LMFDB - Embedded newform 320.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,4,Mod(1,320)] 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("320.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-10,0,0,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.8806112018$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.32456 q^{3} -5.00000 q^{5} -18.9737 q^{7} +13.0000 q^{9} -12.6491 q^{11} -38.0000 q^{13} -31.6228 q^{15} +34.0000 q^{17} -101.193 q^{19} -120.000 q^{21} -82.2192 q^{23} +25.0000 q^{25} -88.5438 q^{27} -270.000 q^{29} +341.526 q^{31} -80.0000 q^{33} +94.8683 q^{35} -206.000 q^{37} -240.333 q^{39} -270.000 q^{41} +537.587 q^{43} -65.0000 q^{45} +132.816 q^{47} +17.0000 q^{49} +215.035 q^{51} +258.000 q^{53} +63.2456 q^{55} -640.000 q^{57} -75.8947 q^{59} +250.000 q^{61} -246.658 q^{63} +190.000 q^{65} -815.868 q^{67} -520.000 q^{69} +645.105 q^{71} -1078.00 q^{73} +158.114 q^{75} +240.000 q^{77} +278.280 q^{79} -911.000 q^{81} -1106.80 q^{83} -170.000 q^{85} -1707.63 q^{87} +890.000 q^{89} +720.999 q^{91} +2160.00 q^{93} +505.964 q^{95} -254.000 q^{97} -164.438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} + 26 q^{9} - 76 q^{13} + 68 q^{17} - 240 q^{21} + 50 q^{25} - 540 q^{29} - 160 q^{33} - 412 q^{37} - 540 q^{41} - 130 q^{45} + 34 q^{49} + 516 q^{53} - 1280 q^{57} + 500 q^{61} + 380 q^{65}+ \cdots - 508 q^{97}+O(q^{100}) $ 2 * q - 10 * q^5 + 26 * q^9 - 76 * q^13 + 68 * q^17 - 240 * q^21 + 50 * q^25 - 540 * q^29 - 160 * q^33 - 412 * q^37 - 540 * q^41 - 130 * q^45 + 34 * q^49 + 516 * q^53 - 1280 * q^57 + 500 * q^61 + 380 * q^65 - 1040 * q^69 - 2156 * q^73 + 480 * q^77 - 1822 * q^81 - 340 * q^85 + 1780 * q^89 + 4320 * q^93 - 508 * 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
$3$	6.32456	1.21716	0.608581	−	0.793492i	$-0.291739\pi$
$3$	6.32456	1.21716	0.608581	+	0.793492i	$0.291739\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−18.9737	−1.02448	−0.512241	−	0.858842i	$-0.671184\pi$
$7$	−18.9737	−1.02448	−0.512241	+	0.858842i	$0.671184\pi$
$8$	0	0
$9$	13.0000	0.481481
$10$	0	0
$11$	−12.6491	−0.346714	−0.173357	−	0.984859i	$-0.555461\pi$
$11$	−12.6491	−0.346714	−0.173357	+	0.984859i	$0.555461\pi$
$12$	0	0
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	0	0
$15$	−31.6228	−0.544331
$16$	0	0
$17$	34.0000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	34.0000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−101.193	−1.22185	−0.610927	−	0.791687i	$-0.709203\pi$
$19$	−101.193	−1.22185	−0.610927	+	0.791687i	$0.709203\pi$
$20$	0	0
$21$	−120.000	−1.24696
$22$	0	0
$23$	−82.2192	−0.745387	−0.372693	−	0.927955i	$-0.621566\pi$
$23$	−82.2192	−0.745387	−0.372693	+	0.927955i	$0.621566\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−88.5438	−0.631121
$28$	0	0
$29$	−270.000	−1.72889	−0.864444	−	0.502729i	$-0.832329\pi$
$29$	−270.000	−1.72889	−0.864444	+	0.502729i	$0.832329\pi$
$30$	0	0
$31$	341.526	1.97871	0.989353	−	0.145537i	$-0.0464908\pi$
$31$	341.526	1.97871	0.989353	+	0.145537i	$0.0464908\pi$
$32$	0	0
$33$	−80.0000	−0.422006
$34$	0	0
$35$	94.8683	0.458162
$36$	0	0
$37$	−206.000	−0.915302	−0.457651	−	0.889132i	$-0.651309\pi$
$37$	−206.000	−0.915302	−0.457651	+	0.889132i	$0.651309\pi$
$38$	0	0
$39$	−240.333	−0.986772
$40$	0	0
$41$	−270.000	−1.02846	−0.514231	−	0.857652i	$-0.671922\pi$
$41$	−270.000	−1.02846	−0.514231	+	0.857652i	$0.671922\pi$
$42$	0	0
$43$	537.587	1.90654	0.953271	−	0.302117i	$-0.0976935\pi$
$43$	537.587	1.90654	0.953271	+	0.302117i	$0.0976935\pi$
$44$	0	0
$45$	−65.0000	−0.215325
$46$	0	0
$47$	132.816	0.412195	0.206097	−	0.978531i	$-0.433924\pi$
$47$	132.816	0.412195	0.206097	+	0.978531i	$0.433924\pi$
$48$	0	0
$49$	17.0000	0.0495627
$50$	0	0
$51$	215.035	0.590410
$52$	0	0
$53$	258.000	0.668661	0.334330	−	0.942456i	$-0.391490\pi$
$53$	258.000	0.668661	0.334330	+	0.942456i	$0.391490\pi$
$54$	0	0
$55$	63.2456	0.155055
$56$	0	0
$57$	−640.000	−1.48719
$58$	0	0
$59$	−75.8947	−0.167469	−0.0837343	−	0.996488i	$-0.526685\pi$
$59$	−75.8947	−0.167469	−0.0837343	+	0.996488i	$0.526685\pi$
$60$	0	0
$61$	250.000	0.524741	0.262371	−	0.964967i	$-0.415496\pi$
$61$	250.000	0.524741	0.262371	+	0.964967i	$0.415496\pi$
$62$	0	0
$63$	−246.658	−0.493269
$64$	0	0
$65$	190.000	0.362563
$66$	0	0
$67$	−815.868	−1.48767	−0.743837	−	0.668362i	$-0.766996\pi$
$67$	−815.868	−1.48767	−0.743837	+	0.668362i	$0.766996\pi$
$68$	0	0
$69$	−520.000	−0.907256
$70$	0	0
$71$	645.105	1.07831	0.539154	−	0.842207i	$-0.318744\pi$
$71$	645.105	1.07831	0.539154	+	0.842207i	$0.318744\pi$
$72$	0	0
$73$	−1078.00	−1.72836	−0.864181	−	0.503182i	$-0.832163\pi$
$73$	−1078.00	−1.72836	−0.864181	+	0.503182i	$0.832163\pi$
$74$	0	0
