LMFDB - Embedded newform 464.4.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	464.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:	$27.3768862427$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 $ x^5 - x^4 - 34x^3 + 74x^2 + 94*x - 198
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.52813$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.01127 q^{3} +0.397400 q^{5} +14.4223 q^{7} -25.9773 q^{9} +O(q^{10})$	$q+1.01127 q^{3} +0.397400 q^{5} +14.4223 q^{7} -25.9773 q^{9} -52.7632 q^{11} +60.7622 q^{13} +0.401880 q^{15} +0.0555215 q^{17} -100.067 q^{19} +14.5849 q^{21} -15.2721 q^{23} -124.842 q^{25} -53.5746 q^{27} +29.0000 q^{29} -172.413 q^{31} -53.3580 q^{33} +5.73143 q^{35} +305.937 q^{37} +61.4472 q^{39} +318.356 q^{41} -467.752 q^{43} -10.3234 q^{45} -249.449 q^{47} -134.997 q^{49} +0.0561475 q^{51} -201.710 q^{53} -20.9681 q^{55} -101.195 q^{57} +696.469 q^{59} -796.475 q^{61} -374.653 q^{63} +24.1469 q^{65} -828.956 q^{67} -15.4443 q^{69} -676.393 q^{71} -735.147 q^{73} -126.249 q^{75} -760.967 q^{77} -149.457 q^{79} +647.209 q^{81} +947.959 q^{83} +0.0220643 q^{85} +29.3269 q^{87} -80.8893 q^{89} +876.332 q^{91} -174.357 q^{93} -39.7667 q^{95} -1035.24 q^{97} +1370.65 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} + 10 q^{5} - 32 q^{7} + 29 q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 + 10 * q^5 - 32 * q^7 + 29 * q^9	$ 5 q - 4 q^{3} + 10 q^{5} - 32 q^{7} + 29 q^{9} - 36 q^{11} + 26 q^{13} - 88 q^{15} + 82 q^{17} - 156 q^{19} + 72 q^{21} - 336 q^{23} + 151 q^{25} - 352 q^{27} + 145 q^{29} - 432 q^{31} + 108 q^{33} - 600 q^{35} - 18 q^{37} - 688 q^{39} + 82 q^{41} - 340 q^{43} - 146 q^{45} - 680 q^{47} - 115 q^{49} - 608 q^{51} - 102 q^{53} - 736 q^{55} - 576 q^{57} - 924 q^{59} - 618 q^{61} - 584 q^{63} - 704 q^{65} - 44 q^{67} - 1056 q^{69} - 1032 q^{71} - 1078 q^{73} + 468 q^{75} - 888 q^{77} - 200 q^{79} - 1843 q^{81} - 452 q^{83} - 1700 q^{85} - 116 q^{87} - 1790 q^{89} + 1128 q^{91} - 1884 q^{93} - 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 + 10 * q^5 - 32 * q^7 + 29 * q^9 - 36 * q^11 + 26 * q^13 - 88 * q^15 + 82 * q^17 - 156 * q^19 + 72 * q^21 - 336 * q^23 + 151 * q^25 - 352 * q^27 + 145 * q^29 - 432 * q^31 + 108 * q^33 - 600 * q^35 - 18 * q^37 - 688 * q^39 + 82 * q^41 - 340 * q^43 - 146 * q^45 - 680 * q^47 - 115 * q^49 - 608 * q^51 - 102 * q^53 - 736 * q^55 - 576 * q^57 - 924 * q^59 - 618 * q^61 - 584 * q^63 - 704 * q^65 - 44 * q^67 - 1056 * q^69 - 1032 * q^71 - 1078 * q^73 + 468 * q^75 - 888 * q^77 - 200 * q^79 - 1843 * q^81 - 452 * q^83 - 1700 * q^85 - 116 * q^87 - 1790 * q^89 + 1128 * q^91 - 1884 * q^93 - 1024 * q^95 - 2518 * q^97 + 1500 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.01127	0.194620	0.0973098	−	0.995254i	$-0.468976\pi$
$3$	1.01127	0.194620	0.0973098	+	0.995254i	$0.468976\pi$
$4$	0	0
$5$	0.397400	0.0355446	0.0177723	−	0.999842i	$-0.494343\pi$
$5$	0.397400	0.0355446	0.0177723	+	0.999842i	$0.494343\pi$
$6$	0	0
$7$	14.4223	0.778731	0.389366	−	0.921083i	$-0.372694\pi$
$7$	14.4223	0.778731	0.389366	+	0.921083i	$0.372694\pi$
$8$	0	0
$9$	−25.9773	−0.962123
$10$	0	0
$11$	−52.7632	−1.44625	−0.723123	−	0.690719i	$-0.757294\pi$
$11$	−52.7632	−1.44625	−0.723123	+	0.690719i	$0.757294\pi$
$12$	0	0
$13$	60.7622	1.29634	0.648170	−	0.761496i	$-0.275535\pi$
$13$	60.7622	1.29634	0.648170	+	0.761496i	$0.275535\pi$
$14$	0	0
$15$	0.401880	0.00691767
$16$	0	0
$17$	0.0555215	0.000792115 0	0.000396057	−	1.00000i	$-0.499874\pi$
$17$	0.0555215	0.000792115 0	0.000396057		1.00000i	$0.499874\pi$
$18$	0	0
$19$	−100.067	−1.20826	−0.604131	−	0.796885i	$-0.706480\pi$
$19$	−100.067	−1.20826	−0.604131	+	0.796885i	$0.706480\pi$
$20$	0	0
$21$	14.5849	0.151556
$22$	0	0
$23$	−15.2721	−0.138454	−0.0692272	−	0.997601i	$-0.522053\pi$
$23$	−15.2721	−0.138454	−0.0692272	+	0.997601i	$0.522053\pi$
$24$	0	0
$25$	−124.842	−0.998737
$26$	0	0
$27$	−53.5746	−0.381868
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−172.413	−0.998915	−0.499458	−	0.866338i	$-0.666467\pi$
$31$	−172.413	−0.998915	−0.499458	+	0.866338i	$0.666467\pi$
$32$	0	0
$33$	−53.3580	−0.281468
$34$	0	0
$35$	5.73143	0.0276797
$36$	0	0
$37$	305.937	1.35934	0.679672	−	0.733516i	$-0.262122\pi$
$37$	305.937	1.35934	0.679672	+	0.733516i	$0.262122\pi$
$38$	0	0
$39$	61.4472	0.252293
$40$	0	0
$41$	318.356	1.21265	0.606327	−	0.795215i	$-0.292642\pi$
$41$	318.356	1.21265	0.606327	+	0.795215i	$0.292642\pi$
$42$	0	0
$43$	−467.752	−1.65887	−0.829435	−	0.558603i	$-0.811338\pi$
$43$	−467.752	−1.65887	−0.829435	+	0.558603i	$0.811338\pi$
$44$	0	0
$45$	−10.3234	−0.0341982
$46$	0	0
$47$	−249.449	−0.774167	−0.387083	−	0.922045i	$-0.626517\pi$
$47$	−249.449	−0.774167	−0.387083	+	0.922045i	$0.626517\pi$
$48$	0	0
$49$	−134.997	−0.393577
$50$	0	0
$51$	0.0561475	0.000154161 0
$52$	0	0
$53$	−201.710	−0.522775	−0.261387	−	0.965234i	$-0.584180\pi$
