LMFDB - Embedded newform 592.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,4,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, 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("592.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	592.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:	$34.9291307234$
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} - 27x^{3} + 3x^{2} + 176x + 144 $ x^5 - x^4 - 27x^3 + 3x^2 + 176*x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 37)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.56768$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.45274 q^{3} +1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} +O(q^{10})$	$q-9.45274 q^{3} +1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} +3.77312 q^{11} -71.0065 q^{13} -14.2549 q^{15} +38.8146 q^{17} +64.4749 q^{19} +119.947 q^{21} +197.638 q^{23} -122.726 q^{25} -334.195 q^{27} +255.479 q^{29} +97.3830 q^{31} -35.6664 q^{33} -19.1354 q^{35} -37.0000 q^{37} +671.206 q^{39} +350.356 q^{41} -138.813 q^{43} +94.0313 q^{45} -305.653 q^{47} -181.986 q^{49} -366.904 q^{51} -310.167 q^{53} +5.68993 q^{55} -609.465 q^{57} -112.918 q^{59} -595.518 q^{61} -791.221 q^{63} -107.079 q^{65} -899.470 q^{67} -1868.22 q^{69} +450.353 q^{71} +469.956 q^{73} +1160.10 q^{75} -47.8776 q^{77} -263.304 q^{79} +1475.49 q^{81} +314.892 q^{83} +58.5331 q^{85} -2414.98 q^{87} -772.521 q^{89} +901.010 q^{91} -920.536 q^{93} +97.2293 q^{95} +190.846 q^{97} +235.270 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 13 q^{3} + 11 q^{5} - 24 q^{7} + 46 q^{9}+O(q^{10}) $ 5 * q - 13 * q^3 + 11 * q^5 - 24 * q^7 + 46 * q^9	$ 5 q - 13 q^{3} + 11 q^{5} - 24 q^{7} + 46 q^{9} - 61 q^{11} - 37 q^{13} + 116 q^{15} + 130 q^{17} + 22 q^{19} - 44 q^{21} - 73 q^{23} + 26 q^{25} - 472 q^{27} + 271 q^{29} - 363 q^{31} + 198 q^{33} - 604 q^{35} - 185 q^{37} + 65 q^{39} + 381 q^{41} + 408 q^{43} - 704 q^{45} - 276 q^{47} - 949 q^{49} + 38 q^{51} + 156 q^{53} + 843 q^{55} - 1618 q^{57} - 100 q^{59} - 1711 q^{61} - 94 q^{63} - 890 q^{65} - 787 q^{67} - 2335 q^{69} - 1578 q^{71} - 313 q^{73} - 684 q^{75} - 342 q^{77} - 569 q^{79} + 385 q^{81} - 2422 q^{83} - 2210 q^{85} - 2371 q^{87} - 2466 q^{89} + 1678 q^{91} - 1142 q^{93} - 794 q^{95} - 2406 q^{97} - 1746 q^{99}+O(q^{100}) $ 5 * q - 13 * q^3 + 11 * q^5 - 24 * q^7 + 46 * q^9 - 61 * q^11 - 37 * q^13 + 116 * q^15 + 130 * q^17 + 22 * q^19 - 44 * q^21 - 73 * q^23 + 26 * q^25 - 472 * q^27 + 271 * q^29 - 363 * q^31 + 198 * q^33 - 604 * q^35 - 185 * q^37 + 65 * q^39 + 381 * q^41 + 408 * q^43 - 704 * q^45 - 276 * q^47 - 949 * q^49 + 38 * q^51 + 156 * q^53 + 843 * q^55 - 1618 * q^57 - 100 * q^59 - 1711 * q^61 - 94 * q^63 - 890 * q^65 - 787 * q^67 - 2335 * q^69 - 1578 * q^71 - 313 * q^73 - 684 * q^75 - 342 * q^77 - 569 * q^79 + 385 * q^81 - 2422 * q^83 - 2210 * q^85 - 2371 * q^87 - 2466 * q^89 + 1678 * q^91 - 1142 * q^93 - 794 * q^95 - 2406 * q^97 - 1746 * 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
$3$	−9.45274	−1.81918	−0.909590	−	0.415506i	$-0.863604\pi$
$3$	−9.45274	−1.81918	−0.909590	+	0.415506i	$0.863604\pi$
$4$	0	0
$5$	1.50802	0.134881	0.0674405	−	0.997723i	$-0.478517\pi$
$5$	1.50802	0.134881	0.0674405	+	0.997723i	$0.478517\pi$
$6$	0	0
$7$	−12.6891	−0.685148	−0.342574	−	0.939491i	$-0.611299\pi$
$7$	−12.6891	−0.685148	−0.342574	+	0.939491i	$0.611299\pi$
$8$	0	0
$9$	62.3543	2.30942
$10$	0	0
$11$	3.77312	0.103422	0.0517109	−	0.998662i	$-0.483533\pi$
$11$	3.77312	0.103422	0.0517109	+	0.998662i	$0.483533\pi$
$12$	0	0
$13$	−71.0065	−1.51490	−0.757449	−	0.652895i	$-0.773554\pi$
$13$	−71.0065	−1.51490	−0.757449	+	0.652895i	$0.773554\pi$
$14$	0	0
$15$	−14.2549	−0.245373
$16$	0	0
$17$	38.8146	0.553760	0.276880	−	0.960904i	$-0.410700\pi$
$17$	38.8146	0.553760	0.276880	+	0.960904i	$0.410700\pi$
$18$	0	0
$19$	64.4749	0.778503	0.389252	−	0.921131i	$-0.372734\pi$
$19$	64.4749	0.778503	0.389252	+	0.921131i	$0.372734\pi$
$20$	0	0
$21$	119.947	1.24641
$22$	0	0
$23$	197.638	1.79175	0.895877	−	0.444302i	$-0.146548\pi$