$75$	158.114	0.243432
$76$	0	0
$77$	240.000	0.355202
$78$	0	0
$79$	278.280	0.396316	0.198158	−	0.980170i	$-0.436504\pi$
$79$	278.280	0.396316	0.198158	+	0.980170i	$0.436504\pi$
$80$	0	0
$81$	−911.000	−1.24966
$82$	0	0
$83$	−1106.80	−1.46370	−0.731848	−	0.681468i	$-0.761342\pi$
$83$	−1106.80	−1.46370	−0.731848	+	0.681468i	$0.761342\pi$
$84$	0	0
$85$	−170.000	−0.216930
$86$	0	0
$87$	−1707.63	−2.10434
$88$	0	0
$89$	890.000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	890.000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	720.999	0.830563
$92$	0	0
$93$	2160.00	2.40840
$94$	0	0
$95$	505.964	0.546430
$96$	0	0
$97$	−254.000	−0.265874	−0.132937	−	0.991124i	$-0.542441\pi$
$97$	−254.000	−0.265874	−0.132937	+	0.991124i	$0.542441\pi$
$98$	0	0
$99$	−164.438	−0.166936
$100$	0	0
$101$	−598.000	−0.589141	−0.294570	−	0.955630i	$-0.595177\pi$
$101$	−598.000	−0.589141	−0.294570	+	0.955630i	$0.595177\pi$
$102$	0	0
$103$	499.640	0.477971	0.238985	−	0.971023i	$-0.423185\pi$
$103$	499.640	0.477971	0.238985	+	0.971023i	$0.423185\pi$
$104$	0	0
$105$	600.000	0.557657
$106$	0	0
$107$	626.131	0.565704	0.282852	−	0.959164i	$-0.408719\pi$
$107$	626.131	0.565704	0.282852	+	0.959164i	$0.408719\pi$
$108$	0	0
$109$	−854.000	−0.750444	−0.375222	−	0.926935i	$-0.622433\pi$
$109$	−854.000	−0.750444	−0.375222	+	0.926935i	$0.622433\pi$
$110$	0	0
$111$	−1302.86	−1.11407
$112$	0	0
$113$	1698.00	1.41358	0.706789	−	0.707424i	$-0.250143\pi$
$113$	1698.00	1.41358	0.706789	+	0.707424i	$0.250143\pi$
$114$	0	0
$115$	411.096	0.333347
$116$	0	0
$117$	−494.000	−0.390345
$118$	0	0
$119$	−645.105	−0.496947
$120$	0	0
$121$	−1171.00	−0.879790
$122$	0	0
$123$	−1707.63	−1.25180
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	234.009	0.163503	0.0817516	−	0.996653i	$-0.473949\pi$
$127$	234.009	0.163503	0.0817516	+	0.996653i	$0.473949\pi$
$128$	0	0
$129$	3400.00	2.32057
$130$	0	0
$131$	1732.93	1.15578	0.577888	−	0.816116i	$-0.303877\pi$
$131$	1732.93	1.15578	0.577888	+	0.816116i	$0.303877\pi$
$132$	0	0
$133$	1920.00	1.25177
$134$	0	0
$135$	442.719	0.282246
$136$	0	0
$137$	1546.00	0.964115	0.482057	−	0.876140i	$-0.339890\pi$
$137$	1546.00	0.964115	0.482057	+	0.876140i	$0.339890\pi$
$138$	0	0
$139$	328.877	0.200683	0.100342	−	0.994953i	$-0.468006\pi$
$139$	328.877	0.200683	0.100342	+	0.994953i	$0.468006\pi$
$140$	0	0
$141$	840.000	0.501708
$142$	0	0
$143$	480.666	0.281086
$144$	0	0
$145$	1350.00	0.773182
$146$	0	0
$147$	107.517	0.0603258
$148$	0	0
$149$	−3246.00	−1.78472	−0.892358	−	0.451328i	$-0.850950\pi$
$149$	−3246.00	−1.78472	−0.892358	+	0.451328i	$0.850950\pi$
$150$	0	0
$151$	1505.24	0.811225	0.405613	−	0.914045i	$-0.367058\pi$
$151$	1505.24	0.811225	0.405613	+	0.914045i	$0.367058\pi$
$152$	0	0
$153$	442.000	0.233553
$154$	0	0
$155$	−1707.63	−0.884904
$156$	0	0
$157$	1226.00	0.623219	0.311610	−	0.950210i	$-0.399132\pi$
$157$	1226.00	0.623219	0.311610	+	0.950210i	$0.399132\pi$
$158$	0	0
$159$	1631.74	0.813868
$160$	0	0
$161$	1560.00	0.763635
$162$	0	0
$163$	1448.32	0.695960	0.347980	−	0.937502i	$-0.386868\pi$
$163$	1448.32	0.695960	0.347980	+	0.937502i	$0.386868\pi$
$164$	0	0
$165$	400.000	0.188727
$166$	0	0
$167$	2333.76	1.08139	0.540694	−	0.841219i	$-0.318162\pi$
$167$	2333.76	1.08139	0.540694	+	0.841219i	$0.318162\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	0	0
$171$	−1315.51	−0.588300
$172$	0	0
$173$	3098.00	1.36148	0.680742	−	0.732524i	$-0.261658\pi$
$173$	3098.00	1.36148	0.680742	+	0.732524i	$0.261658\pi$
$174$	0	0
$175$	−474.342	−0.204896
$176$	0	0
$177$	−480.000	−0.203836
$178$	0	0
$179$	2352.73	0.982411	0.491206	−	0.871044i	$-0.336556\pi$
$179$	2352.73	0.982411	0.491206	+	0.871044i	$0.336556\pi$
$180$	0	0
$181$	−2182.00	−0.896060	−0.448030	−	0.894019i	$-0.647874\pi$
$181$	−2182.00	−0.896060	−0.448030	+	0.894019i	$0.647874\pi$
$182$	0	0
$183$	1581.14	0.638695
$184$	0	0
$185$	1030.00	0.409336
$186$	0	0
$187$	−430.070	−0.168181
$188$	0	0
$189$	1680.00	0.646572
$190$	0	0
$191$	−3023.14	−1.14527	−0.572635	−	0.819810i	$-0.694079\pi$
$191$	−3023.14	−1.14527	−0.572635	+	0.819810i	$0.694079\pi$
$192$	0	0
$193$	1298.00	0.484104	0.242052	−	0.970263i	$-0.422180\pi$
$193$	1298.00	0.484104	0.242052	+	0.970263i	$0.422180\pi$
$194$	0	0
$195$	1201.67	0.441298
$196$	0	0
$197$	−2846.00	−1.02928	−0.514642	−	0.857405i	$-0.672075\pi$
$197$	−2846.00	−1.02928	−0.514642	+	0.857405i	$0.672075\pi$
$198$	0	0
$199$	−3592.35	−1.27967	−0.639836	−	0.768511i	$-0.720998\pi$
$199$	−3592.35	−1.27967	−0.639836	+	0.768511i	$0.720998\pi$
$200$	0	0
$201$	−5160.00	−1.81074
$202$	0	0
$203$	5122.89	1.77121
$204$	0	0
$205$	1350.00	0.459942
$206$	0	0
$207$	−1068.85	−0.358890
$208$	0	0
$209$	1280.00	0.423634