$53$	−201.710	−0.522775	−0.261387	+	0.965234i	$0.584180\pi$
$54$	0	0
$55$	−20.9681	−0.0514062
$56$	0	0
$57$	−101.195	−0.235152
$58$	0	0
$59$	696.469	1.53682	0.768412	−	0.639956i	$-0.221047\pi$
$59$	696.469	1.53682	0.768412	+	0.639956i	$0.221047\pi$
$60$	0	0
$61$	−796.475	−1.67177	−0.835886	−	0.548903i	$-0.815046\pi$
$61$	−796.475	−1.67177	−0.835886	+	0.548903i	$0.815046\pi$
$62$	0	0
$63$	−374.653	−0.749236
$64$	0	0
$65$	24.1469	0.0460778
$66$	0	0
$67$	−828.956	−1.51154	−0.755769	−	0.654838i	$-0.772737\pi$
$67$	−828.956	−1.51154	−0.755769	+	0.654838i	$0.772737\pi$
$68$	0	0
$69$	−15.4443	−0.0269460
$70$	0	0
$71$	−676.393	−1.13061	−0.565303	−	0.824883i	$-0.691241\pi$
$71$	−676.393	−1.13061	−0.565303	+	0.824883i	$0.691241\pi$
$72$	0	0
$73$	−735.147	−1.17866	−0.589332	−	0.807891i	$-0.700609\pi$
$73$	−735.147	−1.17866	−0.589332	+	0.807891i	$0.700609\pi$
$74$	0	0
$75$	−126.249	−0.194374
$76$	0	0
$77$	−760.967	−1.12624
$78$	0	0
$79$	−149.457	−0.212851	−0.106425	−	0.994321i	$-0.533941\pi$
$79$	−149.457	−0.212851	−0.106425	+	0.994321i	$0.533941\pi$
$80$	0	0
$81$	647.209	0.887804
$82$	0	0
$83$	947.959	1.25364	0.626819	−	0.779165i	$-0.284356\pi$
$83$	947.959	1.25364	0.626819	+	0.779165i	$0.284356\pi$
$84$	0	0
$85$	0.0220643	2.81554e−5 0
$86$	0	0
$87$	29.3269	0.0361400
$88$	0	0
$89$	−80.8893	−0.0963399	−0.0481700	−	0.998839i	$-0.515339\pi$
$89$	−80.8893	−0.0963399	−0.0481700	+	0.998839i	$0.515339\pi$
$90$	0	0
$91$	876.332	1.00950
$92$	0	0
$93$	−174.357	−0.194409
$94$	0	0
$95$	−39.7667	−0.0429471
$96$	0	0
$97$	−1035.24	−1.08364	−0.541819	−	0.840495i	$-0.682264\pi$
$97$	−1035.24	−1.08364	−0.541819	+	0.840495i	$0.682264\pi$
$98$	0	0
$99$	1370.65	1.39147
$100$	0	0
$101$	1752.51	1.72655	0.863273	−	0.504738i	$-0.168411\pi$
$101$	1752.51	1.72655	0.863273	+	0.504738i	$0.168411\pi$
$102$	0	0
$103$	138.551	0.132542	0.0662709	−	0.997802i	$-0.478890\pi$
$103$	138.551	0.132542	0.0662709	+	0.997802i	$0.478890\pi$
$104$	0	0
$105$	5.79604	0.00538701
$106$	0	0
$107$	−899.538	−0.812725	−0.406363	−	0.913712i	$-0.633203\pi$
$107$	−899.538	−0.812725	−0.406363	+	0.913712i	$0.633203\pi$
$108$	0	0
$109$	−1968.23	−1.72957	−0.864783	−	0.502146i	$-0.832544\pi$
$109$	−1968.23	−1.72957	−0.864783	+	0.502146i	$0.832544\pi$
$110$	0	0
$111$	309.386	0.264555
$112$	0	0
$113$	299.097	0.248998	0.124499	−	0.992220i	$-0.460268\pi$
$113$	299.097	0.248998	0.124499	+	0.992220i	$0.460268\pi$
$114$	0	0
$115$	−6.06913	−0.00492130
$116$	0	0
$117$	−1578.44	−1.24724
$118$	0	0
$119$	0.800749	0.000616845 0
$120$	0	0
$121$	1452.96	1.09163
$122$	0	0
$123$	321.945	0.236006
$124$	0	0
$125$	−99.2873	−0.0710442
$126$	0	0
$127$	−491.709	−0.343560	−0.171780	−	0.985135i	$-0.554952\pi$
$127$	−491.709	−0.343560	−0.171780	+	0.985135i	$0.554952\pi$
$128$	0	0
$129$	−473.025	−0.322849
$130$	0	0
$131$	−1203.11	−0.802412	−0.401206	−	0.915988i	$-0.631409\pi$
$131$	−1203.11	−0.802412	−0.401206	+	0.915988i	$0.631409\pi$
$132$	0	0
$133$	−1443.20	−0.940912
$134$	0	0
$135$	−21.2905	−0.0135733
$136$	0	0
$137$	−1520.56	−0.948252	−0.474126	−	0.880457i	$-0.657236\pi$
$137$	−1520.56	−0.948252	−0.474126	+	0.880457i	$0.657236\pi$
$138$	0	0
$139$	1754.02	1.07031	0.535157	−	0.844753i	$-0.320252\pi$
$139$	1754.02	1.07031	0.535157	+	0.844753i	$0.320252\pi$
$140$	0	0
$141$	−252.261	−0.150668
$142$	0	0
$143$	−3206.01	−1.87483
$144$	0	0
$145$	11.5246	0.00660046
$146$	0	0
$147$	−136.519	−0.0765979
$148$	0	0
$149$	685.651	0.376985	0.188492	−	0.982075i	$-0.439640\pi$
$149$	685.651	0.376985	0.188492	+	0.982075i	$0.439640\pi$
$150$	0	0
$151$	3388.88	1.82638	0.913189	−	0.407535i	$-0.133612\pi$
$151$	3388.88	1.82638	0.913189	+	0.407535i	$0.133612\pi$
$152$	0	0
$153$	−1.44230	−0.000762112 0
$154$	0	0
$155$	−68.5172	−0.0355060
$156$	0	0
$157$	3040.88	1.54579	0.772893	−	0.634537i	$-0.218809\pi$
$157$	3040.88	1.54579	0.772893	+	0.634537i	$0.218809\pi$
$158$	0	0
$159$	−203.984	−0.101742
$160$	0	0
$161$	−220.259	−0.107819
$162$	0	0
$163$	−1330.78	−0.639477	−0.319739	−	0.947506i	$-0.603595\pi$
$163$	−1330.78	−0.639477	−0.319739	+	0.947506i	$0.603595\pi$
$164$	0	0
$165$	−21.2045	−0.0100047
$166$	0	0
$167$	2010.84	0.931755	0.465878	−	0.884849i	$-0.345739\pi$
$167$	2010.84	0.931755	0.465878	+	0.884849i	$0.345739\pi$
$168$	0	0
$169$	1495.05	0.680496
$170$	0	0
$171$	2599.48	1.16250
$172$	0	0
$173$	2614.73	1.14910	0.574549	−	0.818470i	$-0.305177\pi$
$173$	2614.73	1.14910	0.574549	+	0.818470i	$0.305177\pi$
$174$	0	0
$175$	−1800.51	−0.777748
$176$	0	0
$177$	704.321	0.299096
$178$	0	0
$179$	2332.61	0.974007	0.487003	−	0.873400i	$-0.338090\pi$
$179$	2332.61	0.974007	0.487003	+	0.873400i	$0.338090\pi$
$180$	0	0
$181$	3225.91	1.32475	0.662375	−	0.749172i	$-0.269549\pi$
$181$	3225.91	1.32475	0.662375	+	0.749172i	$0.269549\pi$
$182$	0	0
$183$	−805.454	−0.325360
$184$	0	0
$185$	121.579	0.0483173
$186$	0	0
$187$	−2.92949	−0.00114559
