LMFDB - Embedded newform 1024.4.a.m.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1024,4,Mod(1,1024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1024.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1024, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,-28,0,54,0,0,0,0,0,-124,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.4179558459$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 36x^{8} + 405x^{6} - 1380x^{4} + 420x^{2} - 32 $ x^10 - 36x^8 + 405x^6 - 1380x^4 + 420x^2 - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.446984$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.62644 q^{3} +17.8826 q^{5} -13.8754 q^{7} -5.59607 q^{9} -2.18771 q^{11} -46.3685 q^{13} +82.7326 q^{15} -18.6531 q^{17} -122.194 q^{19} -64.1936 q^{21} -134.006 q^{23} +194.786 q^{25} -150.804 q^{27} -84.5628 q^{29} +31.5391 q^{31} -10.1213 q^{33} -248.128 q^{35} -126.129 q^{37} -214.521 q^{39} +210.504 q^{41} -168.860 q^{43} -100.072 q^{45} -182.902 q^{47} -150.474 q^{49} -86.2972 q^{51} -37.0020 q^{53} -39.1219 q^{55} -565.323 q^{57} +624.494 q^{59} -246.759 q^{61} +77.6476 q^{63} -829.188 q^{65} -129.763 q^{67} -619.970 q^{69} -348.360 q^{71} +299.436 q^{73} +901.168 q^{75} +30.3553 q^{77} -943.487 q^{79} -546.590 q^{81} +443.034 q^{83} -333.565 q^{85} -391.224 q^{87} +1412.35 q^{89} +643.381 q^{91} +145.914 q^{93} -2185.14 q^{95} +1515.29 q^{97} +12.2426 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 28 q^{7} + 54 q^{9} - 124 q^{15} + 4 q^{17} - 276 q^{23} + 50 q^{25} - 368 q^{31} - 4 q^{33} - 732 q^{39} - 944 q^{47} - 94 q^{49} - 1380 q^{55} + 108 q^{57} - 2628 q^{63} - 492 q^{65} - 3468 q^{71}+ \cdots - 4 q^{97}+O(q^{100}) $ 10 * q - 28 * q^7 + 54 * q^9 - 124 * q^15 + 4 * q^17 - 276 * q^23 + 50 * q^25 - 368 * q^31 - 4 * q^33 - 732 * q^39 - 944 * q^47 - 94 * q^49 - 1380 * q^55 + 108 * q^57 - 2628 * q^63 - 492 * q^65 - 3468 * q^71 - 296 * q^73 - 4416 * q^79 - 482 * q^81 - 6036 * q^87 + 88 * q^89 - 6900 * q^95 - 4 * q^97

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
$3$	4.62644	0.890358	0.445179	−	0.895441i	$-0.353140\pi$
$3$	4.62644	0.890358	0.445179	+	0.895441i	$0.353140\pi$
$4$	0	0
$5$	17.8826	1.59947	0.799733	−	0.600356i	$-0.204974\pi$
$5$	17.8826	1.59947	0.799733	+	0.600356i	$0.204974\pi$
$6$	0	0
$7$	−13.8754	−0.749200	−0.374600	−	0.927187i	$-0.622220\pi$
$7$	−13.8754	−0.749200	−0.374600	+	0.927187i	$0.622220\pi$
$8$	0	0
$9$	−5.59607	−0.207262
$10$	0	0
$11$	−2.18771	−0.0599653	−0.0299827	−	0.999550i	$-0.509545\pi$
$11$	−2.18771	−0.0599653	−0.0299827	+	0.999550i	$0.509545\pi$
$12$	0	0
$13$	−46.3685	−0.989255	−0.494627	−	0.869105i	$-0.664695\pi$
$13$	−46.3685	−0.989255	−0.494627	+	0.869105i	$0.664695\pi$
$14$	0	0
$15$	82.7326	1.42410
$16$	0	0
$17$	−18.6531	−0.266119	−0.133060	−	0.991108i	$-0.542480\pi$
$17$	−18.6531	−0.266119	−0.133060	+	0.991108i	$0.542480\pi$
$18$	0	0
$19$	−122.194	−1.47543	−0.737716	−	0.675111i	$-0.764096\pi$
$19$	−122.194	−1.47543	−0.737716	+	0.675111i	$0.764096\pi$
$20$	0	0
$21$	−64.1936	−0.667057
$22$	0	0
$23$	−134.006	−1.21488	−0.607438	−	0.794367i	$-0.707803\pi$
$23$	−134.006	−1.21488	−0.607438	+	0.794367i	$0.707803\pi$
$24$	0	0
$25$	194.786	1.55829
$26$	0	0
$27$	−150.804	−1.07490
$28$	0	0
$29$	−84.5628	−0.541480	−0.270740	−	0.962653i	$-0.587268\pi$
$29$	−84.5628	−0.541480	−0.270740	+	0.962653i	$0.587268\pi$
$30$	0	0
$31$	31.5391	0.182729	0.0913645	−	0.995818i	$-0.470877\pi$
$31$	31.5391	0.182729	0.0913645	+	0.995818i	$0.470877\pi$
$32$	0	0
$33$	−10.1213	−0.0533906
$34$	0	0
$35$	−248.128	−1.19832
$36$	0	0
$37$	−126.129	−0.560418	−0.280209	−	0.959939i	$-0.590404\pi$
$37$	−126.129	−0.560418	−0.280209	+	0.959939i	$0.590404\pi$
$38$	0	0
$39$	−214.521	−0.880791
$40$	0	0
$41$	210.504	0.801834	0.400917	−	0.916114i	$-0.368692\pi$
$41$	210.504	0.801834	0.400917	+	0.916114i	$0.368692\pi$
$42$	0	0
$43$	−168.860	−0.598857	−0.299429	−	0.954119i	$-0.596796\pi$
$43$	−168.860	−0.598857	−0.299429	+	0.954119i	$0.596796\pi$
$44$	0	0
$45$	−100.072	−0.331508
$46$	0	0
$47$	−182.902	−0.567638	−0.283819	−	0.958878i	$-0.591602\pi$
$47$	−182.902	−0.567638	−0.283819	+	0.958878i	$0.591602\pi$
$48$	0	0
$49$	−150.474	−0.438699
$50$	0	0
$51$	−86.2972	−0.236942
$52$	0	0
$53$	−37.0020	−0.0958984	−0.0479492	−	0.998850i	$-0.515269\pi$
$53$	−37.0020	−0.0958984	−0.0479492	+	0.998850i	$0.515269\pi$
$54$	0	0
$55$	−39.1219	−0.0959125
$56$	0	0
$57$	−565.323	−1.31366
$58$	0	0
$59$	624.494	1.37800	0.689001	−	0.724760i	$-0.258050\pi$
$59$	624.494	1.37800	0.689001	+	0.724760i	$0.258050\pi$
$60$	0	0
$61$	−246.759	−0.517938	−0.258969	−	0.965886i	$-0.583383\pi$
$61$	−246.759	−0.517938	−0.258969	+	0.965886i	$0.583383\pi$
$62$	0	0
$63$	77.6476	0.155281
$64$	0	0
$65$	−829.188	−1.58228
$66$	0	0