$210$	0	0
$211$	−4186.86	−1.36604	−0.683021	−	0.730398i	$-0.739334\pi$
$211$	−4186.86	−1.36604	−0.683021	+	0.730398i	$0.739334\pi$
$212$	0	0
$213$	4080.00	1.31247
$214$	0	0
$215$	−2687.94	−0.852631
$216$	0	0
$217$	−6480.00	−2.02715
$218$	0	0
$219$	−6817.87	−2.10369
$220$	0	0
$221$	−1292.00	−0.393255
$222$	0	0
$223$	−4762.39	−1.43010	−0.715052	−	0.699071i	$-0.753597\pi$
$223$	−4762.39	−1.43010	−0.715052	+	0.699071i	$0.753597\pi$
$224$	0	0
$225$	325.000	0.0962963
$226$	0	0
$227$	1663.36	0.486348	0.243174	−	0.969983i	$-0.421811\pi$
$227$	1663.36	0.486348	0.243174	+	0.969983i	$0.421811\pi$
$228$	0	0
$229$	1050.00	0.302995	0.151498	−	0.988458i	$-0.451590\pi$
$229$	1050.00	0.302995	0.151498	+	0.988458i	$0.451590\pi$
$230$	0	0
$231$	1517.89	0.432338
$232$	0	0
$233$	2778.00	0.781085	0.390543	−	0.920585i	$-0.372287\pi$
$233$	2778.00	0.781085	0.390543	+	0.920585i	$0.372287\pi$
$234$	0	0
$235$	−664.078	−0.184339
$236$	0	0
$237$	1760.00	0.482381
$238$	0	0
$239$	2555.12	0.691536	0.345768	−	0.938320i	$-0.387618\pi$
$239$	2555.12	0.691536	0.345768	+	0.938320i	$0.387618\pi$
$240$	0	0
$241$	−5350.00	−1.42997	−0.714987	−	0.699138i	$-0.753567\pi$
$241$	−5350.00	−1.42997	−0.714987	+	0.699138i	$0.753567\pi$
$242$	0	0
$243$	−3370.99	−0.889913
$244$	0	0
$245$	−85.0000	−0.0221651
$246$	0	0
$247$	3845.33	0.990577
$248$	0	0
$249$	−7000.00	−1.78155
$250$	0	0
$251$	−5881.84	−1.47912	−0.739558	−	0.673093i	$-0.764966\pi$
$251$	−5881.84	−1.47912	−0.739558	+	0.673093i	$0.764966\pi$
$252$	0	0
$253$	1040.00	0.258436
$254$	0	0
$255$	−1075.17	−0.264039
$256$	0	0
$257$	1074.00	0.260678	0.130339	−	0.991469i	$-0.458393\pi$
$257$	1074.00	0.260678	0.130339	+	0.991469i	$0.458393\pi$
$258$	0	0
$259$	3908.58	0.937711
$260$	0	0
$261$	−3510.00	−0.832427
$262$	0	0
$263$	1486.27	0.348469	0.174235	−	0.984704i	$-0.444255\pi$
$263$	1486.27	0.348469	0.174235	+	0.984704i	$0.444255\pi$
$264$	0	0
$265$	−1290.00	−0.299034
$266$	0	0
$267$	5628.85	1.29019
$268$	0	0
$269$	−406.000	−0.0920233	−0.0460116	−	0.998941i	$-0.514651\pi$
$269$	−406.000	−0.0920233	−0.0460116	+	0.998941i	$0.514651\pi$
$270$	0	0
$271$	−392.122	−0.0878957	−0.0439479	−	0.999034i	$-0.513994\pi$
$271$	−392.122	−0.0878957	−0.0439479	+	0.999034i	$0.513994\pi$
$272$	0	0
$273$	4560.00	1.01093
$274$	0	0
$275$	−316.228	−0.0693427
$276$	0	0
$277$	−5934.00	−1.28715	−0.643573	−	0.765385i	$-0.722549\pi$
$277$	−5934.00	−1.28715	−0.643573	+	0.765385i	$0.722549\pi$
$278$	0	0
$279$	4439.84	0.952710
$280$	0	0
$281$	−1870.00	−0.396992	−0.198496	−	0.980102i	$-0.563606\pi$
$281$	−1870.00	−0.396992	−0.198496	+	0.980102i	$0.563606\pi$
$282$	0	0
$283$	4888.88	1.02690	0.513452	−	0.858118i	$-0.328366\pi$
$283$	4888.88	1.02690	0.513452	+	0.858118i	$0.328366\pi$
$284$	0	0
$285$	3200.00	0.665093
$286$	0	0
$287$	5122.89	1.05364
$288$	0	0
$289$	−3757.00	−0.764706
$290$	0	0
$291$	−1606.44	−0.323612
$292$	0	0
$293$	−5198.00	−1.03642	−0.518209	−	0.855254i	$-0.673401\pi$
$293$	−5198.00	−1.03642	−0.518209	+	0.855254i	$0.673401\pi$
$294$	0	0
$295$	379.473	0.0748942
$296$	0	0
$297$	1120.00	0.218818
$298$	0	0
$299$	3124.33	0.604297
$300$	0	0
$301$	−10200.0	−1.95322
$302$	0	0
$303$	−3782.08	−0.717079
$304$	0	0
$305$	−1250.00	−0.234671
$306$	0	0
$307$	−3750.46	−0.697232	−0.348616	−	0.937266i	$-0.613348\pi$
$307$	−3750.46	−0.697232	−0.348616	+	0.937266i	$0.613348\pi$
$308$	0	0
$309$	3160.00	0.581767
$310$	0	0
$311$	−6261.31	−1.14163	−0.570814	−	0.821079i	$-0.693372\pi$
$311$	−6261.31	−1.14163	−0.570814	+	0.821079i	$0.693372\pi$
$312$	0	0
$313$	2218.00	0.400539	0.200270	−	0.979741i	$-0.435818\pi$
$313$	2218.00	0.400539	0.200270	+	0.979741i	$0.435818\pi$
$314$	0	0
$315$	1233.29	0.220597
$316$	0	0
$317$	−4134.00	−0.732456	−0.366228	−	0.930525i	$-0.619351\pi$
$317$	−4134.00	−0.732456	−0.366228	+	0.930525i	$0.619351\pi$
$318$	0	0
$319$	3415.26	0.599429
$320$	0	0
$321$	3960.00	0.688553
$322$	0	0
$323$	−3440.56	−0.592687
$324$	0	0
$325$	−950.000	−0.162143
$326$	0	0
$327$	−5401.17	−0.913411
$328$	0	0
$329$	−2520.00	−0.422286
$330$	0	0
$331$	11953.4	1.98495	0.992476	−	0.122443i	$-0.0390729\pi$
$331$	11953.4	1.98495	0.992476	+	0.122443i	$0.0390729\pi$
$332$	0	0
$333$	−2678.00	−0.440701
$334$	0	0
$335$	4079.34	0.665308
$336$	0	0
$337$	−8014.00	−1.29540	−0.647701	−	0.761895i	$-0.724269\pi$
$337$	−8014.00	−1.29540	−0.647701	+	0.761895i	$0.724269\pi$
$338$	0	0
$339$	10739.1	1.72055
$340$	0	0
$341$	−4320.00	−0.686044
$342$	0	0
$343$	6185.42	0.973706
$344$	0	0
$345$	2600.00	0.405737
$346$	0	0
$347$	−4484.11	−0.693716	−0.346858	−	0.937918i	$-0.612752\pi$
$347$	−4484.11	−0.693716	−0.346858	+	0.937918i	$0.612752\pi$
$348$	0	0
$349$	−910.000	−0.139574	−0.0697868	−	0.997562i	$-0.522232\pi$