$188$	0	0
$189$	−772.669	−0.297372
$190$	0	0
$191$	−3661.21	−1.38699	−0.693497	−	0.720460i	$-0.743931\pi$
$191$	−3661.21	−1.38699	−0.693497	+	0.720460i	$0.743931\pi$
$192$	0	0
$193$	1360.18	0.507295	0.253647	−	0.967297i	$-0.418370\pi$
$193$	1360.18	0.507295	0.253647	+	0.967297i	$0.418370\pi$
$194$	0	0
$195$	24.4192	0.00896765
$196$	0	0
$197$	1753.46	0.634157	0.317078	−	0.948399i	$-0.397298\pi$
$197$	1753.46	0.634157	0.317078	+	0.948399i	$0.397298\pi$
$198$	0	0
$199$	−1750.67	−0.623625	−0.311813	−	0.950144i	$-0.600936\pi$
$199$	−1750.67	−0.623625	−0.311813	+	0.950144i	$0.600936\pi$
$200$	0	0
$201$	−838.301	−0.294175
$202$	0	0
$203$	418.247	0.144607
$204$	0	0
$205$	126.515	0.0431032
$206$	0	0
$207$	396.728	0.133210
$208$	0	0
$209$	5279.87	1.74744
$210$	0	0
$211$	4834.55	1.57736	0.788682	−	0.614801i	$-0.210764\pi$
$211$	4834.55	1.57736	0.788682	+	0.614801i	$0.210764\pi$
$212$	0	0
$213$	−684.018	−0.220038
$214$	0	0
$215$	−185.885	−0.0589638
$216$	0	0
$217$	−2486.60	−0.777887
$218$	0	0
$219$	−743.435	−0.229391
$220$	0	0
$221$	3.37361	0.00102685
$222$	0	0
$223$	2529.15	0.759481	0.379741	−	0.925093i	$-0.376013\pi$
$223$	2529.15	0.759481	0.379741	+	0.925093i	$0.376013\pi$
$224$	0	0
$225$	3243.06	0.960908
$226$	0	0
$227$	−1538.37	−0.449803	−0.224901	−	0.974382i	$-0.572206\pi$
$227$	−1538.37	−0.449803	−0.224901	+	0.974382i	$0.572206\pi$
$228$	0	0
$229$	3164.66	0.913216	0.456608	−	0.889668i	$-0.349064\pi$
$229$	3164.66	0.913216	0.456608	+	0.889668i	$0.349064\pi$
$230$	0	0
$231$	−769.546	−0.219188
$232$	0	0
$233$	255.030	0.0717064	0.0358532	−	0.999357i	$-0.488585\pi$
$233$	255.030	0.0717064	0.0358532	+	0.999357i	$0.488585\pi$
$234$	0	0
$235$	−99.1310	−0.0275174
$236$	0	0
$237$	−151.142	−0.0414250
$238$	0	0
$239$	−1338.89	−0.362367	−0.181183	−	0.983449i	$-0.557993\pi$
$239$	−1338.89	−0.362367	−0.181183	+	0.983449i	$0.557993\pi$
$240$	0	0
$241$	−4025.04	−1.07583	−0.537916	−	0.842998i	$-0.680788\pi$
$241$	−4025.04	−1.07583	−0.537916	+	0.842998i	$0.680788\pi$
$242$	0	0
$243$	2101.02	0.554652
$244$	0	0
$245$	−53.6479	−0.0139895
$246$	0	0
$247$	−6080.31	−1.56632
$248$	0	0
$249$	958.646	0.243983
$250$	0	0
$251$	1488.44	0.374300	0.187150	−	0.982331i	$-0.440075\pi$
$251$	1488.44	0.374300	0.187150	+	0.982331i	$0.440075\pi$
$252$	0	0
$253$	805.805	0.200239
$254$	0	0
$255$	0.0223130	5.47959e−6 0
$256$	0	0
$257$	6116.49	1.48458	0.742289	−	0.670080i	$-0.233740\pi$
$257$	6116.49	1.48458	0.742289	+	0.670080i	$0.233740\pi$
$258$	0	0
$259$	4412.32	1.05856
$260$	0	0
$261$	−753.342	−0.178662
$262$	0	0
$263$	−6639.59	−1.55671	−0.778355	−	0.627825i	$-0.783946\pi$
$263$	−6639.59	−1.55671	−0.778355	+	0.627825i	$0.783946\pi$
$264$	0	0
$265$	−80.1598	−0.0185818
$266$	0	0
$267$	−81.8012	−0.0187496
$268$	0	0
$269$	3585.49	0.812682	0.406341	−	0.913721i	$-0.366804\pi$
$269$	3585.49	0.812682	0.406341	+	0.913721i	$0.366804\pi$
$270$	0	0
$271$	−1491.93	−0.334421	−0.167211	−	0.985921i	$-0.553476\pi$
$271$	−1491.93	−0.334421	−0.167211	+	0.985921i	$0.553476\pi$
$272$	0	0
$273$	886.211	0.196469
$274$	0	0
$275$	6587.07	1.44442
$276$	0	0
$277$	−4465.71	−0.968658	−0.484329	−	0.874886i	$-0.660936\pi$
$277$	−4465.71	−0.968658	−0.484329	+	0.874886i	$0.660936\pi$
$278$	0	0
$279$	4478.84	0.961080
$280$	0	0
$281$	−5266.65	−1.11809	−0.559043	−	0.829139i	$-0.688831\pi$
$281$	−5266.65	−1.11809	−0.559043	+	0.829139i	$0.688831\pi$
$282$	0	0
$283$	1115.91	0.234395	0.117198	−	0.993109i	$-0.462609\pi$
$283$	1115.91	0.234395	0.117198	+	0.993109i	$0.462609\pi$
$284$	0	0
$285$	−40.2150	−0.00835836
$286$	0	0
$287$	4591.42	0.944332
$288$	0	0
$289$	−4913.00	−0.999999
$290$	0	0
$291$	−1046.91	−0.210897
$292$	0	0
$293$	2020.80	0.402923	0.201462	−	0.979496i	$-0.435431\pi$
$293$	2020.80	0.402923	0.201462	+	0.979496i	$0.435431\pi$
$294$	0	0
$295$	276.777	0.0546257
$296$	0	0
$297$	2826.77	0.552275
$298$	0	0
$299$	−927.967	−0.179484
$300$	0	0
$301$	−6746.06	−1.29181
$302$	0	0
$303$	1772.27	0.336020
$304$	0	0
$305$	−316.519	−0.0594224
$306$	0	0
$307$	1992.76	0.370465	0.185233	−	0.982695i	$-0.440696\pi$
$307$	1992.76	0.370465	0.185233	+	0.982695i	$0.440696\pi$
$308$	0	0
$309$	140.113	0.0257953
$310$	0	0
$311$	2845.58	0.518837	0.259418	−	0.965765i	$-0.416469\pi$
$311$	2845.58	0.518837	0.259418	+	0.965765i	$0.416469\pi$
$312$	0	0
$313$	−2951.92	−0.533075	−0.266537	−	0.963825i	$-0.585880\pi$
$313$	−2951.92	−0.533075	−0.266537	+	0.963825i	$0.585880\pi$
$314$	0	0
$315$	−148.887	−0.0266312
$316$	0	0
$317$	470.505	0.0833634	0.0416817	−	0.999131i	$-0.486728\pi$
$317$	470.505	0.0833634	0.0416817	+	0.999131i	$0.486728\pi$
$318$	0	0
$319$	−1530.13	−0.268561
$320$	0	0
$321$	−909.679	−0.158172
$322$	0	0
$323$	−5.55588	−0.000957083 0
$324$	0	0
$325$	−7585.68	−1.29470
$326$	0	0
$327$	−1990.42	−0.336608
$328$	0	0
$329$	−3597.63	−0.602868
$330$	0	0
$331$	−548.936	−0.0911548	−0.0455774	−	0.998961i	$-0.514513\pi$