$67$	−129.763	−0.236613	−0.118306	−	0.992977i	$-0.537747\pi$
$67$	−129.763	−0.236613	−0.118306	+	0.992977i	$0.537747\pi$
$68$	0	0
$69$	−619.970	−1.08168
$70$	0	0
$71$	−348.360	−0.582291	−0.291146	−	0.956679i	$-0.594036\pi$
$71$	−348.360	−0.582291	−0.291146	+	0.956679i	$0.594036\pi$
$72$	0	0
$73$	299.436	0.480087	0.240043	−	0.970762i	$-0.422838\pi$
$73$	299.436	0.480087	0.240043	+	0.970762i	$0.422838\pi$
$74$	0	0
$75$	901.168	1.38744
$76$	0	0
$77$	30.3553	0.0449260
$78$	0	0
$79$	−943.487	−1.34368	−0.671839	−	0.740697i	$-0.734495\pi$
$79$	−943.487	−1.34368	−0.671839	+	0.740697i	$0.734495\pi$
$80$	0	0
$81$	−546.590	−0.749781
$82$	0	0
$83$	443.034	0.585895	0.292947	−	0.956129i	$-0.405364\pi$
$83$	443.034	0.585895	0.292947	+	0.956129i	$0.405364\pi$
$84$	0	0
$85$	−333.565	−0.425649
$86$	0	0
$87$	−391.224	−0.482111
$88$	0	0
$89$	1412.35	1.68212	0.841060	−	0.540942i	$-0.181932\pi$
$89$	1412.35	1.68212	0.841060	+	0.540942i	$0.181932\pi$
$90$	0	0
$91$	643.381	0.741150
$92$	0	0
$93$	145.914	0.162694
$94$	0	0
$95$	−2185.14	−2.35990
$96$	0	0
$97$	1515.29	1.58613	0.793063	−	0.609140i	$-0.208485\pi$
$97$	1515.29	1.58613	0.793063	+	0.609140i	$0.208485\pi$
$98$	0	0
$99$	12.2426	0.0124285
$100$	0	0
$101$	810.631	0.798621	0.399311	−	0.916816i	$-0.369249\pi$
$101$	810.631	0.798621	0.399311	+	0.916816i	$0.369249\pi$
$102$	0	0
$103$	1021.00	0.976717	0.488359	−	0.872643i	$-0.337596\pi$
$103$	1021.00	0.976717	0.488359	+	0.872643i	$0.337596\pi$
$104$	0	0
$105$	−1147.95	−1.06693
$106$	0	0
$107$	1754.75	1.58540	0.792700	−	0.609612i	$-0.208675\pi$
$107$	1754.75	1.58540	0.792700	+	0.609612i	$0.208675\pi$
$108$	0	0
$109$	153.624	0.134996	0.0674980	−	0.997719i	$-0.478498\pi$
$109$	153.624	0.134996	0.0674980	+	0.997719i	$0.478498\pi$
$110$	0	0
$111$	−583.528	−0.498973
$112$	0	0
$113$	1722.22	1.43374	0.716870	−	0.697207i	$-0.245574\pi$
$113$	1722.22	1.43374	0.716870	+	0.697207i	$0.245574\pi$
$114$	0	0
$115$	−2396.37	−1.94315
$116$	0	0
$117$	259.481	0.205035
$118$	0	0
$119$	258.818	0.199377
$120$	0	0
$121$	−1326.21	−0.996404
$122$	0	0
$123$	973.883	0.713919
$124$	0	0
$125$	1247.96	0.892969
$126$	0	0
$127$	−699.127	−0.488484	−0.244242	−	0.969714i	$-0.578539\pi$
$127$	−699.127	−0.488484	−0.244242	+	0.969714i	$0.578539\pi$
$128$	0	0
$129$	−781.219	−0.533198
$130$	0	0
$131$	−279.972	−0.186727	−0.0933636	−	0.995632i	$-0.529762\pi$
$131$	−279.972	−0.186727	−0.0933636	+	0.995632i	$0.529762\pi$
$132$	0	0
$133$	1695.49	1.10539
$134$	0	0
$135$	−2696.76	−1.71926
$136$	0	0
$137$	271.386	0.169242	0.0846209	−	0.996413i	$-0.473032\pi$
$137$	271.386	0.169242	0.0846209	+	0.996413i	$0.473032\pi$
$138$	0	0
$139$	−650.449	−0.396909	−0.198455	−	0.980110i	$-0.563592\pi$
$139$	−650.449	−0.396909	−0.198455	+	0.980110i	$0.563592\pi$
$140$	0	0
$141$	−846.185	−0.505402
$142$	0	0
$143$	101.441	0.0593210
$144$	0	0
$145$	−1512.20	−0.866079
$146$	0	0
$147$	−696.158	−0.390600
$148$	0	0
$149$	856.692	0.471026	0.235513	−	0.971871i	$-0.424323\pi$
$149$	856.692	0.471026	0.235513	+	0.971871i	$0.424323\pi$
$150$	0	0
$151$	−3534.47	−1.90484	−0.952421	−	0.304785i	$-0.901415\pi$
$151$	−3534.47	−1.90484	−0.952421	+	0.304785i	$0.901415\pi$
$152$	0	0
$153$	104.384	0.0551564
$154$	0	0
$155$	564.001	0.292269
$156$	0	0
$157$	−1744.48	−0.886784	−0.443392	−	0.896328i	$-0.646225\pi$
$157$	−1744.48	−0.886784	−0.443392	+	0.896328i	$0.646225\pi$
$158$	0	0
$159$	−171.187	−0.0853839
$160$	0	0
$161$	1859.38	0.910186
$162$	0	0
$163$	−3633.63	−1.74606	−0.873030	−	0.487666i	$-0.837848\pi$
$163$	−3633.63	−1.74606	−0.873030	+	0.487666i	$0.837848\pi$
$164$	0	0
$165$	−180.995	−0.0853965
$166$	0	0
$167$	−3048.39	−1.41252	−0.706262	−	0.707950i	$-0.749620\pi$
$167$	−3048.39	−1.41252	−0.706262	+	0.707950i	$0.749620\pi$
$168$	0	0
$169$	−46.9618	−0.0213754
$170$	0	0
$171$	683.806	0.305801
$172$	0	0
$173$	2152.51	0.945965	0.472983	−	0.881072i	$-0.343177\pi$
$173$	2152.51	0.945965	0.472983	+	0.881072i	$0.343177\pi$
$174$	0	0
$175$	−2702.74	−1.16747
$176$	0	0
$177$	2889.18	1.22692
$178$	0	0
$179$	427.457	0.178489	0.0892447	−	0.996010i	$-0.471555\pi$
$179$	427.457	0.178489	0.0892447	+	0.996010i	$0.471555\pi$
$180$	0	0
$181$	2399.83	0.985515	0.492758	−	0.870167i	$-0.335989\pi$
$181$	2399.83	0.985515	0.492758	+	0.870167i	$0.335989\pi$
$182$	0	0
$183$	−1141.62	−0.461151
$184$	0	0
$185$	−2255.51	−0.896370
$186$	0	0
$187$	40.8074	0.0159579
$188$	0	0
$189$	2092.46	0.805312
$190$	0	0
$191$	4035.31	1.52872	0.764358	−	0.644792i	$-0.223056\pi$
$191$	4035.31	1.52872	0.764358	+	0.644792i	$0.223056\pi$
$192$	0	0
$193$	−886.172	−0.330508	−0.165254	−	0.986251i	$-0.552844\pi$
$193$	−886.172	−0.330508	−0.165254	+	0.986251i	$0.552844\pi$
$194$	0	0
$195$	−3836.19	−1.40880
$196$	0	0
$197$	−4624.65	−1.67255	−0.836276	−	0.548308i	$-0.815272\pi$