$349$	−910.000	−0.139574	−0.0697868	+	0.997562i	$0.522232\pi$
$350$	0	0
$351$	3364.66	0.511659
$352$	0	0
$353$	12962.0	1.95438	0.977192	−	0.212357i	$-0.0681140\pi$
$353$	12962.0	1.95438	0.977192	+	0.212357i	$0.0681140\pi$
$354$	0	0
$355$	−3225.52	−0.482234
$356$	0	0
$357$	−4080.00	−0.604864
$358$	0	0
$359$	−12193.7	−1.79265	−0.896325	−	0.443398i	$-0.853773\pi$
$359$	−12193.7	−1.79265	−0.896325	+	0.443398i	$0.853773\pi$
$360$	0	0
$361$	3381.00	0.492929
$362$	0	0
$363$	−7406.05	−1.07085
$364$	0	0
$365$	5390.00	0.772947
$366$	0	0
$367$	−3434.23	−0.488462	−0.244231	−	0.969717i	$-0.578536\pi$
$367$	−3434.23	−0.488462	−0.244231	+	0.969717i	$0.578536\pi$
$368$	0	0
$369$	−3510.00	−0.495185
$370$	0	0
$371$	−4895.21	−0.685031
$372$	0	0
$373$	−4622.00	−0.641603	−0.320802	−	0.947146i	$-0.603952\pi$
$373$	−4622.00	−0.641603	−0.320802	+	0.947146i	$0.603952\pi$
$374$	0	0
$375$	−790.569	−0.108866
$376$	0	0
$377$	10260.0	1.40164
$378$	0	0
$379$	−8449.61	−1.14519	−0.572595	−	0.819838i	$-0.694063\pi$
$379$	−8449.61	−1.14519	−0.572595	+	0.819838i	$0.694063\pi$
$380$	0	0
$381$	1480.00	0.199010
$382$	0	0
$383$	1815.15	0.242166	0.121083	−	0.992642i	$-0.461363\pi$
$383$	1815.15	0.242166	0.121083	+	0.992642i	$0.461363\pi$
$384$	0	0
$385$	−1200.00	−0.158851
$386$	0	0
$387$	6988.63	0.917964
$388$	0	0
$389$	11106.0	1.44755	0.723774	−	0.690037i	$-0.242406\pi$
$389$	11106.0	1.44755	0.723774	+	0.690037i	$0.242406\pi$
$390$	0	0
$391$	−2795.45	−0.361566
$392$	0	0
$393$	10960.0	1.40677
$394$	0	0
$395$	−1391.40	−0.177238
$396$	0	0
$397$	5754.00	0.727418	0.363709	−	0.931513i	$-0.381510\pi$
$397$	5754.00	0.727418	0.363709	+	0.931513i	$0.381510\pi$
$398$	0	0
$399$	12143.1	1.52360
$400$	0	0
$401$	−1118.00	−0.139228	−0.0696138	−	0.997574i	$-0.522177\pi$
$401$	−1118.00	−0.139228	−0.0696138	+	0.997574i	$0.522177\pi$
$402$	0	0
$403$	−12978.0	−1.60417
$404$	0	0
$405$	4555.00	0.558864
$406$	0	0
$407$	2605.72	0.317348
$408$	0	0
$409$	−11374.0	−1.37508	−0.687540	−	0.726146i	$-0.741310\pi$
$409$	−11374.0	−1.37508	−0.687540	+	0.726146i	$0.741310\pi$
$410$	0	0
$411$	9777.76	1.17348
$412$	0	0
$413$	1440.00	0.171568
$414$	0	0
$415$	5533.99	0.654585
$416$	0	0
$417$	2080.00	0.244264
$418$	0	0
$419$	−12674.4	−1.47777	−0.738885	−	0.673832i	$-0.764647\pi$
$419$	−12674.4	−1.47777	−0.738885	+	0.673832i	$0.764647\pi$
$420$	0	0
$421$	−1150.00	−0.133130	−0.0665648	−	0.997782i	$-0.521204\pi$
$421$	−1150.00	−0.133130	−0.0665648	+	0.997782i	$0.521204\pi$
$422$	0	0
$423$	1726.60	0.198464
$424$	0	0
$425$	850.000	0.0970143
$426$	0	0
$427$	−4743.42	−0.537588
$428$	0	0
$429$	3040.00	0.342127
$430$	0	0
$431$	−1353.45	−0.151261	−0.0756307	−	0.997136i	$-0.524097\pi$
$431$	−1353.45	−0.151261	−0.0756307	+	0.997136i	$0.524097\pi$
$432$	0	0
$433$	−7918.00	−0.878787	−0.439394	−	0.898295i	$-0.644807\pi$
$433$	−7918.00	−0.878787	−0.439394	+	0.898295i	$0.644807\pi$
$434$	0	0
$435$	8538.15	0.941087
$436$	0	0
$437$	8320.00	0.910754
$438$	0	0
$439$	−14217.6	−1.54572	−0.772858	−	0.634579i	$-0.781173\pi$
$439$	−14217.6	−1.54572	−0.772858	+	0.634579i	$0.781173\pi$
$440$	0	0
$441$	221.000	0.0238635
$442$	0	0
$443$	10581.0	1.13480	0.567401	−	0.823441i	$-0.307949\pi$
$443$	10581.0	1.13480	0.567401	+	0.823441i	$0.307949\pi$
$444$	0	0
$445$	−4450.00	−0.474045
$446$	0	0
$447$	−20529.5	−2.17229
$448$	0	0
$449$	4474.00	0.470247	0.235124	−	0.971965i	$-0.424450\pi$
$449$	4474.00	0.470247	0.235124	+	0.971965i	$0.424450\pi$
$450$	0	0
$451$	3415.26	0.356582
$452$	0	0
$453$	9520.00	0.987392
$454$	0	0
$455$	−3605.00	−0.371439
$456$	0	0
$457$	4154.00	0.425199	0.212599	−	0.977139i	$-0.431807\pi$
$457$	4154.00	0.425199	0.212599	+	0.977139i	$0.431807\pi$
$458$	0	0
$459$	−3010.49	−0.306138
$460$	0	0
$461$	11282.0	1.13982	0.569908	−	0.821709i	$-0.306979\pi$
$461$	11282.0	1.13982	0.569908	+	0.821709i	$0.306979\pi$
$462$	0	0
$463$	5458.09	0.547860	0.273930	−	0.961750i	$-0.411676\pi$
$463$	5458.09	0.547860	0.273930	+	0.961750i	$0.411676\pi$
$464$	0	0
$465$	−10800.0	−1.07707
$466$	0	0
$467$	−3775.76	−0.374136	−0.187068	−	0.982347i	$-0.559898\pi$
$467$	−3775.76	−0.374136	−0.187068	+	0.982347i	$0.559898\pi$
$468$	0	0
$469$	15480.0	1.52409
$470$	0	0
$471$	7753.90	0.758559
$472$	0	0
$473$	−6800.00	−0.661024
$474$	0	0
$475$	−2529.82	−0.244371
$476$	0	0
$477$	3354.00	0.321948
$478$	0	0
$479$	−8930.27	−0.851847	−0.425923	−	0.904759i	$-0.640051\pi$
$479$	−8930.27	−0.851847	−0.425923	+	0.904759i	$0.640051\pi$
$480$	0	0
$481$	7828.00	0.742050
$482$	0	0
$483$	9866.31	0.929467
$484$	0	0
$485$	1270.00	0.118903
$486$	0	0
$487$	−2422.30	−0.225390	−0.112695	−	0.993630i	$-0.535948\pi$
$487$	−2422.30	−0.225390	−0.112695	+	0.993630i	$0.535948\pi$
$488$	0	0
$489$	9160.00	0.847095