$331$	−548.936	−0.0911548	−0.0455774	+	0.998961i	$0.514513\pi$
$332$	0	0
$333$	−7947.42	−1.30786
$334$	0	0
$335$	−329.427	−0.0537269
$336$	0	0
$337$	−6452.21	−1.04295	−0.521476	−	0.853266i	$-0.674618\pi$
$337$	−6452.21	−1.04295	−0.521476	+	0.853266i	$0.674618\pi$
$338$	0	0
$339$	302.469	0.0484598
$340$	0	0
$341$	9097.09	1.44468
$342$	0	0
$343$	−6893.82	−1.08522
$344$	0	0
$345$	−6.13756	−0.000957782 0
$346$	0	0
$347$	5601.62	0.866601	0.433301	−	0.901249i	$-0.357349\pi$
$347$	5601.62	0.866601	0.433301	+	0.901249i	$0.357349\pi$
$348$	0	0
$349$	8195.00	1.25693	0.628464	−	0.777838i	$-0.283684\pi$
$349$	8195.00	1.25693	0.628464	+	0.777838i	$0.283684\pi$
$350$	0	0
$351$	−3255.31	−0.495030
$352$	0	0
$353$	5190.74	0.782649	0.391325	−	0.920253i	$-0.372017\pi$
$353$	5190.74	0.782649	0.391325	+	0.920253i	$0.372017\pi$
$354$	0	0
$355$	−268.799	−0.0401869
$356$	0	0
$357$	0.809776	0.000120050 0
$358$	0	0
$359$	−5732.62	−0.842775	−0.421387	−	0.906881i	$-0.638457\pi$
$359$	−5732.62	−0.842775	−0.421387	+	0.906881i	$0.638457\pi$
$360$	0	0
$361$	3154.44	0.459898
$362$	0	0
$363$	1469.34	0.212452
$364$	0	0
$365$	−292.148	−0.0418951
$366$	0	0
$367$	1753.16	0.249358	0.124679	−	0.992197i	$-0.460210\pi$
$367$	1753.16	0.249358	0.124679	+	0.992197i	$0.460210\pi$
$368$	0	0
$369$	−8270.03	−1.16672
$370$	0	0
$371$	−2909.13	−0.407101
$372$	0	0
$373$	−6317.52	−0.876967	−0.438483	−	0.898739i	$-0.644484\pi$
$373$	−6317.52	−0.876967	−0.438483	+	0.898739i	$0.644484\pi$
$374$	0	0
$375$	−100.407	−0.0138266
$376$	0	0
$377$	1762.10	0.240724
$378$	0	0
$379$	9658.52	1.30904	0.654519	−	0.756046i	$-0.272871\pi$
$379$	9658.52	1.30904	0.654519	+	0.756046i	$0.272871\pi$
$380$	0	0
$381$	−497.253	−0.0668636
$382$	0	0
$383$	−897.046	−0.119679	−0.0598393	−	0.998208i	$-0.519059\pi$
$383$	−897.046	−0.119679	−0.0598393	+	0.998208i	$0.519059\pi$
$384$	0	0
$385$	−302.409	−0.0400316
$386$	0	0
$387$	12150.9	1.59604
$388$	0	0
$389$	−6841.32	−0.891693	−0.445847	−	0.895109i	$-0.647097\pi$
$389$	−6841.32	−0.891693	−0.445847	+	0.895109i	$0.647097\pi$
$390$	0	0
$391$	−0.847930	−0.000109672 0
$392$	0	0
$393$	−1216.67	−0.156165
$394$	0	0
$395$	−59.3942	−0.00756569
$396$	0	0
$397$	−6220.78	−0.786428	−0.393214	−	0.919447i	$-0.628637\pi$
$397$	−6220.78	−0.786428	−0.393214	+	0.919447i	$0.628637\pi$
$398$	0	0
$399$	−1459.47	−0.183120
$400$	0	0
$401$	−9466.98	−1.17895	−0.589474	−	0.807787i	$-0.700665\pi$
$401$	−9466.98	−1.17895	−0.589474	+	0.807787i	$0.700665\pi$
$402$	0	0
$403$	−10476.2	−1.29493
$404$	0	0
$405$	257.201	0.0315566
$406$	0	0
$407$	−16142.2	−1.96595
$408$	0	0
$409$	−13393.7	−1.61926	−0.809629	−	0.586941i	$-0.800332\pi$
$409$	−13393.7	−1.61926	−0.809629	+	0.586941i	$0.800332\pi$
$410$	0	0
$411$	−1537.71	−0.184549
$412$	0	0
$413$	10044.7	1.19677
$414$	0	0
$415$	376.719	0.0445600
$416$	0	0
$417$	1773.79	0.208304
$418$	0	0
$419$	−4325.28	−0.504305	−0.252152	−	0.967688i	$-0.581138\pi$
$419$	−4325.28	−0.504305	−0.252152	+	0.967688i	$0.581138\pi$
$420$	0	0
$421$	−825.786	−0.0955971	−0.0477985	−	0.998857i	$-0.515221\pi$
$421$	−825.786	−0.0955971	−0.0477985	+	0.998857i	$0.515221\pi$
$422$	0	0
$423$	6480.01	0.744844
$424$	0	0
$425$	−6.93142	−0.000791114 0
$426$	0	0
$427$	−11487.0	−1.30186
$428$	0	0
$429$	−3242.15	−0.364878
$430$	0	0
$431$	−9172.94	−1.02516	−0.512581	−	0.858639i	$-0.671311\pi$
$431$	−9172.94	−1.02516	−0.512581	+	0.858639i	$0.671311\pi$
$432$	0	0
$433$	6703.63	0.744009	0.372004	−	0.928231i	$-0.378671\pi$
$433$	6703.63	0.744009	0.372004	+	0.928231i	$0.378671\pi$
$434$	0	0
$435$	11.6545	0.00128458
$436$	0	0
$437$	1528.24	0.167289
$438$	0	0
$439$	−3578.82	−0.389084	−0.194542	−	0.980894i	$-0.562322\pi$
$439$	−3578.82	−0.389084	−0.194542	+	0.980894i	$0.562322\pi$
$440$	0	0
$441$	3506.86	0.378670
$442$	0	0
$443$	−5612.22	−0.601907	−0.300953	−	0.953639i	$-0.597305\pi$
$443$	−5612.22	−0.601907	−0.300953	+	0.953639i	$0.597305\pi$
$444$	0	0
$445$	−32.1454	−0.00342436
$446$	0	0
$447$	693.381	0.0733687
$448$	0	0
$449$	7835.33	0.823546	0.411773	−	0.911287i	$-0.364910\pi$
$449$	7835.33	0.823546	0.411773	+	0.911287i	$0.364910\pi$
$450$	0	0
$451$	−16797.5	−1.75380
$452$	0	0
$453$	3427.09	0.355449
$454$	0	0
$455$	348.254	0.0358822
$456$	0	0
$457$	−2670.59	−0.273358	−0.136679	−	0.990615i	$-0.543643\pi$
$457$	−2670.59	−0.273358	−0.136679	+	0.990615i	$0.543643\pi$
$458$	0	0
$459$	−2.97454	−0.000302483 0
$460$	0	0
$461$	−8776.95	−0.886731	−0.443366	−	0.896341i	$-0.646216\pi$
$461$	−8776.95	−0.886731	−0.443366	+	0.896341i	$0.646216\pi$
$462$	0	0
$463$	−1696.52	−0.170290	−0.0851448	−	0.996369i	$-0.527135\pi$
$463$	−1696.52	−0.170290	−0.0851448	+	0.996369i	$0.527135\pi$
$464$	0	0
$465$	−69.2896	−0.00691017
$466$	0	0
$467$	13310.1	1.31889	0.659443	−	0.751755i	$-0.270792\pi$
$467$	13310.1	1.31889	0.659443	+	0.751755i	$0.270792\pi$
$468$	0	0
$469$	−11955.5	−1.17708
$470$	0	0
$471$	3075.16	0.300840