$197$	−4624.65	−1.67255	−0.836276	+	0.548308i	$0.815272\pi$
$198$	0	0
$199$	−222.513	−0.0792639	−0.0396319	−	0.999214i	$-0.512619\pi$
$199$	−222.513	−0.0792639	−0.0396319	+	0.999214i	$0.512619\pi$
$200$	0	0
$201$	−600.340	−0.210670
$202$	0	0
$203$	1173.34	0.405677
$204$	0	0
$205$	3764.35	1.28251
$206$	0	0
$207$	749.907	0.251798
$208$	0	0
$209$	267.325	0.0884748
$210$	0	0
$211$	−4988.58	−1.62762	−0.813810	−	0.581131i	$-0.802610\pi$
$211$	−4988.58	−1.62762	−0.813810	+	0.581131i	$0.802610\pi$
$212$	0	0
$213$	−1611.66	−0.518448
$214$	0	0
$215$	−3019.65	−0.957852
$216$	0	0
$217$	−437.618	−0.136901
$218$	0	0
$219$	1385.32	0.427449
$220$	0	0
$221$	864.914	0.263260
$222$	0	0
$223$	−5841.90	−1.75427	−0.877136	−	0.480242i	$-0.840549\pi$
$223$	−5841.90	−1.75427	−0.877136	+	0.480242i	$0.840549\pi$
$224$	0	0
$225$	−1090.04	−0.322975
$226$	0	0
$227$	1597.41	0.467065	0.233533	−	0.972349i	$-0.424971\pi$
$227$	1597.41	0.467065	0.233533	+	0.972349i	$0.424971\pi$
$228$	0	0
$229$	4108.86	1.18568	0.592840	−	0.805320i	$-0.298006\pi$
$229$	4108.86	1.18568	0.592840	+	0.805320i	$0.298006\pi$
$230$	0	0
$231$	140.437	0.0400003
$232$	0	0
$233$	734.054	0.206393	0.103196	−	0.994661i	$-0.467093\pi$
$233$	734.054	0.206393	0.103196	+	0.994661i	$0.467093\pi$
$234$	0	0
$235$	−3270.76	−0.907918
$236$	0	0
$237$	−4364.98	−1.19635
$238$	0	0
$239$	−511.807	−0.138519	−0.0692595	−	0.997599i	$-0.522064\pi$
$239$	−511.807	−0.138519	−0.0692595	+	0.997599i	$0.522064\pi$
$240$	0	0
$241$	−5920.31	−1.58241	−0.791204	−	0.611552i	$-0.790545\pi$
$241$	−5920.31	−1.58241	−0.791204	+	0.611552i	$0.790545\pi$
$242$	0	0
$243$	1542.93	0.407322
$244$	0	0
$245$	−2690.86	−0.701685
$246$	0	0
$247$	5665.95	1.45958
$248$	0	0
$249$	2049.67	0.521656
$250$	0	0
$251$	−437.462	−0.110009	−0.0550046	−	0.998486i	$-0.517517\pi$
$251$	−437.462	−0.110009	−0.0550046	+	0.998486i	$0.517517\pi$
$252$	0	0
$253$	293.166	0.0728505
$254$	0	0
$255$	−1543.22	−0.378980
$256$	0	0
$257$	323.723	0.0785730	0.0392865	−	0.999228i	$-0.487491\pi$
$257$	323.723	0.0785730	0.0392865	+	0.999228i	$0.487491\pi$
$258$	0	0
$259$	1750.09	0.419865
$260$	0	0
$261$	473.219	0.112228
$262$	0	0
$263$	−2689.15	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$263$	−2689.15	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$264$	0	0
$265$	−661.691	−0.153386
$266$	0	0
$267$	6534.14	1.49769
$268$	0	0
$269$	−6652.15	−1.50776	−0.753882	−	0.657009i	$-0.771821\pi$
$269$	−6652.15	−1.50776	−0.753882	+	0.657009i	$0.771821\pi$
$270$	0	0
$271$	−2018.97	−0.452561	−0.226280	−	0.974062i	$-0.572657\pi$
$271$	−2018.97	−0.452561	−0.226280	+	0.974062i	$0.572657\pi$
$272$	0	0
$273$	2976.56	0.659889
$274$	0	0
$275$	−426.136	−0.0934435
$276$	0	0
$277$	−4356.63	−0.944999	−0.472499	−	0.881331i	$-0.656648\pi$
$277$	−4356.63	−0.944999	−0.472499	+	0.881331i	$0.656648\pi$
$278$	0	0
$279$	−176.495	−0.0378728
$280$	0	0
$281$	3893.51	0.826575	0.413287	−	0.910601i	$-0.364381\pi$
$281$	3893.51	0.826575	0.413287	+	0.910601i	$0.364381\pi$
$282$	0	0
$283$	2865.74	0.601945	0.300972	−	0.953633i	$-0.402689\pi$
$283$	2865.74	0.601945	0.300972	+	0.953633i	$0.402689\pi$
$284$	0	0
$285$	−10109.4	−2.10116
$286$	0	0
$287$	−2920.82	−0.600734
$288$	0	0
$289$	−4565.06	−0.929180
$290$	0	0
$291$	7010.39	1.41222
$292$	0	0
$293$	1416.60	0.282452	0.141226	−	0.989977i	$-0.454896\pi$
$293$	1416.60	0.282452	0.141226	+	0.989977i	$0.454896\pi$
$294$	0	0
$295$	11167.6	2.20407
$296$	0	0
$297$	329.914	0.0644565
$298$	0	0
$299$	6213.65	1.20182
$300$	0	0
$301$	2342.99	0.448664
$302$	0	0
$303$	3750.33	0.711059
$304$	0	0
$305$	−4412.69	−0.828425
$306$	0	0
$307$	4195.32	0.779934	0.389967	−	0.920829i	$-0.372486\pi$
$307$	4195.32	0.779934	0.389967	+	0.920829i	$0.372486\pi$
$308$	0	0
$309$	4723.58	0.869628
$310$	0	0
$311$	−2911.18	−0.530797	−0.265399	−	0.964139i	$-0.585504\pi$
$311$	−2911.18	−0.530797	−0.265399	+	0.964139i	$0.585504\pi$
$312$	0	0
$313$	8287.74	1.49665	0.748324	−	0.663333i	$-0.230859\pi$
$313$	8287.74	1.49665	0.748324	+	0.663333i	$0.230859\pi$
$314$	0	0
$315$	1388.54	0.248366
$316$	0	0
$317$	8120.67	1.43881	0.719404	−	0.694592i	$-0.244415\pi$
$317$	8120.67	1.43881	0.719404	+	0.694592i	$0.244415\pi$
$318$	0	0
$319$	184.999	0.0324700
$320$	0	0
$321$	8118.23	1.41157
$322$	0	0
$323$	2279.29	0.392641
$324$	0	0
$325$	−9031.96	−1.54155
$326$	0	0
$327$	710.734	0.120195
$328$	0	0
$329$	2537.84	0.425275
$330$	0	0
$331$	−6362.92	−1.05661	−0.528305	−	0.849055i	$-0.677172\pi$
$331$	−6362.92	−1.05661	−0.528305	+	0.849055i	$0.677172\pi$
$332$	0	0
$333$	705.827	0.116153
$334$	0	0
$335$	−2320.50	−0.378454
$336$	0	0
$337$	5860.06	0.947234	0.473617	−	0.880731i	$-0.342948\pi$
$337$	5860.06	0.947234	0.473617	+	0.880731i	$0.342948\pi$
$338$	0	0
$339$	7967.73	1.27654
$340$	0	0
$341$	−68.9984	−0.0109574
$342$	0	0
$343$	6847.14	1.07787