$490$	0	0
$491$	8993.52	0.826623	0.413311	−	0.910590i	$-0.364372\pi$
$491$	8993.52	0.826623	0.413311	+	0.910590i	$0.364372\pi$
$492$	0	0
$493$	−9180.00	−0.838634
$494$	0	0
$495$	822.192	0.0746561
$496$	0	0
$497$	−12240.0	−1.10471
$498$	0	0
$499$	3541.75	0.317737	0.158868	−	0.987300i	$-0.449215\pi$
$499$	3541.75	0.317737	0.158868	+	0.987300i	$0.449215\pi$
$500$	0	0
$501$	14760.0	1.31622
$502$	0	0
$503$	−2384.36	−0.211358	−0.105679	−	0.994400i	$-0.533702\pi$
$503$	−2384.36	−0.211358	−0.105679	+	0.994400i	$0.533702\pi$
$504$	0	0
$505$	2990.00	0.263472
$506$	0	0
$507$	−4762.39	−0.417170
$508$	0	0
$509$	−2350.00	−0.204640	−0.102320	−	0.994752i	$-0.532627\pi$
$509$	−2350.00	−0.204640	−0.102320	+	0.994752i	$0.532627\pi$
$510$	0	0
$511$	20453.6	1.77067
$512$	0	0
$513$	8960.00	0.771138
$514$	0	0
$515$	−2498.20	−0.213755
$516$	0	0
$517$	−1680.00	−0.142914
$518$	0	0
$519$	19593.5	1.65714
$520$	0	0
$521$	858.000	0.0721491	0.0360745	−	0.999349i	$-0.488515\pi$
$521$	858.000	0.0721491	0.0360745	+	0.999349i	$0.488515\pi$
$522$	0	0
$523$	5799.62	0.484894	0.242447	−	0.970165i	$-0.422050\pi$
$523$	5799.62	0.484894	0.242447	+	0.970165i	$0.422050\pi$
$524$	0	0
$525$	−3000.00	−0.249392
$526$	0	0
$527$	11611.9	0.959813
$528$	0	0
$529$	−5407.00	−0.444399
$530$	0	0
$531$	−986.631	−0.0806330
$532$	0	0
$533$	10260.0	0.833790
$534$	0	0
$535$	−3130.65	−0.252991
$536$	0	0
$537$	14880.0	1.19575
$538$	0	0
$539$	−215.035	−0.0171841
$540$	0	0
$541$	−20478.0	−1.62739	−0.813695	−	0.581292i	$-0.802547\pi$
$541$	−20478.0	−1.62739	−0.813695	+	0.581292i	$0.802547\pi$
$542$	0	0
$543$	−13800.2	−1.09065
$544$	0	0
$545$	4270.00	0.335609
$546$	0	0
$547$	−10429.2	−0.815210	−0.407605	−	0.913158i	$-0.633636\pi$
$547$	−10429.2	−0.815210	−0.407605	+	0.913158i	$0.633636\pi$
$548$	0	0
$549$	3250.00	0.252653
$550$	0	0
$551$	27322.1	2.11245
$552$	0	0
$553$	−5280.00	−0.406019
$554$	0	0
$555$	6514.29	0.498228
$556$	0	0
$557$	13194.0	1.00368	0.501838	−	0.864962i	$-0.332657\pi$
$557$	13194.0	1.00368	0.501838	+	0.864962i	$0.332657\pi$
$558$	0	0
$559$	−20428.3	−1.54566
$560$	0	0
$561$	−2720.00	−0.204703
$562$	0	0
$563$	9771.44	0.731469	0.365734	−	0.930719i	$-0.380818\pi$
$563$	9771.44	0.731469	0.365734	+	0.930719i	$0.380818\pi$
$564$	0	0
$565$	−8490.00	−0.632172
$566$	0	0
$567$	17285.0	1.28025
$568$	0	0
$569$	4594.00	0.338472	0.169236	−	0.985576i	$-0.445870\pi$
$569$	4594.00	0.338472	0.169236	+	0.985576i	$0.445870\pi$
$570$	0	0
$571$	−4389.24	−0.321688	−0.160844	−	0.986980i	$-0.551422\pi$
$571$	−4389.24	−0.321688	−0.160844	+	0.986980i	$0.551422\pi$
$572$	0	0
$573$	−19120.0	−1.39398
$574$	0	0
$575$	−2055.48	−0.149077
$576$	0	0
$577$	−14926.0	−1.07691	−0.538455	−	0.842654i	$-0.680992\pi$
$577$	−14926.0	−1.07691	−0.538455	+	0.842654i	$0.680992\pi$
$578$	0	0
$579$	8209.27	0.589233
$580$	0	0
$581$	21000.0	1.49953
$582$	0	0
$583$	−3263.47	−0.231834
$584$	0	0
$585$	2470.00	0.174567
$586$	0	0
$587$	−8101.76	−0.569668	−0.284834	−	0.958577i	$-0.591939\pi$
$587$	−8101.76	−0.569668	−0.284834	+	0.958577i	$0.591939\pi$
$588$	0	0
$589$	−34560.0	−2.41769
$590$	0	0
$591$	−17999.7	−1.25281
$592$	0	0
$593$	−26958.0	−1.86683	−0.933417	−	0.358794i	$-0.883188\pi$
$593$	−26958.0	−1.86683	−0.933417	+	0.358794i	$0.883188\pi$
$594$	0	0
$595$	3225.52	0.222241
$596$	0	0
$597$	−22720.0	−1.55757
$598$	0	0
$599$	6349.85	0.433135	0.216568	−	0.976268i	$-0.430514\pi$
$599$	6349.85	0.433135	0.216568	+	0.976268i	$0.430514\pi$
$600$	0	0
$601$	21970.0	1.49114	0.745570	−	0.666427i	$-0.232177\pi$
$601$	21970.0	1.49114	0.745570	+	0.666427i	$0.232177\pi$
$602$	0	0
$603$	−10606.3	−0.716287
$604$	0	0
$605$	5855.00	0.393454
$606$	0	0
$607$	3876.95	0.259243	0.129622	−	0.991564i	$-0.458624\pi$
$607$	3876.95	0.259243	0.129622	+	0.991564i	$0.458624\pi$
$608$	0	0
$609$	32400.0	2.15585
$610$	0	0
$611$	−5047.00	−0.334173
$612$	0	0
$613$	−2878.00	−0.189627	−0.0948135	−	0.995495i	$-0.530225\pi$
$613$	−2878.00	−0.189627	−0.0948135	+	0.995495i	$0.530225\pi$
$614$	0	0
$615$	8538.15	0.559823
$616$	0	0
$617$	27354.0	1.78481	0.892407	−	0.451231i	$-0.149015\pi$
$617$	27354.0	1.78481	0.892407	+	0.451231i	$0.149015\pi$
$618$	0	0
$619$	−12547.9	−0.814771	−0.407386	−	0.913256i	$-0.633559\pi$
$619$	−12547.9	−0.814771	−0.407386	+	0.913256i	$0.633559\pi$
$620$	0	0
$621$	7280.00	0.470429
$622$	0	0
$623$	−16886.6	−1.08595
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	8095.43	0.515631
$628$	0	0
$629$	−7004.00	−0.443987
$630$	0	0
$631$	−30876.5	−1.94798	−0.973988	−	0.226598i	$-0.927240\pi$
$631$	−30876.5	−1.94798	−0.973988	+	0.226598i	$0.927240\pi$
$632$	0	0
$633$	−26480.0	−1.66269
$634$	0	0
$635$	−1170.04	−0.0731208
$636$	0	0
$637$	−646.000	−0.0401812
