32.21	even	8
5120.2.a.c.1.3		4	8.5	even	2
5120.2.a.i.1.2		4	8.3	odd	2
5120.2.a.j.1.2		4	1.1	even	1	trivial
5120.2.a.p.1.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q-1.04953 q^{3} +1.00000 q^{5} +0.463747 q^{7} -1.89848 q^{9} -2.89848 q^{11} +5.58333 q^{13} -1.04953 q^{15} -1.34416 q^{17} -6.99755 q^{19} -0.486719 q^{21} +3.84894 q^{23} +1.00000 q^{25} +5.14112 q^{27} +1.55432 q^{29} -3.17157 q^{31} +3.04205 q^{33} +0.463747 q^{35} +9.75592 q^{37} -5.85990 q^{39} -2.89848 q^{41} -2.29463 q^{43} -1.89848 q^{45} +3.29217 q^{47} -6.78494 q^{49} +1.41074 q^{51} -4.65584 q^{53} -2.89848 q^{55} +7.34416 q^{57} +9.68587 q^{59} -4.86946 q^{61} -0.880415 q^{63} +5.58333 q^{65} -8.39370 q^{67} -4.03960 q^{69} -15.8960 q^{71} +0.556770 q^{73} -1.04953 q^{75} -1.34416 q^{77} +4.23917 q^{79} +0.299657 q^{81} -9.99252 q^{83} -1.34416 q^{85} -1.63131 q^{87} -17.3068 q^{89} +2.58926 q^{91} +3.32867 q^{93} -6.99755 q^{95} +9.04205 q^{97} +5.50270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 8 q^{7} + 8 q^{9} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 4 q^{19} - 16 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} - 8 q^{29} - 24 q^{31} - 12 q^{33} - 8 q^{35} + 8 q^{37} - 32 q^{39} + 4 q^{41}+ \cdots + 52 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 - 8 * q^7 + 8 * q^9 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 4 * q^19 - 16 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 - 8 * q^29 - 24 * q^31 - 12 * q^33 - 8 * q^35 + 8 * q^37 - 32 * q^39 + 4 * q^41 - 12 * q^43 + 8 * q^45 - 8 * q^47 + 8 * q^49 - 20 * q^53 + 4 * q^55 + 28 * q^57 + 12 * q^59 - 8 * q^61 - 12 * q^63 + 4 * q^65 - 28 * q^67 + 32 * q^69 - 24 * q^71 + 12 * q^73 - 4 * q^77 - 20 * q^83 - 4 * q^85 + 12 * q^87 - 24 * q^89 + 16 * q^91 + 8 * q^93 - 4 * q^95 + 12 * q^97 + 52 * q^99

Introduction

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