$23$	197.638	1.79175	0.895877	+	0.444302i	$0.146548\pi$
$24$	0	0
$25$	−122.726	−0.981807
$26$	0	0
$27$	−334.195	−2.38207
$28$	0	0
$29$	255.479	1.63591	0.817954	−	0.575283i	$-0.195108\pi$
$29$	255.479	1.63591	0.817954	+	0.575283i	$0.195108\pi$
$30$	0	0
$31$	97.3830	0.564210	0.282105	−	0.959384i	$-0.408967\pi$
$31$	97.3830	0.564210	0.282105	+	0.959384i	$0.408967\pi$
$32$	0	0
$33$	−35.6664	−0.188143
$34$	0	0
$35$	−19.1354	−0.0924135
$36$	0	0
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	0	0
$39$	671.206	2.75587
$40$	0	0
$41$	350.356	1.33455	0.667273	−	0.744813i	$-0.267461\pi$
$41$	350.356	1.33455	0.667273	+	0.744813i	$0.267461\pi$
$42$	0	0
$43$	−138.813	−0.492296	−0.246148	−	0.969232i	$-0.579165\pi$
$43$	−138.813	−0.492296	−0.246148	+	0.969232i	$0.579165\pi$
$44$	0	0
$45$	94.0313	0.311497
$46$	0	0
$47$	−305.653	−0.948597	−0.474299	−	0.880364i	$-0.657298\pi$
$47$	−305.653	−0.948597	−0.474299	+	0.880364i	$0.657298\pi$
$48$	0	0
$49$	−181.986	−0.530572
$50$	0	0
$51$	−366.904	−1.00739
$52$	0	0
$53$	−310.167	−0.803864	−0.401932	−	0.915670i	$-0.631661\pi$
$53$	−310.167	−0.803864	−0.401932	+	0.915670i	$0.631661\pi$
$54$	0	0
$55$	5.68993	0.0139496
$56$	0	0
$57$	−609.465	−1.41624
$58$	0	0
$59$	−112.918	−0.249163	−0.124582	−	0.992209i	$-0.539759\pi$
$59$	−112.918	−0.249163	−0.124582	+	0.992209i	$0.539759\pi$
$60$	0	0
$61$	−595.518	−1.24997	−0.624986	−	0.780636i	$-0.714895\pi$
$61$	−595.518	−1.24997	−0.624986	+	0.780636i	$0.714895\pi$
$62$	0	0
$63$	−791.221	−1.58229
$64$	0	0
$65$	−107.079	−0.204331
$66$	0	0
$67$	−899.470	−1.64012	−0.820058	−	0.572280i	$-0.806059\pi$
$67$	−899.470	−1.64012	−0.820058	+	0.572280i	$0.806059\pi$
$68$	0	0
$69$	−1868.22	−3.25952
$70$	0	0
$71$	450.353	0.752775	0.376388	−	0.926462i	$-0.377166\pi$
$71$	450.353	0.752775	0.376388	+	0.926462i	$0.377166\pi$
$72$	0	0
$73$	469.956	0.753482	0.376741	−	0.926319i	$-0.377045\pi$
$73$	469.956	0.753482	0.376741	+	0.926319i	$0.377045\pi$
$74$	0	0
$75$	1160.10	1.78608
$76$	0	0
$77$	−47.8776	−0.0708593
$78$	0	0
$79$	−263.304	−0.374987	−0.187493	−	0.982266i	$-0.560036\pi$
$79$	−263.304	−0.374987	−0.187493	+	0.982266i	$0.560036\pi$
$80$	0	0
$81$	1475.49	2.02399
$82$	0	0
$83$	314.892	0.416432	0.208216	−	0.978083i	$-0.433234\pi$
$83$	314.892	0.416432	0.208216	+	0.978083i	$0.433234\pi$
$84$	0	0
$85$	58.5331	0.0746918
$86$	0	0
$87$	−2414.98	−2.97601
$88$	0	0
$89$	−772.521	−0.920079	−0.460039	−	0.887898i	$-0.652165\pi$
$89$	−772.521	−0.920079	−0.460039	+	0.887898i	$0.652165\pi$
$90$	0	0
$91$	901.010	1.03793
$92$	0	0
$93$	−920.536	−1.02640
$94$	0	0
$95$	97.2293	0.105005
$96$	0	0
$97$	190.846	0.199768	0.0998838	−	0.994999i	$-0.468153\pi$
$97$	190.846	0.199768	0.0998838	+	0.994999i	$0.468153\pi$
$98$	0	0
$99$	235.270	0.238844
$100$	0	0
$101$	219.256	0.216008	0.108004	−	0.994150i	$-0.465554\pi$
$101$	219.256	0.216008	0.108004	+	0.994150i	$0.465554\pi$
$102$	0	0
$103$	−1064.84	−1.01866	−0.509330	−	0.860571i	$-0.670107\pi$
$103$	−1064.84	−1.01866	−0.509330	+	0.860571i	$0.670107\pi$
$104$	0	0
$105$	180.882	0.168117
$106$	0	0
$107$	−948.801	−0.857234	−0.428617	−	0.903486i	$-0.640999\pi$
$107$	−948.801	−0.857234	−0.428617	+	0.903486i	$0.640999\pi$
$108$	0	0
$109$	836.883	0.735402	0.367701	−	0.929944i	$-0.380145\pi$
$109$	836.883	0.735402	0.367701	+	0.929944i	$0.380145\pi$
$110$	0	0
$111$	349.751	0.299071
$112$	0	0
$113$	−791.819	−0.659187	−0.329593	−	0.944123i	$-0.606912\pi$
$113$	−791.819	−0.659187	−0.329593	+	0.944123i	$0.606912\pi$
$114$	0	0
$115$	298.041	0.241674
$116$	0	0
$117$	−4427.56	−3.49853
$118$	0	0
$119$	−492.523	−0.379408
$120$	0	0
$121$	−1316.76	−0.989304
$122$	0	0
$123$	−3311.82	−2.42778
$124$	0	0
$125$	−373.575	−0.267308
$126$	0	0
$127$	1562.59	1.09179	0.545896	−	0.837853i	$-0.316189\pi$
$127$	1562.59	1.09179	0.545896	+	0.837853i	$0.316189\pi$
$128$	0	0
$129$	1312.16	0.895575
$130$	0	0
$131$	−784.197	−0.523020	−0.261510	−	0.965201i	$-0.584220\pi$
$131$	−784.197	−0.523020	−0.261510	+	0.965201i	$0.584220\pi$
$132$	0	0
$133$	−818.130	−0.533390
$134$	0	0
$135$	−503.971	−0.321296