$344$	0	0
$345$	−11086.7	−1.73010
$346$	0	0
$347$	−2869.60	−0.443942	−0.221971	−	0.975053i	$-0.571249\pi$
$347$	−2869.60	−0.443942	−0.221971	+	0.975053i	$0.571249\pi$
$348$	0	0
$349$	−2748.19	−0.421511	−0.210755	−	0.977539i	$-0.567592\pi$
$349$	−2748.19	−0.421511	−0.210755	+	0.977539i	$0.567592\pi$
$350$	0	0
$351$	6992.54	1.06335
$352$	0	0
$353$	−7548.63	−1.13817	−0.569084	−	0.822280i	$-0.692702\pi$
$353$	−7548.63	−1.13817	−0.569084	+	0.822280i	$0.692702\pi$
$354$	0	0
$355$	−6229.57	−0.931355
$356$	0	0
$357$	1197.41	0.177517
$358$	0	0
$359$	−5554.15	−0.816537	−0.408269	−	0.912862i	$-0.633867\pi$
$359$	−5554.15	−0.816537	−0.408269	+	0.912862i	$0.633867\pi$
$360$	0	0
$361$	8072.37	1.17690
$362$	0	0
$363$	−6135.65	−0.887157
$364$	0	0
$365$	5354.69	0.767883
$366$	0	0
$367$	−3610.98	−0.513601	−0.256800	−	0.966464i	$-0.582668\pi$
$367$	−3610.98	−0.513601	−0.256800	+	0.966464i	$0.582668\pi$
$368$	0	0
$369$	−1177.99	−0.166190
$370$	0	0
$371$	513.417	0.0718471
$372$	0	0
$373$	1718.96	0.238618	0.119309	−	0.992857i	$-0.461932\pi$
$373$	1718.96	0.238618	0.119309	+	0.992857i	$0.461932\pi$
$374$	0	0
$375$	5773.62	0.795062
$376$	0	0
$377$	3921.05	0.535661
$378$	0	0
$379$	−10391.4	−1.40836	−0.704180	−	0.710021i	$-0.748685\pi$
$379$	−10391.4	−1.40836	−0.704180	+	0.710021i	$0.748685\pi$
$380$	0	0
$381$	−3234.47	−0.434926
$382$	0	0
$383$	7668.98	1.02315	0.511575	−	0.859238i	$-0.329062\pi$
$383$	7668.98	1.02315	0.511575	+	0.859238i	$0.329062\pi$
$384$	0	0
$385$	542.831	0.0718577
$386$	0	0
$387$	944.951	0.124120
$388$	0	0
$389$	284.150	0.0370359	0.0185180	−	0.999829i	$-0.494105\pi$
$389$	284.150	0.0370359	0.0185180	+	0.999829i	$0.494105\pi$
$390$	0	0
$391$	2499.62	0.323302
$392$	0	0
$393$	−1295.27	−0.166254
$394$	0	0
$395$	−16872.0	−2.14917
$396$	0	0
$397$	−9209.66	−1.16428	−0.582141	−	0.813088i	$-0.697785\pi$
$397$	−9209.66	−1.16428	−0.582141	+	0.813088i	$0.697785\pi$
$398$	0	0
$399$	7844.07	0.984197
$400$	0	0
$401$	−5565.10	−0.693036	−0.346518	−	0.938043i	$-0.612636\pi$
$401$	−5565.10	−0.693036	−0.346518	+	0.938043i	$0.612636\pi$
$402$	0	0
$403$	−1462.42	−0.180765
$404$	0	0
$405$	−9774.44	−1.19925
$406$	0	0
$407$	275.933	0.0336057
$408$	0	0
$409$	−12077.6	−1.46014	−0.730072	−	0.683370i	$-0.760514\pi$
$409$	−12077.6	−1.46014	−0.730072	+	0.683370i	$0.760514\pi$
$410$	0	0
$411$	1255.55	0.150686
$412$	0	0
$413$	−8665.08	−1.03240
$414$	0	0
$415$	7922.58	0.937119
$416$	0	0
$417$	−3009.26	−0.353391
$418$	0	0
$419$	2054.89	0.239590	0.119795	−	0.992799i	$-0.461776\pi$
$419$	2054.89	0.239590	0.119795	+	0.992799i	$0.461776\pi$
$420$	0	0
$421$	6819.69	0.789480	0.394740	−	0.918793i	$-0.370835\pi$
$421$	6819.69	0.789480	0.394740	+	0.918793i	$0.370835\pi$
$422$	0	0
$423$	1023.53	0.117650
$424$	0	0
$425$	−3633.36	−0.414692
$426$	0	0
$427$	3423.87	0.388040
$428$	0	0
$429$	469.309	0.0528169
$430$	0	0
$431$	12519.2	1.39914	0.699571	−	0.714563i	$-0.253374\pi$
$431$	12519.2	1.39914	0.699571	+	0.714563i	$0.253374\pi$
$432$	0	0
$433$	2921.40	0.324235	0.162117	−	0.986771i	$-0.448168\pi$
$433$	2921.40	0.324235	0.162117	+	0.986771i	$0.448168\pi$
$434$	0	0
$435$	−6996.10	−0.771120
$436$	0	0
$437$	16374.7	1.79247
$438$	0	0
$439$	1140.50	0.123993	0.0619967	−	0.998076i	$-0.480253\pi$
$439$	1140.50	0.123993	0.0619967	+	0.998076i	$0.480253\pi$
$440$	0	0
$441$	842.062	0.0909256
$442$	0	0
$443$	−2606.46	−0.279541	−0.139771	−	0.990184i	$-0.544637\pi$
$443$	−2606.46	−0.279541	−0.139771	+	0.990184i	$0.544637\pi$
$444$	0	0
$445$	25256.4	2.69049
$446$	0	0
$447$	3963.43	0.419382
$448$	0	0
$449$	1752.13	0.184161	0.0920805	−	0.995752i	$-0.470648\pi$
$449$	1752.13	0.184161	0.0920805	+	0.995752i	$0.470648\pi$
$450$	0	0
$451$	−460.521	−0.0480822
$452$	0	0
$453$	−16352.0	−1.69599
$454$	0	0
$455$	11505.3	1.18544
$456$	0	0
$457$	−12875.6	−1.31794	−0.658968	−	0.752171i	$-0.729007\pi$
$457$	−12875.6	−1.31794	−0.658968	+	0.752171i	$0.729007\pi$
$458$	0	0
$459$	2812.95	0.286051
$460$	0	0
$461$	19346.0	1.95452	0.977259	−	0.212050i	$-0.0680139\pi$
$461$	19346.0	1.95452	0.977259	+	0.212050i	$0.0680139\pi$
$462$	0	0
$463$	15002.4	1.50588	0.752938	−	0.658091i	$-0.228636\pi$
$463$	15002.4	1.50588	0.752938	+	0.658091i	$0.228636\pi$
$464$	0	0
$465$	2609.32	0.260224
$466$	0	0
$467$	13674.6	1.35500	0.677501	−	0.735522i	$-0.263063\pi$
$467$	13674.6	1.35500	0.677501	+	0.735522i	$0.263063\pi$
$468$	0	0
$469$	1800.51	0.177270
$470$	0	0
$471$	−8070.75	−0.789555
$472$	0	0
$473$	369.416	0.0359107
$474$	0	0
$475$	−23801.7	−2.29915
$476$	0	0
$477$	207.066	0.0198761
$478$	0	0
$479$	3072.68	0.293099	0.146550	−	0.989203i	$-0.453183\pi$
$479$	3072.68	0.293099	0.146550	+	0.989203i	$0.453183\pi$
$480$	0	0
$481$	5848.41	0.554396
$482$	0	0
$483$	8602.32	0.810392
$484$	0	0
$485$	27097.3	2.53695
$486$	0	0