$638$	0	0
$639$	8386.36	0.519185
$640$	0	0
$641$	−9430.00	−0.581065	−0.290532	−	0.956865i	$-0.593832\pi$
$641$	−9430.00	−0.581065	−0.290532	+	0.956865i	$0.593832\pi$
$642$	0	0
$643$	9847.33	0.603952	0.301976	−	0.953316i	$-0.402354\pi$
$643$	9847.33	0.603952	0.301976	+	0.953316i	$0.402354\pi$
$644$	0	0
$645$	−17000.0	−1.03779
$646$	0	0
$647$	−30048.0	−1.82582	−0.912911	−	0.408158i	$-0.866171\pi$
$647$	−30048.0	−1.82582	−0.912911	+	0.408158i	$0.866171\pi$
$648$	0	0
$649$	960.000	0.0580636
$650$	0	0
$651$	−40983.1	−2.46737
$652$	0	0
$653$	−18742.0	−1.12317	−0.561586	−	0.827418i	$-0.689809\pi$
$653$	−18742.0	−1.12317	−0.561586	+	0.827418i	$0.689809\pi$
$654$	0	0
$655$	−8664.64	−0.516879
$656$	0	0
$657$	−14014.0	−0.832174
$658$	0	0
$659$	8323.11	0.491992	0.245996	−	0.969271i	$-0.420885\pi$
$659$	8323.11	0.491992	0.245996	+	0.969271i	$0.420885\pi$
$660$	0	0
$661$	−7630.00	−0.448975	−0.224488	−	0.974477i	$-0.572071\pi$
$661$	−7630.00	−0.448975	−0.224488	+	0.974477i	$0.572071\pi$
$662$	0	0
$663$	−8171.33	−0.478655
$664$	0	0
$665$	−9600.00	−0.559808
$666$	0	0
$667$	22199.2	1.28869
$668$	0	0
$669$	−30120.0	−1.74067
$670$	0	0
$671$	−3162.28	−0.181935
$672$	0	0
$673$	−10878.0	−0.623055	−0.311528	−	0.950237i	$-0.600841\pi$
$673$	−10878.0	−0.623055	−0.311528	+	0.950237i	$0.600841\pi$
$674$	0	0
$675$	−2213.59	−0.126224
$676$	0	0
$677$	−126.000	−0.00715299	−0.00357649	−	0.999994i	$-0.501138\pi$
$677$	−126.000	−0.00715299	−0.00357649	+	0.999994i	$0.501138\pi$
$678$	0	0
$679$	4819.31	0.272383
$680$	0	0
$681$	10520.0	0.591964
$682$	0	0
$683$	−16412.2	−0.919467	−0.459734	−	0.888057i	$-0.652055\pi$
$683$	−16412.2	−0.919467	−0.459734	+	0.888057i	$0.652055\pi$
$684$	0	0
$685$	−7730.00	−0.431165
$686$	0	0
$687$	6640.78	0.368794
$688$	0	0
$689$	−9804.00	−0.542094
$690$	0	0
$691$	13193.0	0.726319	0.363159	−	0.931727i	$-0.381698\pi$
$691$	13193.0	0.726319	0.363159	+	0.931727i	$0.381698\pi$
$692$	0	0
$693$	3120.00	0.171023
$694$	0	0
$695$	−1644.38	−0.0897483
$696$	0	0
$697$	−9180.00	−0.498877
$698$	0	0
$699$	17569.6	0.950707
$700$	0	0
$701$	22010.0	1.18589	0.592943	−	0.805244i	$-0.297966\pi$
$701$	22010.0	1.18589	0.592943	+	0.805244i	$0.297966\pi$
$702$	0	0
$703$	20845.7	1.11837
$704$	0	0
$705$	−4200.00	−0.224370
$706$	0	0
$707$	11346.3	0.603564
$708$	0	0
$709$	−550.000	−0.0291335	−0.0145668	−	0.999894i	$-0.504637\pi$
$709$	−550.000	−0.0291335	−0.0145668	+	0.999894i	$0.504637\pi$
$710$	0	0
$711$	3617.65	0.190819
$712$	0	0
$713$	−28080.0	−1.47490
$714$	0	0
$715$	−2403.33	−0.125706
$716$	0	0
$717$	16160.0	0.841710
$718$	0	0
$719$	17936.4	0.930343	0.465171	−	0.885221i	$-0.345993\pi$
$719$	17936.4	0.930343	0.465171	+	0.885221i	$0.345993\pi$
$720$	0	0
$721$	−9480.00	−0.489672
$722$	0	0
$723$	−33836.4	−1.74051
$724$	0	0
$725$	−6750.00	−0.345778
$726$	0	0
$727$	16728.4	0.853403	0.426701	−	0.904393i	$-0.359676\pi$
$727$	16728.4	0.853403	0.426701	+	0.904393i	$0.359676\pi$
$728$	0	0
$729$	3277.00	0.166489
$730$	0	0
$731$	18278.0	0.924808
$732$	0	0
$733$	−2422.00	−0.122044	−0.0610222	−	0.998136i	$-0.519436\pi$
$733$	−2422.00	−0.122044	−0.0610222	+	0.998136i	$0.519436\pi$
$734$	0	0
$735$	−537.587	−0.0269785
$736$	0	0
$737$	10320.0	0.515797
$738$	0	0
$739$	−19555.5	−0.973426	−0.486713	−	0.873562i	$-0.661804\pi$
$739$	−19555.5	−0.973426	−0.486713	+	0.873562i	$0.661804\pi$
$740$	0	0
$741$	24320.0	1.20569
$742$	0	0
$743$	31059.9	1.53362	0.766808	−	0.641876i	$-0.221844\pi$
$743$	31059.9	1.53362	0.766808	+	0.641876i	$0.221844\pi$
$744$	0	0
$745$	16230.0	0.798149
$746$	0	0
$747$	−14388.4	−0.704743
$748$	0	0
$749$	−11880.0	−0.579554
$750$	0	0
$751$	−12155.8	−0.590641	−0.295320	−	0.955398i	$-0.595426\pi$
$751$	−12155.8	−0.590641	−0.295320	+	0.955398i	$0.595426\pi$
$752$	0	0
$753$	−37200.0	−1.80032
$754$	0	0
$755$	−7526.22	−0.362791
$756$	0	0
$757$	19346.0	0.928854	0.464427	−	0.885611i	$-0.346260\pi$
$757$	19346.0	0.928854	0.464427	+	0.885611i	$0.346260\pi$
$758$	0	0
$759$	6577.54	0.314558
$760$	0	0
$761$	−33078.0	−1.57566	−0.787830	−	0.615893i	$-0.788795\pi$
$761$	−33078.0	−1.57566	−0.787830	+	0.615893i	$0.788795\pi$
$762$	0	0
$763$	16203.5	0.768816
$764$	0	0
$765$	−2210.00	−0.104448
$766$	0	0
$767$	2884.00	0.135769
$768$	0	0
$769$	32530.0	1.52544	0.762719	−	0.646730i	$-0.223864\pi$
$769$	32530.0	1.52544	0.762719	+	0.646730i	$0.223864\pi$
$770$	0	0
$771$	6792.57	0.317287
$772$	0	0
$773$	12002.0	0.558450	0.279225	−	0.960226i	$-0.409922\pi$
$773$	12002.0	0.558450	0.279225	+	0.960226i	$0.409922\pi$
$774$	0	0
$775$	8538.15	0.395741
$776$	0	0
$777$	24720.0	1.14134
$778$	0	0
$779$	27322.1	1.25663
$780$	0	0
$781$	−8160.00	−0.373864
$782$	0	0
$783$	23906.8	1.09114
$784$	0	0
$785$	−6130.00	−0.278712