$472$	0	0
$473$	24680.1	2.39914
$474$	0	0
$475$	12492.6	1.20674
$476$	0	0
$477$	5239.90	0.502974
$478$	0	0
$479$	13545.1	1.29205	0.646023	−	0.763318i	$-0.276431\pi$
$479$	13545.1	1.29205	0.646023	+	0.763318i	$0.276431\pi$
$480$	0	0
$481$	18589.4	1.76217
$482$	0	0
$483$	−222.742	−0.0209837
$484$	0	0
$485$	−411.406	−0.0385175
$486$	0	0
$487$	−6253.55	−0.581879	−0.290940	−	0.956741i	$-0.593968\pi$
$487$	−6253.55	−0.581879	−0.290940	+	0.956741i	$0.593968\pi$
$488$	0	0
$489$	−1345.78	−0.124455
$490$	0	0
$491$	−19651.9	−1.80627	−0.903135	−	0.429358i	$-0.858740\pi$
$491$	−19651.9	−1.80627	−0.903135	+	0.429358i	$0.858740\pi$
$492$	0	0
$493$	1.61012	0.000147092 0
$494$	0	0
$495$	544.695	0.0494591
$496$	0	0
$497$	−9755.14	−0.880438
$498$	0	0
$499$	8240.86	0.739302	0.369651	−	0.929171i	$-0.379477\pi$
$499$	8240.86	0.739302	0.369651	+	0.929171i	$0.379477\pi$
$500$	0	0
$501$	2033.51	0.181338
$502$	0	0
$503$	5183.95	0.459525	0.229762	−	0.973247i	$-0.426205\pi$
$503$	5183.95	0.459525	0.229762	+	0.973247i	$0.426205\pi$
$504$	0	0
$505$	696.447	0.0613693
$506$	0	0
$507$	1511.90	0.132438
$508$	0	0
$509$	−5768.83	−0.502355	−0.251178	−	0.967941i	$-0.580818\pi$
$509$	−5768.83	−0.502355	−0.251178	+	0.967941i	$0.580818\pi$
$510$	0	0
$511$	−10602.5	−0.917863
$512$	0	0
$513$	5361.06	0.461397
$514$	0	0
$515$	55.0601	0.00471114
$516$	0	0
$517$	13161.7	1.11964
$518$	0	0
$519$	2644.20	0.223637
$520$	0	0
$521$	8352.54	0.702364	0.351182	−	0.936307i	$-0.385780\pi$
$521$	8352.54	0.702364	0.351182	+	0.936307i	$0.385780\pi$
$522$	0	0
$523$	8984.23	0.751153	0.375577	−	0.926791i	$-0.377445\pi$
$523$	8984.23	0.751153	0.375577	+	0.926791i	$0.377445\pi$
$524$	0	0
$525$	−1820.81	−0.151365
$526$	0	0
$527$	−9.57266	−0.000791256 0
$528$	0	0
$529$	−11933.8	−0.980830
$530$	0	0
$531$	−18092.4	−1.47861
$532$	0	0
$533$	19344.0	1.57201
$534$	0	0
$535$	−357.477	−0.0288880
$536$	0	0
$537$	2358.90	0.189561
$538$	0	0
$539$	7122.88	0.569210
$540$	0	0
$541$	−20311.2	−1.61414	−0.807068	−	0.590459i	$-0.798947\pi$
$541$	−20311.2	−1.61414	−0.807068	+	0.590459i	$0.798947\pi$
$542$	0	0
$543$	3262.27	0.257822
$544$	0	0
$545$	−782.177	−0.0614766
$546$	0	0
$547$	−15772.1	−1.23285	−0.616424	−	0.787415i	$-0.711419\pi$
$547$	−15772.1	−1.23285	−0.616424	+	0.787415i	$0.711419\pi$
$548$	0	0
$549$	20690.3	1.60845
$550$	0	0
$551$	−2901.95	−0.224369
$552$	0	0
$553$	−2155.51	−0.165754
$554$	0	0
$555$	122.950	0.00940349
$556$	0	0
$557$	−22007.3	−1.67411	−0.837053	−	0.547121i	$-0.815724\pi$
$557$	−22007.3	−1.67411	−0.837053	+	0.547121i	$0.815724\pi$
$558$	0	0
$559$	−28421.6	−2.15046
$560$	0	0
$561$	−2.96252	−0.000222955 0
$562$	0	0
$563$	−10458.9	−0.782933	−0.391466	−	0.920192i	$-0.628032\pi$
$563$	−10458.9	−0.782933	−0.391466	+	0.920192i	$0.628032\pi$
$564$	0	0
$565$	118.861	0.00885051
$566$	0	0
$567$	9334.25	0.691361
$568$	0	0
$569$	−23223.1	−1.71101	−0.855505	−	0.517795i	$-0.826753\pi$
$569$	−23223.1	−1.71101	−0.855505	+	0.517795i	$0.826753\pi$
$570$	0	0
$571$	19270.6	1.41235	0.706174	−	0.708038i	$-0.250420\pi$
$571$	19270.6	1.41235	0.706174	+	0.708038i	$0.250420\pi$
$572$	0	0
$573$	−3702.48	−0.269936
$574$	0	0
$575$	1906.60	0.138280
$576$	0	0
$577$	−9604.90	−0.692993	−0.346497	−	0.938051i	$-0.612629\pi$
$577$	−9604.90	−0.692993	−0.346497	+	0.938051i	$0.612629\pi$
$578$	0	0
$579$	1375.51	0.0987296
$580$	0	0
$581$	13671.8	0.976248
$582$	0	0
$583$	10642.9	0.756061
$584$	0	0
$585$	−627.273	−0.0443325
$586$	0	0
$587$	−16362.2	−1.15050	−0.575249	−	0.817978i	$-0.695095\pi$
$587$	−16362.2	−1.15050	−0.575249	+	0.817978i	$0.695095\pi$
$588$	0	0
$589$	17252.9	1.20695
$590$	0	0
$591$	1773.23	0.123419
$592$	0	0
$593$	13326.7	0.922873	0.461436	−	0.887173i	$-0.347334\pi$
$593$	13326.7	0.922873	0.461436	+	0.887173i	$0.347334\pi$
$594$	0	0
$595$	0.318218	2.19255e−5 0
$596$	0	0
$597$	−1770.40	−0.121370
$598$	0	0
$599$	13513.2	0.921762	0.460881	−	0.887462i	$-0.347533\pi$
$599$	13513.2	0.921762	0.460881	+	0.887462i	$0.347533\pi$
$600$	0	0
$601$	−16678.5	−1.13200	−0.565998	−	0.824406i	$-0.691509\pi$
$601$	−16678.5	−1.13200	−0.565998	+	0.824406i	$0.691509\pi$
$602$	0	0
$603$	21534.0	1.45429
$604$	0	0
$605$	577.405	0.0388014
$606$	0	0
$607$	−4843.46	−0.323871	−0.161936	−	0.986801i	$-0.551774\pi$
$607$	−4843.46	−0.323871	−0.161936	+	0.986801i	$0.551774\pi$
$608$	0	0
$609$	422.962	0.0281433
$610$	0	0
$611$	−15157.1	−1.00358
$612$	0	0
$613$	17193.6	1.13286	0.566431	−	0.824110i	$-0.308324\pi$
$613$	17193.6	1.13286	0.566431	+	0.824110i	$0.308324\pi$
$614$	0	0
$615$	127.941	0.00838874
$616$	0	0
$617$	6436.63	0.419982	0.209991	−	0.977703i	$-0.432657\pi$
$617$	6436.63	0.419982	0.209991	+	0.977703i	$0.432657\pi$
$618$	0	0
$619$	21330.7	1.38506	0.692532	−	0.721388i	$-0.256495\pi$
$619$	21330.7	1.38506	0.692532	+	0.721388i	$0.256495\pi$
$620$	0	0
$621$	818.196	0.0528713
$622$	0	0
$623$	−1166.61	−0.0750229
$624$	0	0