$487$	−8689.64	−0.808553	−0.404276	−	0.914637i	$-0.632477\pi$
$487$	−8689.64	−0.808553	−0.404276	+	0.914637i	$0.632477\pi$
$488$	0	0
$489$	−16810.8	−1.55462
$490$	0	0
$491$	−15831.2	−1.45509	−0.727547	−	0.686057i	$-0.759340\pi$
$491$	−15831.2	−1.45509	−0.727547	+	0.686057i	$0.759340\pi$
$492$	0	0
$493$	1577.35	0.144098
$494$	0	0
$495$	218.929	0.0198790
$496$	0	0
$497$	4833.62	0.436253
$498$	0	0
$499$	−2309.01	−0.207146	−0.103573	−	0.994622i	$-0.533027\pi$
$499$	−2309.01	−0.207146	−0.103573	+	0.994622i	$0.533027\pi$
$500$	0	0
$501$	−14103.2	−1.25765
$502$	0	0
$503$	−6901.81	−0.611802	−0.305901	−	0.952063i	$-0.598958\pi$
$503$	−6901.81	−0.611802	−0.305901	+	0.952063i	$0.598958\pi$
$504$	0	0
$505$	14496.2	1.27737
$506$	0	0
$507$	−217.266	−0.0190318
$508$	0	0
$509$	131.481	0.0114495	0.00572475	−	0.999984i	$-0.498178\pi$
$509$	131.481	0.0114495	0.00572475	+	0.999984i	$0.498178\pi$
$510$	0	0
$511$	−4154.79	−0.359681
$512$	0	0
$513$	18427.3	1.58594
$514$	0	0
$515$	18258.1	1.56223
$516$	0	0
$517$	400.136	0.0340386
$518$	0	0
$519$	9958.43	0.842248
$520$	0	0
$521$	11931.3	1.00330	0.501649	−	0.865071i	$-0.332727\pi$
$521$	11931.3	1.00330	0.501649	+	0.865071i	$0.332727\pi$
$522$	0	0
$523$	13721.4	1.14721	0.573607	−	0.819131i	$-0.305544\pi$
$523$	13721.4	1.14721	0.573607	+	0.819131i	$0.305544\pi$
$524$	0	0
$525$	−12504.0	−1.03947
$526$	0	0
$527$	−588.301	−0.0486277
$528$	0	0
$529$	5790.59	0.475926
$530$	0	0
$531$	−3494.71	−0.285607
$532$	0	0
$533$	−9760.75	−0.793218
$534$	0	0
$535$	31379.4	2.53579
$536$	0	0
$537$	1977.60	0.158920
$538$	0	0
$539$	329.193	0.0263067
$540$	0	0
$541$	−12101.0	−0.961665	−0.480833	−	0.876812i	$-0.659666\pi$
$541$	−12101.0	−0.961665	−0.480833	+	0.876812i	$0.659666\pi$
$542$	0	0
$543$	11102.7	0.877462
$544$	0	0
$545$	2747.20	0.215921
$546$	0	0
$547$	63.9157	0.00499605	0.00249803	−	0.999997i	$-0.499205\pi$
$547$	63.9157	0.00499605	0.00249803	+	0.999997i	$0.499205\pi$
$548$	0	0
$549$	1380.88	0.107349
$550$	0	0
$551$	10333.1	0.798917
$552$	0	0
$553$	13091.2	1.00668
$554$	0	0
$555$	−10435.0	−0.798090
$556$	0	0
$557$	−6052.26	−0.460400	−0.230200	−	0.973143i	$-0.573938\pi$
$557$	−6052.26	−0.460400	−0.230200	+	0.973143i	$0.573938\pi$
$558$	0	0
$559$	7829.77	0.592422
$560$	0	0
$561$	188.793	0.0142083
$562$	0	0
$563$	20638.9	1.54498	0.772491	−	0.635026i	$-0.219011\pi$
$563$	20638.9	1.54498	0.772491	+	0.635026i	$0.219011\pi$
$564$	0	0
$565$	30797.7	2.29322
$566$	0	0
$567$	7584.14	0.561736
$568$	0	0
$569$	−21728.1	−1.60086	−0.800431	−	0.599425i	$-0.795396\pi$
$569$	−21728.1	−1.60086	−0.800431	+	0.599425i	$0.795396\pi$
$570$	0	0
$571$	22737.8	1.66646	0.833229	−	0.552929i	$-0.186490\pi$
$571$	22737.8	1.66646	0.833229	+	0.552929i	$0.186490\pi$
$572$	0	0
$573$	18669.1	1.36110
$574$	0	0
$575$	−26102.5	−1.89313
$576$	0	0
$577$	−26648.2	−1.92267	−0.961335	−	0.275383i	$-0.911195\pi$
$577$	−26648.2	−1.92267	−0.961335	+	0.275383i	$0.911195\pi$
$578$	0	0
$579$	−4099.82	−0.294270
$580$	0	0
$581$	−6147.26	−0.438952
$582$	0	0
$583$	80.9495	0.00575058
$584$	0	0
$585$	4640.20	0.327946
$586$	0	0
$587$	−1898.75	−0.133509	−0.0667544	−	0.997769i	$-0.521264\pi$
$587$	−1898.75	−0.133509	−0.0667544	+	0.997769i	$0.521264\pi$
$588$	0	0
$589$	−3853.89	−0.269604
$590$	0	0
$591$	−21395.7	−1.48917
$592$	0	0
$593$	−4474.79	−0.309878	−0.154939	−	0.987924i	$-0.549518\pi$
$593$	−4474.79	−0.309878	−0.154939	+	0.987924i	$0.549518\pi$
$594$	0	0
$595$	4628.34	0.318896
$596$	0	0
$597$	−1029.44	−0.0705733
$598$	0	0
$599$	12603.8	0.859725	0.429863	−	0.902894i	$-0.358562\pi$
$599$	12603.8	0.859725	0.429863	+	0.902894i	$0.358562\pi$
$600$	0	0
$601$	7220.64	0.490077	0.245038	−	0.969513i	$-0.421199\pi$
$601$	7220.64	0.490077	0.245038	+	0.969513i	$0.421199\pi$
$602$	0	0
$603$	726.163	0.0490408
$604$	0	0
$605$	−23716.1	−1.59371
$606$	0	0
$607$	13695.6	0.915796	0.457898	−	0.889005i	$-0.348603\pi$
$607$	13695.6	0.915796	0.457898	+	0.889005i	$0.348603\pi$
$608$	0	0
$609$	5428.39	0.361198
$610$	0	0
$611$	8480.89	0.561539
$612$	0	0
$613$	3335.20	0.219751	0.109876	−	0.993945i	$-0.464955\pi$
$613$	3335.20	0.219751	0.109876	+	0.993945i	$0.464955\pi$
$614$	0	0
$615$	17415.5	1.14189
$616$	0	0
$617$	−4186.39	−0.273157	−0.136579	−	0.990629i	$-0.543611\pi$
$617$	−4186.39	−0.273157	−0.136579	+	0.990629i	$0.543611\pi$
$618$	0	0
$619$	6788.94	0.440825	0.220412	−	0.975407i	$-0.429260\pi$
$619$	6788.94	0.440825	0.220412	+	0.975407i	$0.429260\pi$
$620$	0	0
$621$	20208.6	1.30587
$622$	0	0
$623$	−19596.9	−1.26024
$624$	0	0
$625$	−2031.54	−0.130018
$626$	0	0
$627$	1236.76	0.0787743
$628$	0	0
$629$	2352.69	0.149138
$630$	0	0
$631$	−16106.3	−1.01614	−0.508069	−	0.861316i	$-0.669640\pi$
$631$	−16106.3	−1.01614	−0.508069	+	0.861316i	$0.669640\pi$
$632$	0	0
$633$	−23079.3	−1.44917
$634$	0	0