$136$	0	0
$137$	2107.26	1.31413	0.657064	−	0.753835i	$-0.271798\pi$
$137$	2107.26	1.31413	0.657064	+	0.753835i	$0.271798\pi$
$138$	0	0
$139$	−2025.93	−1.23624	−0.618119	−	0.786085i	$-0.712105\pi$
$139$	−2025.93	−1.23624	−0.618119	+	0.786085i	$0.712105\pi$
$140$	0	0
$141$	2889.26	1.72567
$142$	0	0
$143$	−267.916	−0.156673
$144$	0	0
$145$	385.267	0.220653
$146$	0	0
$147$	1720.27	0.965206
$148$	0	0
$149$	745.838	0.410077	0.205038	−	0.978754i	$-0.434268\pi$
$149$	745.838	0.410077	0.205038	+	0.978754i	$0.434268\pi$
$150$	0	0
$151$	−658.206	−0.354729	−0.177365	−	0.984145i	$-0.556757\pi$
$151$	−658.206	−0.354729	−0.177365	+	0.984145i	$0.556757\pi$
$152$	0	0
$153$	2420.26	1.27886
$154$	0	0
$155$	146.855	0.0761012
$156$	0	0
$157$	−2059.43	−1.04688	−0.523441	−	0.852062i	$-0.675352\pi$
$157$	−2059.43	−1.04688	−0.523441	+	0.852062i	$0.675352\pi$
$158$	0	0
$159$	2931.93	1.46237
$160$	0	0
$161$	−2507.85	−1.22762
$162$	0	0
$163$	−1229.58	−0.590848	−0.295424	−	0.955366i	$-0.595461\pi$
$163$	−1229.58	−0.590848	−0.295424	+	0.955366i	$0.595461\pi$
$164$	0	0
$165$	−53.7855	−0.0253769
$166$	0	0
$167$	152.311	0.0705761	0.0352881	−	0.999377i	$-0.488765\pi$
$167$	152.311	0.0705761	0.0352881	+	0.999377i	$0.488765\pi$
$168$	0	0
$169$	2844.93	1.29491
$170$	0	0
$171$	4020.29	1.79789
$172$	0	0
$173$	1156.14	0.508089	0.254044	−	0.967193i	$-0.418239\pi$
$173$	1156.14	0.508089	0.254044	+	0.967193i	$0.418239\pi$
$174$	0	0
$175$	1557.28	0.672683
$176$	0	0
$177$	1067.38	0.453273
$178$	0	0
$179$	−332.020	−0.138639	−0.0693194	−	0.997595i	$-0.522083\pi$
$179$	−332.020	−0.138639	−0.0693194	+	0.997595i	$0.522083\pi$
$180$	0	0
$181$	−3340.60	−1.37185	−0.685926	−	0.727671i	$-0.740603\pi$
$181$	−3340.60	−1.37185	−0.685926	+	0.727671i	$0.740603\pi$
$182$	0	0
$183$	5629.28	2.27393
$184$	0	0
$185$	−55.7966	−0.0221743
$186$	0	0
$187$	146.452	0.0572709
$188$	0	0
$189$	4240.64	1.63207
$190$	0	0
$191$	−172.721	−0.0654328	−0.0327164	−	0.999465i	$-0.510416\pi$
$191$	−172.721	−0.0654328	−0.0327164	+	0.999465i	$0.510416\pi$
$192$	0	0
$193$	3533.01	1.31768	0.658838	−	0.752285i	$-0.271048\pi$
$193$	3533.01	1.31768	0.658838	+	0.752285i	$0.271048\pi$
$194$	0	0
$195$	1012.19	0.371715
$196$	0	0
$197$	2038.45	0.737227	0.368613	−	0.929583i	$-0.379833\pi$
$197$	2038.45	0.737227	0.368613	+	0.929583i	$0.379833\pi$
$198$	0	0
$199$	−3418.61	−1.21778	−0.608891	−	0.793254i	$-0.708385\pi$
$199$	−3418.61	−1.21778	−0.608891	+	0.793254i	$0.708385\pi$
$200$	0	0
$201$	8502.46	2.98367
$202$	0	0
$203$	−3241.81	−1.12084
$204$	0	0
$205$	528.342	0.180005
$206$	0	0
$207$	12323.6	4.13791
$208$	0	0
$209$	243.272	0.0805142
$210$	0	0
$211$	4651.74	1.51772	0.758861	−	0.651253i	$-0.225756\pi$
$211$	4651.74	1.51772	0.758861	+	0.651253i	$0.225756\pi$
$212$	0	0
$213$	−4257.07	−1.36943
$214$	0	0
$215$	−209.332	−0.0664014
$216$	0	0
$217$	−1235.70	−0.386567
$218$	0	0
$219$	−4442.37	−1.37072
$220$	0	0
$221$	−2756.09	−0.838890
$222$	0	0
$223$	−280.807	−0.0843240	−0.0421620	−	0.999111i	$-0.513425\pi$
$223$	−280.807	−0.0843240	−0.0421620	+	0.999111i	$0.513425\pi$
$224$	0	0
$225$	−7652.48	−2.26740
$226$	0	0
$227$	−3749.27	−1.09624	−0.548122	−	0.836398i	$-0.684657\pi$
$227$	−3749.27	−1.09624	−0.548122	+	0.836398i	$0.684657\pi$
$228$	0	0
$229$	−5265.97	−1.51959	−0.759793	−	0.650165i	$-0.774700\pi$
$229$	−5265.97	−1.51959	−0.759793	+	0.650165i	$0.774700\pi$
$230$	0	0
$231$	452.575	0.128906
$232$	0	0
$233$	−4555.42	−1.28084	−0.640420	−	0.768025i	$-0.721240\pi$
$233$	−4555.42	−1.28084	−0.640420	+	0.768025i	$0.721240\pi$
$234$	0	0
$235$	−460.930	−0.127948
$236$	0	0
$237$	2488.94	0.682169
$238$	0	0
$239$	2746.85	0.743426	0.371713	−	0.928348i	$-0.378770\pi$
$239$	2746.85	0.743426	0.371713	+	0.928348i	$0.378770\pi$
$240$	0	0
$241$	2224.32	0.594528	0.297264	−	0.954795i	$-0.403926\pi$
$241$	2224.32	0.594528	0.297264	+	0.954795i	$0.403926\pi$
$242$	0	0
$243$	−4924.17	−1.29994
$244$	0	0
$245$	−274.438	−0.0715641
$246$	0	0
$247$	−4578.14	−1.17935
$248$	0	0
$249$	−2976.59	−0.757565
$250$	0	0
$251$	−3928.85	−0.987996	−0.493998	−	0.869463i	$-0.664465\pi$