$625$	15565.8	0.996211
$626$	0	0
$627$	5339.39	0.340087
$628$	0	0
$629$	16.9861	0.00107676
$630$	0	0
$631$	21812.9	1.37616	0.688080	−	0.725635i	$-0.258454\pi$
$631$	21812.9	1.37616	0.688080	+	0.725635i	$0.258454\pi$
$632$	0	0
$633$	4889.05	0.306986
$634$	0	0
$635$	−195.405	−0.0122117
$636$	0	0
$637$	−8202.72	−0.510210
$638$	0	0
$639$	17570.9	1.08778
$640$	0	0
$641$	6216.36	0.383044	0.191522	−	0.981488i	$-0.438658\pi$
$641$	6216.36	0.383044	0.191522	+	0.981488i	$0.438658\pi$
$642$	0	0
$643$	−17137.5	−1.05107	−0.525535	−	0.850772i	$-0.676135\pi$
$643$	−17137.5	−1.05107	−0.525535	+	0.850772i	$0.676135\pi$
$644$	0	0
$645$	−187.980	−0.0114755
$646$	0	0
$647$	25615.7	1.55650	0.778250	−	0.627955i	$-0.216108\pi$
$647$	25615.7	1.55650	0.778250	+	0.627955i	$0.216108\pi$
$648$	0	0
$649$	−36748.0	−2.22262
$650$	0	0
$651$	−2514.63	−0.151392
$652$	0	0
$653$	6999.67	0.419477	0.209738	−	0.977758i	$-0.432739\pi$
$653$	6999.67	0.419477	0.209738	+	0.977758i	$0.432739\pi$
$654$	0	0
$655$	−478.115	−0.0285214
$656$	0	0
$657$	19097.2	1.13402
$658$	0	0
$659$	25604.3	1.51351	0.756755	−	0.653699i	$-0.226784\pi$
$659$	25604.3	1.51351	0.756755	+	0.653699i	$0.226784\pi$
$660$	0	0
$661$	−6767.95	−0.398249	−0.199125	−	0.979974i	$-0.563810\pi$
$661$	−6767.95	−0.398249	−0.199125	+	0.979974i	$0.563810\pi$
$662$	0	0
$663$	3.41165	0.000199845 0
$664$	0	0
$665$	−573.528	−0.0334443
$666$	0	0
$667$	−442.891	−0.0257103
$668$	0	0
$669$	2557.66	0.147810
$670$	0	0
$671$	42024.6	2.41779
$672$	0	0
$673$	−9068.65	−0.519422	−0.259711	−	0.965686i	$-0.583627\pi$
$673$	−9068.65	−0.519422	−0.259711	+	0.965686i	$0.583627\pi$
$674$	0	0
$675$	6688.36	0.381385
$676$	0	0
$677$	7776.78	0.441486	0.220743	−	0.975332i	$-0.429152\pi$
$677$	7776.78	0.441486	0.220743	+	0.975332i	$0.429152\pi$
$678$	0	0
$679$	−14930.6	−0.843864
$680$	0	0
$681$	−1555.71	−0.0875405
$682$	0	0
$683$	22719.6	1.27283	0.636414	−	0.771348i	$-0.280417\pi$
$683$	22719.6	1.27283	0.636414	+	0.771348i	$0.280417\pi$
$684$	0	0
$685$	−604.273	−0.0337052
$686$	0	0
$687$	3200.34	0.177730
$688$	0	0
$689$	−12256.4	−0.677694
$690$	0	0
$691$	−8639.29	−0.475621	−0.237810	−	0.971312i	$-0.576430\pi$
$691$	−8639.29	−0.475621	−0.237810	+	0.971312i	$0.576430\pi$
$692$	0	0
$693$	19767.9	1.08358
$694$	0	0
$695$	697.046	0.0380438
$696$	0	0
$697$	17.6756	0.000960561 0
$698$	0	0
$699$	257.905	0.0139555
$700$	0	0
$701$	−21727.2	−1.17065	−0.585324	−	0.810799i	$-0.699033\pi$
$701$	−21727.2	−1.17065	−0.585324	+	0.810799i	$0.699033\pi$
$702$	0	0
$703$	−30614.2	−1.64244
$704$	0	0
$705$	−100.249	−0.00535543
$706$	0	0
$707$	25275.2	1.34452
$708$	0	0
$709$	−9645.70	−0.510934	−0.255467	−	0.966818i	$-0.582229\pi$
$709$	−9645.70	−0.510934	−0.255467	+	0.966818i	$0.582229\pi$
$710$	0	0
$711$	3882.49	0.204789
$712$	0	0
$713$	2633.12	0.138304
$714$	0	0
$715$	−1274.07	−0.0666399
$716$	0	0
$717$	−1353.99	−0.0705237
$718$	0	0
$719$	9752.94	0.505874	0.252937	−	0.967483i	$-0.418603\pi$
$719$	9752.94	0.505874	0.252937	+	0.967483i	$0.418603\pi$
$720$	0	0
$721$	1998.22	0.103215
$722$	0	0
$723$	−4070.42	−0.209378
$724$	0	0
$725$	−3620.42	−0.185461
$726$	0	0
$727$	−14814.1	−0.755743	−0.377872	−	0.925858i	$-0.623344\pi$
$727$	−14814.1	−0.755743	−0.377872	+	0.925858i	$0.623344\pi$
$728$	0	0
$729$	−15349.9	−0.779858
$730$	0	0
$731$	−25.9703	−0.00131402
$732$	0	0
$733$	−1757.82	−0.0885767	−0.0442884	−	0.999019i	$-0.514102\pi$
$733$	−1757.82	−0.0885767	−0.0442884	+	0.999019i	$0.514102\pi$
$734$	0	0
$735$	−54.2527	−0.00272264
$736$	0	0
$737$	43738.4	2.18606
$738$	0	0
$739$	−6626.94	−0.329873	−0.164936	−	0.986304i	$-0.552742\pi$
$739$	−6626.94	−0.329873	−0.164936	+	0.986304i	$0.552742\pi$
$740$	0	0
$741$	−6148.85	−0.304836
$742$	0	0
$743$	−5832.98	−0.288010	−0.144005	−	0.989577i	$-0.545998\pi$
$743$	−5832.98	−0.288010	−0.144005	+	0.989577i	$0.545998\pi$
$744$	0	0
$745$	272.478	0.0133998
$746$	0	0
$747$	−24625.4	−1.20615
$748$	0	0
$749$	−12973.4	−0.632895
$750$	0	0
$751$	15771.0	0.766302	0.383151	−	0.923686i	$-0.374839\pi$
$751$	15771.0	0.766302	0.383151	+	0.923686i	$0.374839\pi$
$752$	0	0
$753$	1505.22	0.0728461
$754$	0	0
$755$	1346.74	0.0649178
$756$	0	0
$757$	−10779.2	−0.517541	−0.258770	−	0.965939i	$-0.583317\pi$
$757$	−10779.2	−0.517541	−0.258770	+	0.965939i	$0.583317\pi$
$758$	0	0
$759$	814.889	0.0389705
$760$	0	0
$761$	34744.5	1.65504	0.827520	−	0.561436i	$-0.189751\pi$
$761$	34744.5	1.65504	0.827520	+	0.561436i	$0.189751\pi$
$762$	0	0
$763$	−28386.5	−1.34687
$764$	0	0
$765$	−0.573171	−2.70889e−5 0
$766$	0	0
$767$	42319.0	1.99224
$768$	0	0
$769$	35660.8	1.67225	0.836125	−	0.548539i	$-0.184816\pi$
$769$	35660.8	1.67225	0.836125	+	0.548539i	$0.184816\pi$
$770$	0	0
$771$	6185.45	0.288928
$772$	0	0
$773$	−25790.3	−1.20002	−0.600009	−	0.799994i	$-0.704836\pi$
$773$	−25790.3	−1.20002	−0.600009	+	0.799994i	$0.704836\pi$
$774$	0	0
$775$	21524.5	0.997653
$776$	0	0