$786$	0	0
$787$	19954.0	0.903789	0.451895	−	0.892071i	$-0.350748\pi$
$787$	19954.0	0.903789	0.451895	+	0.892071i	$0.350748\pi$
$788$	0	0
$789$	9400.00	0.424143
$790$	0	0
$791$	−32217.3	−1.44819
$792$	0	0
$793$	−9500.00	−0.425416
$794$	0	0
$795$	−8158.68	−0.363973
$796$	0	0
$797$	32666.0	1.45181	0.725903	−	0.687797i	$-0.241422\pi$
$797$	32666.0	1.45181	0.725903	+	0.687797i	$0.241422\pi$
$798$	0	0
$799$	4515.73	0.199944
$800$	0	0
$801$	11570.0	0.510369
$802$	0	0
$803$	13635.7	0.599246
$804$	0	0
$805$	−7800.00	−0.341508
$806$	0	0
$807$	−2567.77	−0.112007
$808$	0	0
$809$	−23110.0	−1.00433	−0.502166	−	0.864771i	$-0.667463\pi$
$809$	−23110.0	−1.00433	−0.502166	+	0.864771i	$0.667463\pi$
$810$	0	0
$811$	−35632.5	−1.54282	−0.771411	−	0.636338i	$-0.780448\pi$
$811$	−35632.5	−1.54282	−0.771411	+	0.636338i	$0.780448\pi$
$812$	0	0
$813$	−2480.00	−0.106983
$814$	0	0
$815$	−7241.62	−0.311243
$816$	0	0
$817$	−54400.0	−2.32952
$818$	0	0
$819$	9372.99	0.399901
$820$	0	0
$821$	8850.00	0.376208	0.188104	−	0.982149i	$-0.439766\pi$
$821$	8850.00	0.376208	0.188104	+	0.982149i	$0.439766\pi$
$822$	0	0
$823$	−13895.0	−0.588519	−0.294259	−	0.955726i	$-0.595073\pi$
$823$	−13895.0	−0.588519	−0.294259	+	0.955726i	$0.595073\pi$
$824$	0	0
$825$	−2000.00	−0.0844013
$826$	0	0
$827$	35841.3	1.50704	0.753520	−	0.657425i	$-0.228354\pi$
$827$	35841.3	1.50704	0.753520	+	0.657425i	$0.228354\pi$
$828$	0	0
$829$	23034.0	0.965023	0.482511	−	0.875890i	$-0.339725\pi$
$829$	23034.0	0.965023	0.482511	+	0.875890i	$0.339725\pi$
$830$	0	0
$831$	−37529.9	−1.56666
$832$	0	0
$833$	578.000	0.0240414
$834$	0	0
$835$	−11668.8	−0.483612
$836$	0	0
$837$	−30240.0	−1.24880
$838$	0	0
$839$	−36960.7	−1.52089	−0.760444	−	0.649403i	$-0.775019\pi$
$839$	−36960.7	−1.52089	−0.760444	+	0.649403i	$0.775019\pi$
$840$	0	0
$841$	48511.0	1.98905
$842$	0	0
$843$	−11826.9	−0.483204
$844$	0	0
$845$	3765.00	0.153278
$846$	0	0
$847$	22218.2	0.901328
$848$	0	0
$849$	30920.0	1.24991
$850$	0	0
$851$	16937.2	0.682254
$852$	0	0
$853$	19122.0	0.767555	0.383778	−	0.923425i	$-0.374623\pi$
$853$	19122.0	0.767555	0.383778	+	0.923425i	$0.374623\pi$
$854$	0	0
$855$	6577.54	0.263096
$856$	0	0
$857$	17786.0	0.708936	0.354468	−	0.935068i	$-0.384662\pi$
$857$	17786.0	0.708936	0.354468	+	0.935068i	$0.384662\pi$
$858$	0	0
$859$	28713.5	1.14050	0.570251	−	0.821470i	$-0.306846\pi$
$859$	28713.5	1.14050	0.570251	+	0.821470i	$0.306846\pi$
$860$	0	0
$861$	32400.0	1.28245
$862$	0	0
$863$	23748.7	0.936750	0.468375	−	0.883530i	$-0.344840\pi$
$863$	23748.7	0.936750	0.468375	+	0.883530i	$0.344840\pi$
$864$	0	0
$865$	−15490.0	−0.608874
$866$	0	0
$867$	−23761.4	−0.930770
$868$	0	0
$869$	−3520.00	−0.137408
$870$	0	0
$871$	31003.0	1.20608
$872$	0	0
$873$	−3302.00	−0.128013
$874$	0	0
$875$	2371.71	0.0916324
$876$	0	0
$877$	7706.00	0.296708	0.148354	−	0.988934i	$-0.452602\pi$
$877$	7706.00	0.296708	0.148354	+	0.988934i	$0.452602\pi$
$878$	0	0
$879$	−32875.0	−1.26149
$880$	0	0
$881$	10410.0	0.398095	0.199048	−	0.979990i	$-0.436215\pi$
$881$	10410.0	0.398095	0.199048	+	0.979990i	$0.436215\pi$
$882$	0	0
$883$	26822.4	1.02225	0.511125	−	0.859506i	$-0.329229\pi$
$883$	26822.4	1.02225	0.511125	+	0.859506i	$0.329229\pi$
$884$	0	0
$885$	2400.00	0.0911583
$886$	0	0
$887$	21130.3	0.799873	0.399937	−	0.916543i	$-0.369032\pi$
$887$	21130.3	0.799873	0.399937	+	0.916543i	$0.369032\pi$
$888$	0	0
$889$	−4440.00	−0.167506
$890$	0	0
$891$	11523.3	0.433273
$892$	0	0
$893$	−13440.0	−0.503642
$894$	0	0
$895$	−11763.7	−0.439348
$896$	0	0
$897$	19760.0	0.735526
$898$	0	0
$899$	−92212.0	−3.42096
$900$	0	0
$901$	8772.00	0.324348
$902$	0	0
$903$	−64510.5	−2.37738
$904$	0	0
$905$	10910.0	0.400730
$906$	0	0
$907$	19220.3	0.703639	0.351819	−	0.936068i	$-0.385563\pi$
$907$	19220.3	0.703639	0.351819	+	0.936068i	$0.385563\pi$
$908$	0	0
$909$	−7774.00	−0.283660
$910$	0	0
$911$	39402.0	1.43298	0.716491	−	0.697597i	$-0.245747\pi$
$911$	39402.0	1.43298	0.716491	+	0.697597i	$0.245747\pi$
$912$	0	0
$913$	14000.0	0.507483
$914$	0	0
$915$	−7905.69	−0.285633
$916$	0	0
$917$	−32880.0	−1.18407
$918$	0	0
$919$	42931.1	1.54099	0.770493	−	0.637449i	$-0.220010\pi$
$919$	42931.1	1.54099	0.770493	+	0.637449i	$0.220010\pi$
$920$	0	0
$921$	−23720.0	−0.848643
$922$	0	0
$923$	−24514.0	−0.874201
$924$	0	0
$925$	−5150.00	−0.183060
$926$	0	0
$927$	6495.32	0.230134
$928$	0	0
$929$	34746.0	1.22710	0.613552	−	0.789654i	$-0.289740\pi$
$929$	34746.0	1.22710	0.613552	+	0.789654i	$0.289740\pi$
$930$	0	0
$931$	−1720.28	−0.0605584
$932$	0	0
$933$	−39600.0	−1.38955
$934$	0	0
$935$	2150.35	0.0752128
$936$	0	0
$937$	21594.0	0.752876	0.376438	−	0.926442i	$-0.377149\pi$
$937$	21594.0	0.752876	0.376438	+	0.926442i	$0.377149\pi$