$251$	−3928.85	−0.987996	−0.493998	+	0.869463i	$0.664465\pi$
$252$	0	0
$253$	745.712	0.185306
$254$	0	0
$255$	−553.298	−0.135878
$256$	0	0
$257$	−3962.37	−0.961734	−0.480867	−	0.876794i	$-0.659678\pi$
$257$	−3962.37	−0.961734	−0.480867	+	0.876794i	$0.659678\pi$
$258$	0	0
$259$	469.498	0.112638
$260$	0	0
$261$	15930.2	3.77800
$262$	0	0
$263$	−37.4419	−0.00877857	−0.00438929	−	0.999990i	$-0.501397\pi$
$263$	−37.4419	−0.00877857	−0.00438929	+	0.999990i	$0.501397\pi$
$264$	0	0
$265$	−467.738	−0.108426
$266$	0	0
$267$	7302.44	1.67379
$268$	0	0
$269$	−2913.59	−0.660390	−0.330195	−	0.943913i	$-0.607114\pi$
$269$	−2913.59	−0.660390	−0.330195	+	0.943913i	$0.607114\pi$
$270$	0	0
$271$	6604.08	1.48033	0.740165	−	0.672425i	$-0.234747\pi$
$271$	6604.08	1.48033	0.740165	+	0.672425i	$0.234747\pi$
$272$	0	0
$273$	−8517.02	−1.88818
$274$	0	0
$275$	−463.060	−0.101540
$276$	0	0
$277$	1587.03	0.344242	0.172121	−	0.985076i	$-0.444938\pi$
$277$	1587.03	0.344242	0.172121	+	0.985076i	$0.444938\pi$
$278$	0	0
$279$	6072.25	1.30300
$280$	0	0
$281$	8590.39	1.82370	0.911850	−	0.410524i	$-0.134654\pi$
$281$	8590.39	1.82370	0.911850	+	0.410524i	$0.134654\pi$
$282$	0	0
$283$	−613.606	−0.128887	−0.0644437	−	0.997921i	$-0.520527\pi$
$283$	−613.606	−0.128887	−0.0644437	+	0.997921i	$0.520527\pi$
$284$	0	0
$285$	−919.083	−0.191024
$286$	0	0
$287$	−4445.71	−0.914362
$288$	0	0
$289$	−3406.43	−0.693350
$290$	0	0
$291$	−1804.02	−0.363413
$292$	0	0
$293$	4502.56	0.897755	0.448878	−	0.893593i	$-0.351824\pi$
$293$	4502.56	0.897755	0.448878	+	0.893593i	$0.351824\pi$
$294$	0	0
$295$	−170.282	−0.0336074
$296$	0	0
$297$	−1260.96	−0.246358
$298$	0	0
$299$	−14033.6	−2.71432
$300$	0	0
$301$	1761.41	0.337296
$302$	0	0
$303$	−2072.57	−0.392958
$304$	0	0
$305$	−898.051	−0.168598
$306$	0	0
$307$	−1490.79	−0.277146	−0.138573	−	0.990352i	$-0.544252\pi$
$307$	−1490.79	−0.277146	−0.138573	+	0.990352i	$0.544252\pi$
$308$	0	0
$309$	10065.7	1.85313
$310$	0	0
$311$	−213.697	−0.0389636	−0.0194818	−	0.999810i	$-0.506202\pi$
$311$	−213.697	−0.0389636	−0.0194818	+	0.999810i	$0.506202\pi$
$312$	0	0
$313$	−8886.22	−1.60472	−0.802362	−	0.596838i	$-0.796424\pi$
$313$	−8886.22	−1.60472	−0.802362	+	0.596838i	$0.796424\pi$
$314$	0	0
$315$	−1193.17	−0.213421
$316$	0	0
$317$	2408.56	0.426745	0.213373	−	0.976971i	$-0.431555\pi$
$317$	2408.56	0.426745	0.213373	+	0.976971i	$0.431555\pi$
$318$	0	0
$319$	963.956	0.169189
$320$	0	0
$321$	8968.76	1.55946
$322$	0	0
$323$	2502.57	0.431104
$324$	0	0
$325$	8714.34	1.48734
$326$	0	0
$327$	−7910.84	−1.33783
$328$	0	0
$329$	3878.47	0.649930
$330$	0	0
$331$	1592.84	0.264503	0.132252	−	0.991216i	$-0.457779\pi$
$331$	1592.84	0.264503	0.132252	+	0.991216i	$0.457779\pi$
$332$	0	0
$333$	−2307.11	−0.379666
$334$	0	0
$335$	−1356.42	−0.221221
$336$	0	0
$337$	−4151.95	−0.671131	−0.335565	−	0.942017i	$-0.608927\pi$
$337$	−4151.95	−0.671131	−0.335565	+	0.942017i	$0.608927\pi$
$338$	0	0
$339$	7484.86	1.19918
$340$	0	0
$341$	367.438	0.0583516
$342$	0	0
$343$	6661.61	1.04867
$344$	0	0
$345$	−2817.31	−0.439648
$346$	0	0
$347$	−6076.73	−0.940103	−0.470051	−	0.882639i	$-0.655765\pi$
$347$	−6076.73	−0.940103	−0.470051	+	0.882639i	$0.655765\pi$
$348$	0	0
$349$	−1581.40	−0.242551	−0.121276	−	0.992619i	$-0.538698\pi$
$349$	−1581.40	−0.242551	−0.121276	+	0.992619i	$0.538698\pi$
$350$	0	0
$351$	23730.0	3.60859
$352$	0	0
$353$	−2015.99	−0.303967	−0.151984	−	0.988383i	$-0.548566\pi$
$353$	−2015.99	−0.303967	−0.151984	+	0.988383i	$0.548566\pi$
$354$	0	0
$355$	679.140	0.101535
$356$	0	0
$357$	4655.69	0.690211
$358$	0	0
$359$	−3176.91	−0.467050	−0.233525	−	0.972351i	$-0.575026\pi$
$359$	−3176.91	−0.467050	−0.233525	+	0.972351i	$0.575026\pi$
$360$	0	0
$361$	−2701.98	−0.393932
$362$	0	0
$363$	12447.0	1.79972
$364$	0	0
$365$	708.701	0.101630
$366$	0	0
$367$	−3963.25	−0.563706	−0.281853	−	0.959458i	$-0.590949\pi$
$367$	−3963.25	−0.563706	−0.281853	+	0.959458i	$0.590949\pi$
$368$	0	0
$369$	21846.2	3.08202
$370$	0	0
$371$	3935.75	0.550766
$372$	0	0
$373$	−11018.6	−1.52955	−0.764776	−	0.644296i	$-0.777150\pi$