$777$	4462.06	0.206017
$778$	0	0
$779$	−31857.0	−1.46520
$780$	0	0
$781$	35688.6	1.63513
$782$	0	0
$783$	−1553.66	−0.0709111
$784$	0	0
$785$	1208.44	0.0549443
$786$	0	0
$787$	−34725.4	−1.57284	−0.786421	−	0.617691i	$-0.788068\pi$
$787$	−34725.4	−1.57284	−0.786421	+	0.617691i	$0.788068\pi$
$788$	0	0
$789$	−6714.44	−0.302966
$790$	0	0
$791$	4313.68	0.193902
$792$	0	0
$793$	−48395.6	−2.16718
$794$	0	0
$795$	−81.0635	−0.00361638
$796$	0	0
$797$	−41090.9	−1.82624	−0.913121	−	0.407688i	$-0.866335\pi$
$797$	−41090.9	−1.82624	−0.913121	+	0.407688i	$0.866335\pi$
$798$	0	0
$799$	−13.8498	−0.000613229 0
$800$	0	0
$801$	2101.29	0.0926909
$802$	0	0
$803$	38788.7	1.70464
$804$	0	0
$805$	−87.5309	−0.00383237
$806$	0	0
$807$	3625.92	0.158164
$808$	0	0
$809$	−38012.1	−1.65196	−0.825980	−	0.563700i	$-0.809378\pi$
$809$	−38012.1	−1.65196	−0.825980	+	0.563700i	$0.809378\pi$
$810$	0	0
$811$	9915.01	0.429301	0.214650	−	0.976691i	$-0.431139\pi$
$811$	9915.01	0.429301	0.214650	+	0.976691i	$0.431139\pi$
$812$	0	0
$813$	−1508.75	−0.0650850
$814$	0	0
$815$	−528.853	−0.0227299
$816$	0	0
$817$	46806.6	2.00435
$818$	0	0
$819$	−22764.8	−0.971264
$820$	0	0
$821$	36205.0	1.53905	0.769527	−	0.638614i	$-0.220492\pi$
$821$	36205.0	1.53905	0.769527	+	0.638614i	$0.220492\pi$
$822$	0	0
$823$	35060.9	1.48499	0.742494	−	0.669852i	$-0.233643\pi$
$823$	35060.9	1.48499	0.742494	+	0.669852i	$0.233643\pi$
$824$	0	0
$825$	6661.33	0.281112
$826$	0	0
$827$	−45893.2	−1.92970	−0.964850	−	0.262801i	$-0.915354\pi$
$827$	−45893.2	−1.92970	−0.964850	+	0.262801i	$0.915354\pi$
$828$	0	0
$829$	1929.98	0.0808576	0.0404288	−	0.999182i	$-0.487128\pi$
$829$	1929.98	0.0808576	0.0404288	+	0.999182i	$0.487128\pi$
$830$	0	0
$831$	−4516.05	−0.188520
$832$	0	0
$833$	−7.49525	−0.000311759 0
$834$	0	0
$835$	799.107	0.0331188
$836$	0	0
$837$	9236.98	0.381454
$838$	0	0
$839$	10217.0	0.420419	0.210210	−	0.977656i	$-0.432585\pi$
$839$	10217.0	0.420419	0.210210	+	0.977656i	$0.432585\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−5326.03	−0.217601
$844$	0	0
$845$	594.133	0.0241879
$846$	0	0
$847$	20955.0	0.850085
$848$	0	0
$849$	1128.49	0.0456179
$850$	0	0
$851$	−4672.30	−0.188207
$852$	0	0
$853$	37522.3	1.50614	0.753071	−	0.657940i	$-0.228572\pi$
$853$	37522.3	1.50614	0.753071	+	0.657940i	$0.228572\pi$
$854$	0	0
$855$	1033.03	0.0413204
$856$	0	0
$857$	−43735.7	−1.74327	−0.871635	−	0.490155i	$-0.836940\pi$
$857$	−43735.7	−1.74327	−0.871635	+	0.490155i	$0.836940\pi$
$858$	0	0
$859$	−12167.8	−0.483304	−0.241652	−	0.970363i	$-0.577689\pi$
$859$	−12167.8	−0.483304	−0.241652	+	0.970363i	$0.577689\pi$
$860$	0	0
$861$	4643.19	0.183786
$862$	0	0
$863$	−4186.87	−0.165148	−0.0825739	−	0.996585i	$-0.526314\pi$
$863$	−4186.87	−0.165148	−0.0825739	+	0.996585i	$0.526314\pi$
$864$	0	0
$865$	1039.09	0.0408442
$866$	0	0
$867$	−4968.38	−0.194620
$868$	0	0
$869$	7885.83	0.307835
$870$	0	0
$871$	−50369.2	−1.95947
$872$	0	0
$873$	26892.8	1.04259
$874$	0	0
$875$	−1431.95	−0.0553244
$876$	0	0
$877$	24101.9	0.928007	0.464003	−	0.885833i	$-0.346413\pi$
$877$	24101.9	0.928007	0.464003	+	0.885833i	$0.346413\pi$
$878$	0	0
$879$	2043.58	0.0784169
$880$	0	0
$881$	17395.9	0.665246	0.332623	−	0.943060i	$-0.392066\pi$
$881$	17395.9	0.665246	0.332623	+	0.943060i	$0.392066\pi$
$882$	0	0
$883$	−24764.3	−0.943812	−0.471906	−	0.881649i	$-0.656434\pi$
$883$	−24764.3	−0.943812	−0.471906	+	0.881649i	$0.656434\pi$
$884$	0	0
$885$	279.897	0.0106312
$886$	0	0
$887$	−50038.4	−1.89417	−0.947083	−	0.320989i	$-0.895985\pi$
$887$	−50038.4	−1.89417	−0.947083	+	0.320989i	$0.895985\pi$
$888$	0	0
$889$	−7091.58	−0.267541
$890$	0	0
$891$	−34148.8	−1.28398
$892$	0	0
$893$	24961.6	0.935396
$894$	0	0
$895$	926.978	0.0346206
$896$	0	0
$897$	−938.428	−0.0349311
$898$	0	0
$899$	−4999.99	−0.185494
$900$	0	0
$901$	−11.1993	−0.000414098 0
$902$	0	0
$903$	−6822.11	−0.251413
$904$	0	0
$905$	1281.98	0.0470877
$906$	0	0
$907$	−15480.5	−0.566727	−0.283364	−	0.959013i	$-0.591450\pi$
$907$	−15480.5	−0.566727	−0.283364	+	0.959013i	$0.591450\pi$
$908$	0	0
$909$	−45525.5	−1.66115
$910$	0	0
$911$	15453.8	0.562028	0.281014	−	0.959704i	$-0.409329\pi$
$911$	15453.8	0.562028	0.281014	+	0.959704i	$0.409329\pi$
$912$	0	0
$913$	−50017.4	−1.81307
$914$	0	0
$915$	−320.087	−0.0115648
$916$	0	0
$917$	−17351.6	−0.624863
$918$	0	0
$919$	46473.1	1.66812	0.834062	−	0.551671i	$-0.186010\pi$
$919$	46473.1	1.66812	0.834062	+	0.551671i	$0.186010\pi$
$920$	0	0
$921$	2015.23	0.0720999
$922$	0	0
$923$	−41099.1	−1.46565
$924$	0	0
$925$	−38193.8	−1.35763
$926$	0	0
$927$	−3599.18	−0.127522
$928$	0	0
$929$	9778.33	0.345335	0.172668	−	0.984980i	$-0.444761\pi$
$929$	9778.33	0.345335	0.172668	+	0.984980i	$0.444761\pi$
$930$	0	0
$931$	13508.8	0.475545
$932$	0	0
$933$	2877.66	0.100976
$934$	0	0
$935$	−1.16418	−4.07196e−5 0
$936$	0	0
$937$	−11449.9	−0.399201	−0.199601	−	0.979877i	$-0.563964\pi$