$938$	0	0
$939$	14027.9	0.487521
$940$	0	0
$941$	20018.0	0.693484	0.346742	−	0.937961i	$-0.387288\pi$
$941$	20018.0	0.693484	0.346742	+	0.937961i	$0.387288\pi$
$942$	0	0
$943$	22199.2	0.766601
$944$	0	0
$945$	−8400.00	−0.289156
$946$	0	0
$947$	−46150.3	−1.58361	−0.791807	−	0.610771i	$-0.790859\pi$
$947$	−46150.3	−1.58361	−0.791807	+	0.610771i	$0.790859\pi$
$948$	0	0
$949$	40964.0	1.40121
$950$	0	0
$951$	−26145.7	−0.891517
$952$	0	0
$953$	−342.000	−0.0116248	−0.00581242	−	0.999983i	$-0.501850\pi$
$953$	−342.000	−0.0116248	−0.00581242	+	0.999983i	$0.501850\pi$
$954$	0	0
$955$	15115.7	0.512180
$956$	0	0
$957$	21600.0	0.729602
$958$	0	0
$959$	−29333.3	−0.987718
$960$	0	0
$961$	86849.0	2.91528
$962$	0	0
$963$	8139.70	0.272376
$964$	0	0
$965$	−6490.00	−0.216498
$966$	0	0
$967$	−51728.5	−1.72025	−0.860123	−	0.510087i	$-0.829613\pi$
$967$	−51728.5	−1.72025	−0.860123	+	0.510087i	$0.829613\pi$
$968$	0	0
$969$	−21760.0	−0.721395
$970$	0	0
$971$	3099.03	0.102423	0.0512115	−	0.998688i	$-0.483692\pi$
$971$	3099.03	0.102423	0.0512115	+	0.998688i	$0.483692\pi$
$972$	0	0
$973$	−6240.00	−0.205596
$974$	0	0
$975$	−6008.33	−0.197354
$976$	0	0
$977$	26226.0	0.858796	0.429398	−	0.903115i	$-0.358726\pi$
$977$	26226.0	0.858796	0.429398	+	0.903115i	$0.358726\pi$
$978$	0	0
$979$	−11257.7	−0.367516
$980$	0	0
$981$	−11102.0	−0.361325
$982$	0	0
$983$	11049.0	0.358503	0.179251	−	0.983803i	$-0.442632\pi$
$983$	11049.0	0.358503	0.179251	+	0.983803i	$0.442632\pi$
$984$	0	0
$985$	14230.0	0.460310
$986$	0	0
$987$	−15937.9	−0.513990
$988$	0	0
$989$	−44200.0	−1.42111
$990$	0	0
$991$	45928.9	1.47223	0.736115	−	0.676856i	$-0.236658\pi$
$991$	45928.9	1.47223	0.736115	+	0.676856i	$0.236658\pi$
$992$	0	0
$993$	75600.0	2.41601
$994$	0	0
$995$	17961.7	0.572287
$996$	0	0
$997$	31026.0	0.985560	0.492780	−	0.870154i	$-0.335981\pi$
$997$	31026.0	0.985560	0.492780	+	0.870154i	$0.335981\pi$
$998$	0	0
$999$	18240.0	0.577666

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.a.p.1.2		2
4.3	odd	2	inner	320.4.a.p.1.1		2
5.4	even	2		1600.4.a.ch.1.1		2
8.3	odd	2		160.4.a.f.1.2	yes	2
8.5	even	2		160.4.a.f.1.1	✓	2
16.3	odd	4		1280.4.d.u.641.2		4
16.5	even	4		1280.4.d.u.641.1		4
16.11	odd	4		1280.4.d.u.641.3		4
16.13	even	4		1280.4.d.u.641.4		4
20.19	odd	2		1600.4.a.ch.1.2		2
24.5	odd	2		1440.4.a.v.1.1		2
24.11	even	2		1440.4.a.v.1.2		2
40.3	even	4		800.4.c.j.449.3		4
40.13	odd	4		800.4.c.j.449.2		4
40.19	odd	2		800.4.a.p.1.1		2
40.27	even	4		800.4.c.j.449.1		4
40.29	even	2		800.4.a.p.1.2		2
40.37	odd	4		800.4.c.j.449.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.f.1.1	✓	2	8.5	even	2
160.4.a.f.1.2	yes	2	8.3	odd	2
320.4.a.p.1.1		2	4.3	odd	2	inner
320.4.a.p.1.2		2	1.1	even	1	trivial
800.4.a.p.1.1		2	40.19	odd	2
800.4.a.p.1.2		2	40.29	even	2
800.4.c.j.449.1		4	40.27	even	4
800.4.c.j.449.2		4	40.13	odd	4
800.4.c.j.449.3		4	40.3	even	4
800.4.c.j.449.4		4	40.37	odd	4
1280.4.d.u.641.1		4	16.5	even	4
1280.4.d.u.641.2		4	16.3	odd	4
1280.4.d.u.641.3		4	16.11	odd	4
1280.4.d.u.641.4		4	16.13	even	4
1440.4.a.v.1.1		2	24.5	odd	2
1440.4.a.v.1.2		2	24.11	even	2
1600.4.a.ch.1.1		2	5.4	even	2
1600.4.a.ch.1.2		2	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q+6.32456 q^{3} -5.00000 q^{5} -18.9737 q^{7} +13.0000 q^{9} -12.6491 q^{11} -38.0000 q^{13} -31.6228 q^{15} +34.0000 q^{17} -101.193 q^{19} -120.000 q^{21} -82.2192 q^{23} +25.0000 q^{25} -88.5438 q^{27} -270.000 q^{29} +341.526 q^{31} -80.0000 q^{33} +94.8683 q^{35} -206.000 q^{37} -240.333 q^{39} -270.000 q^{41} +537.587 q^{43} -65.0000 q^{45} +132.816 q^{47} +17.0000 q^{49} +215.035 q^{51} +258.000 q^{53} +63.2456 q^{55} -640.000 q^{57} -75.8947 q^{59} +250.000 q^{61} -246.658 q^{63} +190.000 q^{65} -815.868 q^{67} -520.000 q^{69} +645.105 q^{71} -1078.00 q^{73} +158.114 q^{75} +240.000 q^{77} +278.280 q^{79} -911.000 q^{81} -1106.80 q^{83} -170.000 q^{85} -1707.63 q^{87} +890.000 q^{89} +720.999 q^{91} +2160.00 q^{93} +505.964 q^{95} -254.000 q^{97} -164.438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} + 26 q^{9} - 76 q^{13} + 68 q^{17} - 240 q^{21} + 50 q^{25} - 540 q^{29} - 160 q^{33} - 412 q^{37} - 540 q^{41} - 130 q^{45} + 34 q^{49} + 516 q^{53} - 1280 q^{57} + 500 q^{61} + 380 q^{65}+ \cdots - 508 q^{97}+O(q^{100}) \) 2 * q - 10 * q^5 + 26 * q^9 - 76 * q^13 + 68 * q^17 - 240 * q^21 + 50 * q^25 - 540 * q^29 - 160 * q^33 - 412 * q^37 - 540 * q^41 - 130 * q^45 + 34 * q^49 + 516 * q^53 - 1280 * q^57 + 500 * q^61 + 380 * q^65 - 1040 * q^69 - 2156 * q^73 + 480 * q^77 - 1822 * q^81 - 340 * q^85 + 1780 * q^89 + 4320 * q^93 - 508 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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