$373$	−11018.6	−1.52955	−0.764776	+	0.644296i	$0.777150\pi$
$374$	0	0
$375$	3531.30	0.486282
$376$	0	0
$377$	−18140.7	−2.47823
$378$	0	0
$379$	1215.73	0.164769	0.0823847	−	0.996601i	$-0.473746\pi$
$379$	1215.73	0.164769	0.0823847	+	0.996601i	$0.473746\pi$
$380$	0	0
$381$	−14770.8	−1.98617
$382$	0	0
$383$	−8873.92	−1.18391	−0.591953	−	0.805972i	$-0.701643\pi$
$383$	−8873.92	−1.18391	−0.591953	+	0.805972i	$0.701643\pi$
$384$	0	0
$385$	−72.2003	−0.00955757
$386$	0	0
$387$	−8655.56	−1.13692
$388$	0	0
$389$	−5901.38	−0.769182	−0.384591	−	0.923087i	$-0.625657\pi$
$389$	−5901.38	−0.769182	−0.384591	+	0.923087i	$0.625657\pi$
$390$	0	0
$391$	7671.24	0.992202
$392$	0	0
$393$	7412.81	0.951468
$394$	0	0
$395$	−397.066	−0.0505786
$396$	0	0
$397$	13938.7	1.76212	0.881060	−	0.473004i	$-0.156830\pi$
$397$	13938.7	1.76212	0.881060	+	0.473004i	$0.156830\pi$
$398$	0	0
$399$	7733.57	0.970333
$400$	0	0
$401$	−11862.8	−1.47730	−0.738652	−	0.674087i	$-0.764537\pi$
$401$	−11862.8	−1.47730	−0.738652	+	0.674087i	$0.764537\pi$
$402$	0	0
$403$	−6914.83	−0.854720
$404$	0	0
$405$	2225.06	0.272998
$406$	0	0
$407$	−139.606	−0.0170024
$408$	0	0
$409$	−15104.9	−1.82614	−0.913068	−	0.407808i	$-0.866293\pi$
$409$	−15104.9	−1.82614	−0.913068	+	0.407808i	$0.866293\pi$
$410$	0	0
$411$	−19919.4	−2.39064
$412$	0	0
$413$	1432.83	0.170714
$414$	0	0
$415$	474.862	0.0561688
$416$	0	0
$417$	19150.6	2.24894
$418$	0	0
$419$	16878.5	1.96794	0.983970	−	0.178334i	$-0.0570708\pi$
$419$	16878.5	1.96794	0.983970	+	0.178334i	$0.0570708\pi$
$420$	0	0
$421$	9479.38	1.09738	0.548690	−	0.836026i	$-0.315127\pi$
$421$	9479.38	1.09738	0.548690	+	0.836026i	$0.315127\pi$
$422$	0	0
$423$	−19058.8	−2.19071
$424$	0	0
$425$	−4763.56	−0.543686
$426$	0	0
$427$	7556.60	0.856416
$428$	0	0
$429$	2532.54	0.285017
$430$	0	0
$431$	11023.1	1.23193	0.615965	−	0.787774i	$-0.288766\pi$
$431$	11023.1	1.23193	0.615965	+	0.787774i	$0.288766\pi$
$432$	0	0
$433$	2955.66	0.328037	0.164018	−	0.986457i	$-0.447554\pi$
$433$	2955.66	0.328037	0.164018	+	0.986457i	$0.447554\pi$
$434$	0	0
$435$	−3641.83	−0.401408
$436$	0	0
$437$	12742.7	1.39489
$438$	0	0
$439$	−4573.78	−0.497255	−0.248627	−	0.968599i	$-0.579979\pi$
$439$	−4573.78	−0.497255	−0.248627	+	0.968599i	$0.579979\pi$
$440$	0	0
$441$	−11347.6	−1.22531
$442$	0	0
$443$	1029.07	0.110367	0.0551836	−	0.998476i	$-0.482426\pi$
$443$	1029.07	0.110367	0.0551836	+	0.998476i	$0.482426\pi$
$444$	0	0
$445$	−1164.97	−0.124101
$446$	0	0
$447$	−7050.21	−0.746003
$448$	0	0
$449$	−1172.94	−0.123284	−0.0616420	−	0.998098i	$-0.519634\pi$
$449$	−1172.94	−0.123284	−0.0616420	+	0.998098i	$0.519634\pi$
$450$	0	0
$451$	1321.94	0.138021
$452$	0	0
$453$	6221.85	0.645316
$454$	0	0
$455$	1358.74	0.139997
$456$	0	0
$457$	−11020.8	−1.12808	−0.564040	−	0.825747i	$-0.690754\pi$
$457$	−11020.8	−1.12808	−0.564040	+	0.825747i	$0.690754\pi$
$458$	0	0
$459$	−12971.6	−1.31909
$460$	0	0
$461$	−2568.56	−0.259501	−0.129750	−	0.991547i	$-0.541418\pi$
$461$	−2568.56	−0.259501	−0.129750	+	0.991547i	$0.541418\pi$
$462$	0	0
$463$	−2479.09	−0.248841	−0.124420	−	0.992230i	$-0.539707\pi$
$463$	−2479.09	−0.248841	−0.124420	+	0.992230i	$0.539707\pi$
$464$	0	0
$465$	−1388.18	−0.138442
$466$	0	0
$467$	9158.15	0.907470	0.453735	−	0.891137i	$-0.350091\pi$
$467$	9158.15	0.907470	0.453735	+	0.891137i	$0.350091\pi$
$468$	0	0
$469$	11413.5	1.12372
$470$	0	0
$471$	19467.3	1.90447
$472$	0	0
$473$	−523.757	−0.0509141
$474$	0	0
$475$	−7912.74	−0.764340
$476$	0	0
$477$	−19340.3	−1.85646
$478$	0	0
$479$	9855.94	0.940144	0.470072	−	0.882628i	$-0.344228\pi$
$479$	9855.94	0.940144	0.470072	+	0.882628i	$0.344228\pi$
$480$	0	0
$481$	2627.24	0.249048
$482$	0	0
$483$	23706.1	2.23326
$484$	0	0
$485$	287.799	0.0269449
$486$	0	0
$487$	−9612.71	−0.894443	−0.447221	−	0.894423i	$-0.647586\pi$
$487$	−9612.71	−0.894443	−0.447221	+	0.894423i	$0.647586\pi$
$488$	0	0
$489$	11622.9	1.07486
$490$	0	0
$491$	−19063.0	−1.75214	−0.876071	−	0.482183i	$-0.839844\pi$
$491$	−19063.0	−1.75214	−0.876071	+	0.482183i	$0.839844\pi$
$492$	0	0