$937$	−11449.9	−0.399201	−0.199601	+	0.979877i	$0.563964\pi$
$938$	0	0
$939$	−2985.20	−0.103747
$940$	0	0
$941$	−30749.6	−1.06526	−0.532629	−	0.846349i	$-0.678796\pi$
$941$	−30749.6	−1.06526	−0.532629	+	0.846349i	$0.678796\pi$
$942$	0	0
$943$	−4861.96	−0.167897
$944$	0	0
$945$	−307.059	−0.0105700
$946$	0	0
$947$	9461.65	0.324670	0.162335	−	0.986736i	$-0.448098\pi$
$947$	9461.65	0.324670	0.162335	+	0.986736i	$0.448098\pi$
$948$	0	0
$949$	−44669.2	−1.52795
$950$	0	0
$951$	475.810	0.0162242
$952$	0	0
$953$	48729.1	1.65634	0.828170	−	0.560477i	$-0.189382\pi$
$953$	48729.1	1.65634	0.828170	+	0.560477i	$0.189382\pi$
$954$	0	0
$955$	−1454.96	−0.0493001
$956$	0	0
$957$	−1547.38	−0.0522673
$958$	0	0
$959$	−21930.0	−0.738434
$960$	0	0
$961$	−64.5926	−0.00216819
$962$	0	0
$963$	23367.6	0.781942
$964$	0	0
$965$	540.536	0.0180316
$966$	0	0
$967$	−28715.6	−0.954945	−0.477473	−	0.878647i	$-0.658447\pi$
$967$	−28715.6	−0.954945	−0.477473	+	0.878647i	$0.658447\pi$
$968$	0	0
$969$	−5.61852	−0.000186267 0
$970$	0	0
$971$	−21535.3	−0.711741	−0.355870	−	0.934535i	$-0.615816\pi$
$971$	−21535.3	−0.711741	−0.355870	+	0.934535i	$0.615816\pi$
$972$	0	0
$973$	25297.0	0.833487
$974$	0	0
$975$	−7671.20	−0.251974
$976$	0	0
$977$	−33512.3	−1.09739	−0.548697	−	0.836021i	$-0.684876\pi$
$977$	−33512.3	−1.09739	−0.548697	+	0.836021i	$0.684876\pi$
$978$	0	0
$979$	4267.98	0.139331
$980$	0	0
$981$	51129.5	1.66406
$982$	0	0
$983$	23009.3	0.746574	0.373287	−	0.927716i	$-0.378231\pi$
$983$	23009.3	0.746574	0.373287	+	0.927716i	$0.378231\pi$
$984$	0	0
$985$	696.826	0.0225408
$986$	0	0
$987$	−3638.18	−0.117330
$988$	0	0
$989$	7143.55	0.229678
$990$	0	0
$991$	−18411.2	−0.590163	−0.295081	−	0.955472i	$-0.595347\pi$
$991$	−18411.2	−0.590163	−0.295081	+	0.955472i	$0.595347\pi$
$992$	0	0
$993$	−555.124	−0.0177405
$994$	0	0
$995$	−695.715	−0.0221665
$996$	0	0
$997$	47191.7	1.49907	0.749537	−	0.661962i	$-0.230276\pi$
$997$	47191.7	1.49907	0.749537	+	0.661962i	$0.230276\pi$
$998$	0	0
$999$	−16390.4	−0.519090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.m.1.3		5
4.3	odd	2		232.4.a.d.1.3	✓	5
8.3	odd	2		1856.4.a.z.1.3		5
8.5	even	2		1856.4.a.ba.1.3		5
12.11	even	2		2088.4.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.d.1.3	✓	5	4.3	odd	2
464.4.a.m.1.3		5	1.1	even	1	trivial
1856.4.a.z.1.3		5	8.3	odd	2
1856.4.a.ba.1.3		5	8.5	even	2
2088.4.a.f.1.3		5	12.11	even	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q+1.01127 q^{3} +0.397400 q^{5} +14.4223 q^{7} -25.9773 q^{9} +O(q^{10})\)	\(q+1.01127 q^{3} +0.397400 q^{5} +14.4223 q^{7} -25.9773 q^{9} -52.7632 q^{11} +60.7622 q^{13} +0.401880 q^{15} +0.0555215 q^{17} -100.067 q^{19} +14.5849 q^{21} -15.2721 q^{23} -124.842 q^{25} -53.5746 q^{27} +29.0000 q^{29} -172.413 q^{31} -53.3580 q^{33} +5.73143 q^{35} +305.937 q^{37} +61.4472 q^{39} +318.356 q^{41} -467.752 q^{43} -10.3234 q^{45} -249.449 q^{47} -134.997 q^{49} +0.0561475 q^{51} -201.710 q^{53} -20.9681 q^{55} -101.195 q^{57} +696.469 q^{59} -796.475 q^{61} -374.653 q^{63} +24.1469 q^{65} -828.956 q^{67} -15.4443 q^{69} -676.393 q^{71} -735.147 q^{73} -126.249 q^{75} -760.967 q^{77} -149.457 q^{79} +647.209 q^{81} +947.959 q^{83} +0.0220643 q^{85} +29.3269 q^{87} -80.8893 q^{89} +876.332 q^{91} -174.357 q^{93} -39.7667 q^{95} -1035.24 q^{97} +1370.65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} + 10 q^{5} - 32 q^{7} + 29 q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 + 10 * q^5 - 32 * q^7 + 29 * q^9	\( 5 q - 4 q^{3} + 10 q^{5} - 32 q^{7} + 29 q^{9} - 36 q^{11} + 26 q^{13} - 88 q^{15} + 82 q^{17} - 156 q^{19} + 72 q^{21} - 336 q^{23} + 151 q^{25} - 352 q^{27} + 145 q^{29} - 432 q^{31} + 108 q^{33} - 600 q^{35} - 18 q^{37} - 688 q^{39} + 82 q^{41} - 340 q^{43} - 146 q^{45} - 680 q^{47} - 115 q^{49} - 608 q^{51} - 102 q^{53} - 736 q^{55} - 576 q^{57} - 924 q^{59} - 618 q^{61} - 584 q^{63} - 704 q^{65} - 44 q^{67} - 1056 q^{69} - 1032 q^{71} - 1078 q^{73} + 468 q^{75} - 888 q^{77} - 200 q^{79} - 1843 q^{81} - 452 q^{83} - 1700 q^{85} - 116 q^{87} - 1790 q^{89} + 1128 q^{91} - 1884 q^{93} - 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 + 10 * q^5 - 32 * q^7 + 29 * q^9 - 36 * q^11 + 26 * q^13 - 88 * q^15 + 82 * q^17 - 156 * q^19 + 72 * q^21 - 336 * q^23 + 151 * q^25 - 352 * q^27 + 145 * q^29 - 432 * q^31 + 108 * q^33 - 600 * q^35 - 18 * q^37 - 688 * q^39 + 82 * q^41 - 340 * q^43 - 146 * q^45 - 680 * q^47 - 115 * q^49 - 608 * q^51 - 102 * q^53 - 736 * q^55 - 576 * q^57 - 924 * q^59 - 618 * q^61 - 584 * q^63 - 704 * q^65 - 44 * q^67 - 1056 * q^69 - 1032 * q^71 - 1078 * q^73 + 468 * q^75 - 888 * q^77 - 200 * q^79 - 1843 * q^81 - 452 * q^83 - 1700 * q^85 - 116 * q^87 - 1790 * q^89 + 1128 * q^91 - 1884 * q^93 - 1024 * q^95 - 2518 * q^97 + 1500 * q^99

Introduction

L-functions

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