$635$	−12502.2	−0.781314
$636$	0	0
$637$	6977.25	0.433985
$638$	0	0
$639$	1949.45	0.120687
$640$	0	0
$641$	−6682.21	−0.411749	−0.205875	−	0.978578i	$-0.566004\pi$
$641$	−6682.21	−0.411749	−0.205875	+	0.978578i	$0.566004\pi$
$642$	0	0
$643$	7047.69	0.432245	0.216123	−	0.976366i	$-0.430659\pi$
$643$	7047.69	0.432245	0.216123	+	0.976366i	$0.430659\pi$
$644$	0	0
$645$	−13970.2	−0.852832
$646$	0	0
$647$	−5078.45	−0.308585	−0.154292	−	0.988025i	$-0.549310\pi$
$647$	−5078.45	−0.308585	−0.154292	+	0.988025i	$0.549310\pi$
$648$	0	0
$649$	−1366.21	−0.0826324
$650$	0	0
$651$	−2024.61	−0.121891
$652$	0	0
$653$	−8753.86	−0.524602	−0.262301	−	0.964986i	$-0.584481\pi$
$653$	−8753.86	−0.524602	−0.262301	+	0.964986i	$0.584481\pi$
$654$	0	0
$655$	−5006.62	−0.298664
$656$	0	0
$657$	−1675.67	−0.0995037
$658$	0	0
$659$	8133.42	0.480778	0.240389	−	0.970677i	$-0.422725\pi$
$659$	8133.42	0.480778	0.240389	+	0.970677i	$0.422725\pi$
$660$	0	0
$661$	−8917.15	−0.524716	−0.262358	−	0.964971i	$-0.584500\pi$
$661$	−8917.15	−0.524716	−0.262358	+	0.964971i	$0.584500\pi$
$662$	0	0
$663$	4001.47	0.234396
$664$	0	0
$665$	30319.7	1.76804
$666$	0	0
$667$	11331.9	0.657831
$668$	0	0
$669$	−27027.2	−1.56193
$670$	0	0
$671$	539.837	0.0310584
$672$	0	0
$673$	14664.4	0.839925	0.419963	−	0.907541i	$-0.362043\pi$
$673$	14664.4	0.839925	0.419963	+	0.907541i	$0.362043\pi$
$674$	0	0
$675$	−29374.5	−1.67500
$676$	0	0
$677$	−8195.60	−0.465262	−0.232631	−	0.972565i	$-0.574733\pi$
$677$	−8195.60	−0.465262	−0.232631	+	0.972565i	$0.574733\pi$
$678$	0	0
$679$	−21025.2	−1.18833
$680$	0	0
$681$	7390.32	0.415855
$682$	0	0
$683$	18574.4	1.04060	0.520300	−	0.853983i	$-0.325820\pi$
$683$	18574.4	1.04060	0.520300	+	0.853983i	$0.325820\pi$
$684$	0	0
$685$	4853.09	0.270696
$686$	0	0
$687$	19009.4	1.05568
$688$	0	0
$689$	1715.73	0.0948679
$690$	0	0
$691$	20257.2	1.11523	0.557613	−	0.830101i	$-0.311718\pi$
$691$	20257.2	1.11523	0.557613	+	0.830101i	$0.311718\pi$
$692$	0	0
$693$	−169.870	−0.00931146
$694$	0	0
$695$	−11631.7	−0.634843
$696$	0	0
$697$	−3926.54	−0.213383
$698$	0	0
$699$	3396.06	0.183763
$700$	0	0
$701$	22609.3	1.21818	0.609089	−	0.793102i	$-0.291535\pi$
$701$	22609.3	1.21818	0.609089	+	0.793102i	$0.291535\pi$
$702$	0	0
$703$	15412.2	0.826859
$704$	0	0
$705$	−15132.0	−0.808373
$706$	0	0
$707$	−11247.8	−0.598327
$708$	0	0
$709$	−27690.9	−1.46679	−0.733395	−	0.679802i	$-0.762066\pi$
$709$	−27690.9	−1.46679	−0.733395	+	0.679802i	$0.762066\pi$
$710$	0	0
$711$	5279.82	0.278493
$712$	0	0
$713$	−4226.43	−0.221993
$714$	0	0
$715$	1814.02	0.0948819
$716$	0	0
$717$	−2367.84	−0.123331
$718$	0	0
$719$	−2111.24	−0.109507	−0.0547537	−	0.998500i	$-0.517437\pi$
$719$	−2111.24	−0.109507	−0.0547537	+	0.998500i	$0.517437\pi$
$720$	0	0
$721$	−14166.7	−0.731757
$722$	0	0
$723$	−27389.9	−1.40891
$724$	0	0
$725$	−16471.7	−0.843784
$726$	0	0
$727$	−14763.6	−0.753164	−0.376582	−	0.926383i	$-0.622901\pi$
$727$	−14763.6	−0.753164	−0.376582	+	0.926383i	$0.622901\pi$
$728$	0	0
$729$	21896.2	1.11244
$730$	0	0
$731$	3149.75	0.159368
$732$	0	0
$733$	4836.29	0.243700	0.121850	−	0.992549i	$-0.461117\pi$
$733$	4836.29	0.243700	0.121850	+	0.992549i	$0.461117\pi$
$734$	0	0
$735$	−12449.1	−0.624751
$736$	0	0
$737$	283.883	0.0141886
$738$	0	0
$739$	16015.7	0.797224	0.398612	−	0.917120i	$-0.369492\pi$
$739$	16015.7	0.797224	0.398612	+	0.917120i	$0.369492\pi$
$740$	0	0
$741$	26213.2	1.29955
$742$	0	0
$743$	−20313.3	−1.00299	−0.501495	−	0.865160i	$-0.667217\pi$
$743$	−20313.3	−1.00299	−0.501495	+	0.865160i	$0.667217\pi$
$744$	0	0
$745$	15319.9	0.753391
$746$	0	0
$747$	−2479.25	−0.121434
$748$	0	0
$749$	−24347.8	−1.18778
$750$	0	0
$751$	28755.3	1.39720	0.698598	−	0.715514i	$-0.253808\pi$
$751$	28755.3	1.39720	0.698598	+	0.715514i	$0.253808\pi$
$752$	0	0
$753$	−2023.89	−0.0979477
$754$	0	0
$755$	−63205.4	−3.04673
$756$	0	0
$757$	−32535.4	−1.56211	−0.781057	−	0.624460i	$-0.785319\pi$
$757$	−32535.4	−1.56211	−0.781057	+	0.624460i	$0.785319\pi$
$758$	0	0
$759$	1356.31	0.0648631
$760$	0	0
$761$	−9298.53	−0.442932	−0.221466	−	0.975168i	$-0.571084\pi$
$761$	−9298.53	−0.442932	−0.221466	+	0.975168i	$0.571084\pi$
$762$	0	0
$763$	−2131.60	−0.101139
$764$	0	0
$765$	1866.65	0.0882208
$766$	0	0
$767$	−28956.8	−1.36319
$768$	0	0
$769$	−20402.0	−0.956717	−0.478358	−	0.878165i	$-0.658768\pi$
$769$	−20402.0	−0.956717	−0.478358	+	0.878165i	$0.658768\pi$
$770$	0	0
$771$	1497.68	0.0699582
$772$	0	0
$773$	−10376.1	−0.482798	−0.241399	−	0.970426i	$-0.577606\pi$
$773$	−10376.1	−0.482798	−0.241399	+	0.970426i	$0.577606\pi$
$774$	0	0
$775$	6143.40	0.284745
$776$	0	0
$777$	8096.67	0.373831
$778$	0	0
$779$	−25722.3	−1.18305
$780$	0	0
$781$	762.109	0.0349173
$782$	0	0
$783$	12752.4	0.582034
$784$	0	0
$785$	−31195.9	−1.41838
$786$	0	0