$493$	9916.33	0.905901
$494$	0	0
$495$	354.792	0.0322156
$496$	0	0
$497$	−5714.58	−0.515763
$498$	0	0
$499$	6598.57	0.591969	0.295985	−	0.955193i	$-0.404352\pi$
$499$	6598.57	0.591969	0.295985	+	0.955193i	$0.404352\pi$
$500$	0	0
$501$	−1439.76	−0.128391
$502$	0	0
$503$	4191.61	0.371560	0.185780	−	0.982591i	$-0.440519\pi$
$503$	4191.61	0.371560	0.185780	+	0.982591i	$0.440519\pi$
$504$	0	0
$505$	330.642	0.0291354
$506$	0	0
$507$	−26892.3	−2.35568
$508$	0	0
$509$	−9553.58	−0.831935	−0.415967	−	0.909379i	$-0.636557\pi$
$509$	−9553.58	−0.831935	−0.415967	+	0.909379i	$0.636557\pi$
$510$	0	0
$511$	−5963.33	−0.516247
$512$	0	0
$513$	−21547.2	−1.85445
$514$	0	0
$515$	−1605.80	−0.137398
$516$	0	0
$517$	−1153.27	−0.0981057
$518$	0	0
$519$	−10928.6	−0.924305
$520$	0	0
$521$	1074.39	0.0903456	0.0451728	−	0.998979i	$-0.485616\pi$
$521$	1074.39	0.0903456	0.0451728	+	0.998979i	$0.485616\pi$
$522$	0	0
$523$	−5493.14	−0.459270	−0.229635	−	0.973277i	$-0.573753\pi$
$523$	−5493.14	−0.459270	−0.229635	+	0.973277i	$0.573753\pi$
$524$	0	0
$525$	−14720.6	−1.22373
$526$	0	0
$527$	3779.88	0.312437
$528$	0	0
$529$	26893.7	2.21038
$530$	0	0
$531$	−7040.90	−0.575422
$532$	0	0
$533$	−24877.5	−2.02170
$534$	0	0
$535$	−1430.81	−0.115625
$536$	0	0
$537$	3138.50	0.252209
$538$	0	0
$539$	−686.656	−0.0548727
$540$	0	0
$541$	−3776.15	−0.300091	−0.150046	−	0.988679i	$-0.547942\pi$
$541$	−3776.15	−0.300091	−0.150046	+	0.988679i	$0.547942\pi$
$542$	0	0
$543$	31577.9	2.49565
$544$	0	0
$545$	1262.03	0.0991919
$546$	0	0
$547$	−502.622	−0.0392880	−0.0196440	−	0.999807i	$-0.506253\pi$
$547$	−502.622	−0.0392880	−0.0196440	+	0.999807i	$0.506253\pi$
$548$	0	0
$549$	−37133.1	−2.88671
$550$	0	0
$551$	16472.0	1.27356
$552$	0	0
$553$	3341.09	0.256922
$554$	0	0
$555$	527.431	0.0403391
$556$	0	0
$557$	2600.54	0.197825	0.0989125	−	0.995096i	$-0.468464\pi$
$557$	2600.54	0.197825	0.0989125	+	0.995096i	$0.468464\pi$
$558$	0	0
$559$	9856.60	0.745777
$560$	0	0
$561$	−1384.38	−0.104186
$562$	0	0
$563$	−10974.5	−0.821530	−0.410765	−	0.911741i	$-0.634738\pi$
$563$	−10974.5	−0.821530	−0.410765	+	0.911741i	$0.634738\pi$
$564$	0	0
$565$	−1194.08	−0.0889118
$566$	0	0
$567$	−18722.7	−1.38673
$568$	0	0
$569$	−6044.57	−0.445346	−0.222673	−	0.974893i	$-0.571478\pi$
$569$	−6044.57	−0.445346	−0.222673	+	0.974893i	$0.571478\pi$
$570$	0	0
$571$	1398.76	0.102515	0.0512577	−	0.998685i	$-0.483677\pi$
$571$	1398.76	0.102515	0.0512577	+	0.998685i	$0.483677\pi$
$572$	0	0
$573$	1632.69	0.119034
$574$	0	0
$575$	−24255.3	−1.75916
$576$	0	0
$577$	10841.3	0.782199	0.391100	−	0.920348i	$-0.372095\pi$
$577$	10841.3	0.782199	0.391100	+	0.920348i	$0.372095\pi$
$578$	0	0
$579$	−33396.6	−2.39709
$580$	0	0
$581$	−3995.70	−0.285318
$582$	0	0
$583$	−1170.30	−0.0831370
$584$	0	0
$585$	−6676.83	−0.471886
$586$	0	0
$587$	−19183.2	−1.34885	−0.674424	−	0.738344i	$-0.735608\pi$
$587$	−19183.2	−1.34885	−0.674424	+	0.738344i	$0.735608\pi$
$588$	0	0
$589$	6278.76	0.439239
$590$	0	0
$591$	−19269.0	−1.34115
$592$	0	0
$593$	−2984.87	−0.206701	−0.103351	−	0.994645i	$-0.532956\pi$
$593$	−2984.87	−0.206701	−0.103351	+	0.994645i	$0.532956\pi$
$594$	0	0
$595$	−742.733	−0.0511749
$596$	0	0
$597$	32315.2	2.21536
$598$	0	0
$599$	−26488.6	−1.80684	−0.903418	−	0.428760i	$-0.858950\pi$
$599$	−26488.6	−1.80684	−0.903418	+	0.428760i	$0.858950\pi$
$600$	0	0
$601$	−11038.5	−0.749202	−0.374601	−	0.927186i	$-0.622220\pi$
$601$	−11038.5	−0.749202	−0.374601	+	0.927186i	$0.622220\pi$
$602$	0	0
$603$	−56085.8	−3.78771
$604$	0	0
$605$	−1985.70	−0.133438
$606$	0	0
$607$	−1147.97	−0.0767621	−0.0383811	−	0.999263i	$-0.512220\pi$
$607$	−1147.97	−0.0767621	−0.0383811	+	0.999263i	$0.512220\pi$
$608$	0	0
$609$	30644.0	2.03901
$610$	0	0
$611$	21703.4	1.43703
$612$	0	0
$613$	15341.5	1.01083	0.505413	−	0.862878i	$-0.331340\pi$
$613$	15341.5	1.01083	0.505413	+	0.862878i	$0.331340\pi$
$614$	0	0
$615$	−4994.28	−0.327462
$616$	0	0
$617$	−4831.63	−0.315258	−0.157629	−	0.987498i	$-0.550385\pi$
$617$	−4831.63	−0.315258	−0.157629	+	0.987498i	$0.550385\pi$
$618$	0	0
$619$	13437.2	0.872514	0.436257	−	0.899822i	$-0.356304\pi$
$619$	13437.2	0.872514	0.436257	+	0.899822i	$0.356304\pi$
$620$	0	0
$621$	−66049.5	−4.26808
$622$	0	0
$623$	9802.61	0.630390
$624$	0	0
$625$	14777.4	0.945752
$626$	0	0
$627$	−2299.59	−0.146470
$628$	0	0
$629$	−1436.14	−0.0910376
$630$	0	0
$631$	−2826.33	−0.178311	−0.0891556	−	0.996018i	$-0.528417\pi$
$631$	−2826.33	−0.178311	−0.0891556	+	0.996018i	$0.528417\pi$
$632$	0	0
$633$	−43971.7	−2.76101
$634$	0	0
$635$	2356.41	0.147262
$636$	0	0
$637$	12922.2	0.803762
$638$	0	0
$639$	28081.4	1.73847
$640$	0	0
$641$	−26104.6	−1.60853	−0.804266	−	0.594270i	$-0.797441\pi$
$641$	−26104.6	−1.60853	−0.804266	+	0.594270i	$0.797441\pi$
$642$	0	0
$643$	−8811.39	−0.540416	−0.270208	−	0.962802i	$-0.587092\pi$
$643$	−8811.39	−0.540416	−0.270208	+	0.962802i	$0.587092\pi$
$644$	0	0
$645$	1978.76	0.120796
$646$	0	0
$647$	−20932.0	−1.27191	−0.635953	−	0.771728i	$-0.719393\pi$
$647$	−20932.0	−1.27191	−0.635953	+	0.771728i	$0.719393\pi$
$648$	0	0
$649$	−426.052	−0.0257689
$650$	0	0
$651$	11680.8	0.703236
$652$	0	0
$653$	−15214.9	−0.911802	−0.455901	−	0.890031i	$-0.650683\pi$
$653$	−15214.9	−0.911802	−0.455901	+	0.890031i	$0.650683\pi$
$654$	0	0
$655$	−1182.58	−0.0705455
$656$	0	0
$657$	29303.8	1.74010
$658$	0	0
$659$	−1499.89	−0.0886609	−0.0443305	−	0.999017i	$-0.514115\pi$
$659$	−1499.89	−0.0886609	−0.0443305	+	0.999017i	$0.514115\pi$
$660$	0	0
$661$	−5665.19	−0.333359	−0.166680	−	0.986011i	$-0.553305\pi$
$661$	−5665.19	−0.333359	−0.166680	+	0.986011i	$0.553305\pi$
$662$	0	0
$663$	26052.6	1.52609
$664$	0	0
$665$	−1233.75	−0.0719443
$666$	0	0
$667$	50492.4	2.93115
$668$	0	0
$669$	2654.40	0.153401
$670$	0	0
$671$	−2246.96	−0.129274
$672$	0	0
$673$	−10083.0	−0.577522	−0.288761	−	0.957401i	$-0.593243\pi$
$673$	−10083.0	−0.577522	−0.288761	+	0.957401i	$0.593243\pi$
$674$	0	0
$675$	41014.3	2.33873
$676$	0	0
$677$	15720.0	0.892421	0.446210	−	0.894928i	$-0.352773\pi$
$677$	15720.0	0.892421	0.446210	+	0.894928i	$0.352773\pi$
$678$	0	0
$679$	−2421.67	−0.136870
$680$	0	0
$681$	35440.8	1.99427
$682$	0	0
$683$	−12308.4	−0.689557	−0.344778	−	0.938684i	$-0.612046\pi$
$683$	−12308.4	−0.689557	−0.344778	+	0.938684i	$0.612046\pi$
$684$	0	0
$685$	3177.79	0.177251
$686$	0	0
$687$	49777.9	2.76440
$688$	0	0
$689$	22023.9	1.21777
$690$	0	0
$691$	−23345.1	−1.28522	−0.642611	−	0.766193i	$-0.722149\pi$
$691$	−23345.1	−1.28522	−0.642611	+	0.766193i	$0.722149\pi$
$692$	0	0
$693$	−2985.38	−0.163644
$694$	0	0
$695$	−3055.13	−0.166745
$696$	0	0
$697$	13598.9	0.739018
$698$	0	0
$699$	43061.2	2.33008
$700$	0	0
$701$	23813.7	1.28307	0.641534	−	0.767094i	$-0.278298\pi$
$701$	23813.7	1.28307	0.641534	+	0.767094i	$0.278298\pi$
$702$	0	0
$703$	−2385.57	−0.127985
$704$	0	0
$705$	4357.05	0.232760
$706$	0	0
$707$	−2782.17	−0.147998
$708$	0	0
$709$	−17351.3	−0.919101	−0.459550	−	0.888152i	$-0.651989\pi$
$709$	−17351.3	−0.919101	−0.459550	+	0.888152i	$0.651989\pi$
$710$	0	0
$711$	−16418.1	−0.866001
$712$	0	0
$713$	19246.6	1.01093
$714$	0	0
$715$	−404.022	−0.0211323
$716$	0	0
$717$	−25965.3	−1.35243
$718$	0	0
$719$	−8786.46	−0.455743	−0.227872	−	0.973691i	$-0.573177\pi$
$719$	−8786.46	−0.455743	−0.227872	+	0.973691i	$0.573177\pi$
$720$	0	0
$721$	13511.9	0.697933
$722$	0	0
$723$	−21026.0	−1.08155
$724$	0	0
$725$	−31353.9	−1.60615
$726$	0	0
$727$	−1973.85	−0.100696	−0.0503480	−	0.998732i	$-0.516033\pi$
$727$	−1973.85	−0.100696	−0.0503480	+	0.998732i	$0.516033\pi$
$728$	0	0
$729$	6708.62	0.340833
$730$	0	0
$731$	−5387.95	−0.272614
$732$	0	0
$733$	−16371.2	−0.824943	−0.412471	−	0.910971i	$-0.635334\pi$
$733$	−16371.2	−0.824943	−0.412471	+	0.910971i	$0.635334\pi$
$734$	0	0
$735$	2594.19	0.130188
$736$	0	0
$737$	−3393.81	−0.169624
$738$	0	0
$739$	28981.5	1.44263	0.721314	−	0.692608i	$-0.243538\pi$
$739$	28981.5	1.44263	0.721314	+	0.692608i	$0.243538\pi$
$740$	0	0
$741$	43276.0	2.14546
$742$	0	0
$743$	25088.6	1.23878	0.619389	−	0.785084i	$-0.287380\pi$
$743$	25088.6	1.23878	0.619389	+	0.785084i	$0.287380\pi$
$744$	0	0
$745$	1124.74	0.0553116
$746$	0	0
$747$	19634.8	0.961715