$787$	−16869.6	−0.764088	−0.382044	−	0.924144i	$-0.624780\pi$
$787$	−16869.6	−0.764088	−0.382044	+	0.924144i	$0.624780\pi$
$788$	0	0
$789$	−12441.2	−0.561367
$790$	0	0
$791$	−23896.4	−1.07416
$792$	0	0
$793$	11441.8	0.512373
$794$	0	0
$795$	−3061.27	−0.136569
$796$	0	0
$797$	−9300.12	−0.413334	−0.206667	−	0.978411i	$-0.566262\pi$
$797$	−9300.12	−0.413334	−0.206667	+	0.978411i	$0.566262\pi$
$798$	0	0
$799$	3411.68	0.151060
$800$	0	0
$801$	−7903.61	−0.348639
$802$	0	0
$803$	−655.079	−0.0287886
$804$	0	0
$805$	33250.6	1.45581
$806$	0	0
$807$	−30775.8	−1.34245
$808$	0	0
$809$	−29320.9	−1.27425	−0.637126	−	0.770760i	$-0.719877\pi$
$809$	−29320.9	−1.27425	−0.637126	+	0.770760i	$0.719877\pi$
$810$	0	0
$811$	20488.9	0.887132	0.443566	−	0.896242i	$-0.353713\pi$
$811$	20488.9	0.887132	0.443566	+	0.896242i	$0.353713\pi$
$812$	0	0
$813$	−9340.66	−0.402941
$814$	0	0
$815$	−64978.7	−2.79276
$816$	0	0
$817$	20633.6	0.883574
$818$	0	0
$819$	−3600.40	−0.153612
$820$	0	0
$821$	28650.9	1.21793	0.608966	−	0.793196i	$-0.291585\pi$
$821$	28650.9	1.21793	0.608966	+	0.793196i	$0.291585\pi$
$822$	0	0
$823$	−24605.9	−1.04217	−0.521086	−	0.853504i	$-0.674473\pi$
$823$	−24605.9	−1.04217	−0.521086	+	0.853504i	$0.674473\pi$
$824$	0	0
$825$	−1971.49	−0.0831982
$826$	0	0
$827$	−34076.6	−1.43284	−0.716421	−	0.697668i	$-0.754221\pi$
$827$	−34076.6	−1.43284	−0.716421	+	0.697668i	$0.754221\pi$
$828$	0	0
$829$	−1293.46	−0.0541904	−0.0270952	−	0.999633i	$-0.508626\pi$
$829$	−1293.46	−0.0541904	−0.0270952	+	0.999633i	$0.508626\pi$
$830$	0	0
$831$	−20155.7	−0.841388
$832$	0	0
$833$	2806.80	0.116746
$834$	0	0
$835$	−54513.1	−2.25929
$836$	0	0
$837$	−4756.22	−0.196415
$838$	0	0
$839$	−28847.5	−1.18704	−0.593521	−	0.804819i	$-0.702263\pi$
$839$	−28847.5	−1.18704	−0.593521	+	0.804819i	$0.702263\pi$
$840$	0	0
$841$	−17238.1	−0.706800
$842$	0	0
$843$	18013.1	0.735948
$844$	0	0
$845$	−839.798	−0.0341893
$846$	0	0
$847$	18401.7	0.746506
$848$	0	0
$849$	13258.2	0.535947
$850$	0	0
$851$	16902.0	0.680839
$852$	0	0
$853$	58.7693	0.00235900	0.00117950	−	0.999999i	$-0.499625\pi$
$853$	58.7693	0.00235900	0.00117950	+	0.999999i	$0.499625\pi$
$854$	0	0
$855$	12228.2	0.489118
$856$	0	0
$857$	−20953.6	−0.835194	−0.417597	−	0.908632i	$-0.637128\pi$
$857$	−20953.6	−0.835194	−0.417597	+	0.908632i	$0.637128\pi$
$858$	0	0
$859$	41459.5	1.64677	0.823387	−	0.567481i	$-0.192082\pi$
$859$	41459.5	1.64677	0.823387	+	0.567481i	$0.192082\pi$
$860$	0	0
$861$	−13513.0	−0.534868
$862$	0	0
$863$	3389.59	0.133700	0.0668499	−	0.997763i	$-0.478705\pi$
$863$	3389.59	0.133700	0.0668499	+	0.997763i	$0.478705\pi$
$864$	0	0
$865$	38492.3	1.51304
$866$	0	0
$867$	−21120.0	−0.827304
$868$	0	0
$869$	2064.07	0.0805741
$870$	0	0
$871$	6016.91	0.234070
$872$	0	0
$873$	−8479.66	−0.328743
$874$	0	0
$875$	−17315.9	−0.669012
$876$	0	0
$877$	19675.3	0.757568	0.378784	−	0.925485i	$-0.376342\pi$
$877$	19675.3	0.757568	0.378784	+	0.925485i	$0.376342\pi$
$878$	0	0
$879$	6553.79	0.251484
$880$	0	0
$881$	1497.48	0.0572660	0.0286330	−	0.999590i	$-0.490885\pi$
$881$	1497.48	0.0572660	0.0286330	+	0.999590i	$0.490885\pi$
$882$	0	0
$883$	−5860.23	−0.223344	−0.111672	−	0.993745i	$-0.535621\pi$
$883$	−5860.23	−0.223344	−0.111672	+	0.993745i	$0.535621\pi$
$884$	0	0
$885$	51666.0	1.96241
$886$	0	0
$887$	−18058.0	−0.683573	−0.341787	−	0.939778i	$-0.611032\pi$
$887$	−18058.0	−0.683573	−0.341787	+	0.939778i	$0.611032\pi$
$888$	0	0
$889$	9700.66	0.365973
$890$	0	0
$891$	1195.78	0.0449609
$892$	0	0
$893$	22349.5	0.837512
$894$	0	0
$895$	7644.03	0.285488
$896$	0	0
$897$	28747.1	1.07005
$898$	0	0
$899$	−2667.04	−0.0989441
$900$	0	0
$901$	690.200	0.0255204
$902$	0	0
$903$	10839.7	0.399472
$904$	0	0
$905$	42915.2	1.57630
$906$	0	0
$907$	30297.4	1.10916	0.554580	−	0.832130i	$-0.312879\pi$
$907$	30297.4	1.10916	0.554580	+	0.832130i	$0.312879\pi$
$908$	0	0
$909$	−4536.35	−0.165524
$910$	0	0
$911$	−31977.7	−1.16297	−0.581487	−	0.813556i	$-0.697529\pi$
$911$	−31977.7	−1.16297	−0.581487	+	0.813556i	$0.697529\pi$
$912$	0	0
$913$	−969.228	−0.0351334
$914$	0	0
$915$	−20415.0	−0.737595
$916$	0	0
$917$	3884.72	0.139896
$918$	0	0
$919$	−40696.7	−1.46078	−0.730391	−	0.683029i	$-0.760662\pi$
$919$	−40696.7	−1.46078	−0.730391	+	0.683029i	$0.760662\pi$
$920$	0	0
$921$	19409.4	0.694421
$922$	0	0
$923$	16152.9	0.576034
$924$	0	0
$925$	−24568.2	−0.873295
$926$	0	0
$927$	−5713.57	−0.202436
$928$	0	0
$929$	−11467.5	−0.404989	−0.202495	−	0.979283i	$-0.564905\pi$
$929$	−11467.5	−0.404989	−0.202495	+	0.979283i	$0.564905\pi$
$930$	0	0
$931$	18387.0	0.647271
$932$	0	0
$933$	−13468.4	−0.472600
$934$	0	0
$935$	729.742	0.0255242
$936$	0	0
$937$	14100.2	0.491603	0.245802	−	0.969320i	$-0.420949\pi$
$937$	14100.2	0.491603	0.245802	+	0.969320i	$0.420949\pi$
$938$	0	0
$939$	38342.7	1.33255
$940$	0	0
$941$	29781.6	1.03172	0.515862	−	0.856672i	$-0.327472\pi$
$941$	29781.6	1.03172	0.515862	+	0.856672i	$0.327472\pi$
$942$	0	0
$943$	−28208.8	−0.974129
$944$	0	0
$945$	37418.5	1.28807
$946$	0	0
$947$	35353.8	1.21314	0.606571	−	0.795030i	$-0.292545\pi$
$947$	35353.8	1.21314	0.606571	+	0.795030i	$0.292545\pi$
$948$	0	0
$949$	−13884.4	−0.474928
$950$	0	0
$951$	37569.8	1.28106
$952$	0	0
$953$	6456.01	0.219445	0.109722	−	0.993962i	$-0.465004\pi$
$953$	6456.01	0.219445	0.109722	+	0.993962i	$0.465004\pi$
$954$	0	0
$955$	72161.7	2.44513
$956$	0	0
$957$	855.885	0.0289100
$958$	0	0
$959$	−3765.59	−0.126796
$960$	0	0
$961$	−28796.3	−0.966610
$962$	0	0
$963$	−9819.69	−0.328593
$964$	0	0
$965$	−15847.0	−0.528636
$966$	0	0
$967$	−15099.9	−0.502153	−0.251076	−	0.967967i	$-0.580785\pi$
$967$	−15099.9	−0.502153	−0.251076	+	0.967967i	$0.580785\pi$
$968$	0	0
$969$	10545.0	0.349591
$970$	0	0
$971$	8800.82	0.290867	0.145433	−	0.989368i	$-0.453542\pi$
$971$	8800.82	0.290867	0.145433	+	0.989368i	$0.453542\pi$
$972$	0	0
$973$	9025.23	0.297364
$974$	0	0
$975$	−41785.8	−1.37253
$976$	0	0
$977$	34900.0	1.14284	0.571418	−	0.820659i	$-0.306393\pi$
$977$	34900.0	1.14284	0.571418	+	0.820659i	$0.306393\pi$
$978$	0	0
$979$	−3089.81	−0.100869
$980$	0	0
$981$	−859.693	−0.0279795
$982$	0	0
$983$	21221.5	0.688567	0.344283	−	0.938866i	$-0.388122\pi$
$983$	21221.5	0.688567	0.344283	+	0.938866i	$0.388122\pi$
$984$	0	0
$985$	−82700.7	−2.67519
$986$	0	0
$987$	11741.1	0.378647
$988$	0	0
$989$	22628.2	0.727538
$990$	0	0
$991$	−23985.3	−0.768838	−0.384419	−	0.923159i	$-0.625598\pi$
$991$	−23985.3	−0.768838	−0.384419	+	0.923159i	$0.625598\pi$
$992$	0	0
$993$	−29437.7	−0.940761
$994$	0	0
$995$	−3979.10	−0.126780
$996$	0	0
$997$	−15222.3	−0.483547	−0.241773	−	0.970333i	$-0.577729\pi$
$997$	−15222.3	−0.483547	−0.241773	+	0.970333i	$0.577729\pi$
$998$	0	0
$999$	19020.7	0.602391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.a.m.1.8		10
4.3	odd	2		1024.4.a.n.1.3		10
8.3	odd	2		1024.4.a.n.1.8		10
8.5	even	2	inner	1024.4.a.m.1.3		10
16.3	odd	4		1024.4.b.j.513.3		10
16.5	even	4		1024.4.b.k.513.3		10
16.11	odd	4		1024.4.b.j.513.8		10
16.13	even	4		1024.4.b.k.513.8		10
32.3	odd	8		16.4.e.a.13.5	yes	10
32.5	even	8		128.4.e.a.97.2		10
32.11	odd	8		16.4.e.a.5.5	✓	10
32.13	even	8		128.4.e.a.33.2		10
32.19	odd	8		128.4.e.b.33.4		10
32.21	even	8		64.4.e.a.49.4		10
32.27	odd	8		128.4.e.b.97.4		10
32.29	even	8		64.4.e.a.17.4		10
96.11	even	8		144.4.k.a.37.1		10
96.29	odd	8		576.4.k.a.145.5		10
96.35	even	8		144.4.k.a.109.1		10
96.53	odd	8		576.4.k.a.433.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.5	✓	10	32.11	odd	8
16.4.e.a.13.5	yes	10	32.3	odd	8
64.4.e.a.17.4		10	32.29	even	8
64.4.e.a.49.4		10	32.21	even	8
128.4.e.a.33.2		10	32.13	even	8
128.4.e.a.97.2		10	32.5	even	8
128.4.e.b.33.4		10	32.19	odd	8
128.4.e.b.97.4		10	32.27	odd	8
144.4.k.a.37.1		10	96.11	even	8
144.4.k.a.109.1		10	96.35	even	8
576.4.k.a.145.5		10	96.29	odd	8
576.4.k.a.433.5		10	96.53	odd	8
1024.4.a.m.1.3		10	8.5	even	2	inner
1024.4.a.m.1.8		10	1.1	even	1	trivial
1024.4.a.n.1.3		10	4.3	odd	2
1024.4.a.n.1.8		10	8.3	odd	2
1024.4.b.j.513.3		10	16.3	odd	4
1024.4.b.j.513.8		10	16.11	odd	4
1024.4.b.k.513.3		10	16.5	even	4
1024.4.b.k.513.8		10	16.13	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q+4.62644 q^{3} +17.8826 q^{5} -13.8754 q^{7} -5.59607 q^{9} -2.18771 q^{11} -46.3685 q^{13} +82.7326 q^{15} -18.6531 q^{17} -122.194 q^{19} -64.1936 q^{21} -134.006 q^{23} +194.786 q^{25} -150.804 q^{27} -84.5628 q^{29} +31.5391 q^{31} -10.1213 q^{33} -248.128 q^{35} -126.129 q^{37} -214.521 q^{39} +210.504 q^{41} -168.860 q^{43} -100.072 q^{45} -182.902 q^{47} -150.474 q^{49} -86.2972 q^{51} -37.0020 q^{53} -39.1219 q^{55} -565.323 q^{57} +624.494 q^{59} -246.759 q^{61} +77.6476 q^{63} -829.188 q^{65} -129.763 q^{67} -619.970 q^{69} -348.360 q^{71} +299.436 q^{73} +901.168 q^{75} +30.3553 q^{77} -943.487 q^{79} -546.590 q^{81} +443.034 q^{83} -333.565 q^{85} -391.224 q^{87} +1412.35 q^{89} +643.381 q^{91} +145.914 q^{93} -2185.14 q^{95} +1515.29 q^{97} +12.2426 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{7} + 54 q^{9} - 124 q^{15} + 4 q^{17} - 276 q^{23} + 50 q^{25} - 368 q^{31} - 4 q^{33} - 732 q^{39} - 944 q^{47} - 94 q^{49} - 1380 q^{55} + 108 q^{57} - 2628 q^{63} - 492 q^{65} - 3468 q^{71}+ \cdots - 4 q^{97}+O(q^{100}) \) 10 * q - 28 * q^7 + 54 * q^9 - 124 * q^15 + 4 * q^17 - 276 * q^23 + 50 * q^25 - 368 * q^31 - 4 * q^33 - 732 * q^39 - 944 * q^47 - 94 * q^49 - 1380 * q^55 + 108 * q^57 - 2628 * q^63 - 492 * q^65 - 3468 * q^71 - 296 * q^73 - 4416 * q^79 - 482 * q^81 - 6036 * q^87 + 88 * q^89 - 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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