$748$	0	0
$749$	12039.4	0.587332
$750$	0	0
$751$	16568.9	0.805069	0.402534	−	0.915405i	$-0.368129\pi$
$751$	16568.9	0.805069	0.402534	+	0.915405i	$0.368129\pi$
$752$	0	0
$753$	37138.4	1.79734
$754$	0	0
$755$	−992.586	−0.0478462
$756$	0	0
$757$	5430.63	0.260739	0.130370	−	0.991465i	$-0.458384\pi$
$757$	5430.63	0.260739	0.130370	+	0.991465i	$0.458384\pi$
$758$	0	0
$759$	−7049.02	−0.337106
$760$	0	0
$761$	−7111.25	−0.338742	−0.169371	−	0.985552i	$-0.554174\pi$
$761$	−7111.25	−0.338742	−0.169371	+	0.985552i	$0.554174\pi$
$762$	0	0
$763$	−10619.3	−0.503860
$764$	0	0
$765$	3649.79	0.172495
$766$	0	0
$767$	8017.89	0.377457
$768$	0	0
$769$	−31140.3	−1.46027	−0.730136	−	0.683302i	$-0.760543\pi$
$769$	−31140.3	−1.46027	−0.730136	+	0.683302i	$0.760543\pi$
$770$	0	0
$771$	37455.2	1.74957
$772$	0	0
$773$	34520.2	1.60622	0.803109	−	0.595832i	$-0.203178\pi$
$773$	34520.2	1.60622	0.803109	+	0.595832i	$0.203178\pi$
$774$	0	0
$775$	−11951.4	−0.553945
$776$	0	0
$777$	−4438.04	−0.204908
$778$	0	0
$779$	22589.2	1.03895
$780$	0	0
$781$	1699.24	0.0778534
$782$	0	0
$783$	−85379.9	−3.89684
$784$	0	0
$785$	−3105.66	−0.141205
$786$	0	0
$787$	31271.2	1.41639	0.708195	−	0.706017i	$-0.249510\pi$
$787$	31271.2	1.41639	0.708195	+	0.706017i	$0.249510\pi$
$788$	0	0
$789$	353.928	0.0159698
$790$	0	0
$791$	10047.5	0.451641
$792$	0	0
$793$	42285.7	1.89358
$794$	0	0
$795$	4421.40	0.197246
$796$	0	0
$797$	37179.6	1.65241	0.826205	−	0.563370i	$-0.190495\pi$
$797$	37179.6	1.65241	0.826205	+	0.563370i	$0.190495\pi$
$798$	0	0
$799$	−11863.8	−0.525296
$800$	0	0
$801$	−48170.0	−2.12485
$802$	0	0
$803$	1773.20	0.0779264
$804$	0	0
$805$	−3781.88	−0.165582
$806$	0	0
$807$	27541.4	1.20137
$808$	0	0
$809$	2118.84	0.0920820	0.0460410	−	0.998940i	$-0.485340\pi$
$809$	2118.84	0.0920820	0.0460410	+	0.998940i	$0.485340\pi$
$810$	0	0
$811$	−18596.6	−0.805198	−0.402599	−	0.915377i	$-0.631893\pi$
$811$	−18596.6	−0.805198	−0.402599	+	0.915377i	$0.631893\pi$
$812$	0	0
$813$	−62426.7	−2.69299
$814$	0	0
$815$	−1854.23	−0.0796942
$816$	0	0
$817$	−8949.93	−0.383254
$818$	0	0
$819$	56181.9	2.39701
$820$	0	0
$821$	2969.97	0.126252	0.0631258	−	0.998006i	$-0.479893\pi$
$821$	2969.97	0.126252	0.0631258	+	0.998006i	$0.479893\pi$
$822$	0	0
$823$	−21859.2	−0.925835	−0.462918	−	0.886401i	$-0.653197\pi$
$823$	−21859.2	−0.925835	−0.462918	+	0.886401i	$0.653197\pi$
$824$	0	0
$825$	4377.19	0.184720
$826$	0	0
$827$	40220.4	1.69117	0.845587	−	0.533837i	$-0.179250\pi$
$827$	40220.4	1.69117	0.845587	+	0.533837i	$0.179250\pi$
$828$	0	0
$829$	16095.9	0.674346	0.337173	−	0.941443i	$-0.390529\pi$
$829$	16095.9	0.674346	0.337173	+	0.941443i	$0.390529\pi$
$830$	0	0
$831$	−15001.7	−0.626239
$832$	0	0
$833$	−7063.72	−0.293810
$834$	0	0
$835$	229.688	0.00951938
$836$	0	0
$837$	−32544.9	−1.34399
$838$	0	0
$839$	7811.35	0.321428	0.160714	−	0.987001i	$-0.448620\pi$
$839$	7811.35	0.321428	0.160714	+	0.987001i	$0.448620\pi$
$840$	0	0
$841$	40880.7	1.67620
$842$	0	0
$843$	−81202.7	−3.31764
$844$	0	0
$845$	4290.19	0.174659
$846$	0	0
$847$	16708.6	0.677820
$848$	0	0
$849$	5800.26	0.234469
$850$	0	0
$851$	−7312.60	−0.294563
$852$	0	0
$853$	−24196.0	−0.971224	−0.485612	−	0.874175i	$-0.661403\pi$
$853$	−24196.0	−0.971224	−0.485612	+	0.874175i	$0.661403\pi$
$854$	0	0
$855$	6062.66	0.242501
$856$	0	0
$857$	−18625.1	−0.742380	−0.371190	−	0.928557i	$-0.621050\pi$
$857$	−18625.1	−0.742380	−0.371190	+	0.928557i	$0.621050\pi$
$858$	0	0
$859$	−31480.8	−1.25042	−0.625210	−	0.780457i	$-0.714987\pi$
$859$	−31480.8	−1.25042	−0.625210	+	0.780457i	$0.714987\pi$
$860$	0	0
$861$	42024.1	1.66339
$862$	0	0
$863$	8968.69	0.353763	0.176882	−	0.984232i	$-0.443399\pi$
$863$	8968.69	0.353763	0.176882	+	0.984232i	$0.443399\pi$
$864$	0	0
$865$	1743.47	0.0685316
$866$	0	0
$867$	32200.1	1.26133
$868$	0	0
$869$	−993.477	−0.0387818
$870$	0	0
$871$	63868.2	2.48461
$872$	0	0
$873$	11900.1	0.461347
$874$	0	0
$875$	4740.33	0.183146
$876$	0	0
$877$	44884.3	1.72820	0.864101	−	0.503318i	$-0.167887\pi$
$877$	44884.3	1.72820	0.864101	+	0.503318i	$0.167887\pi$
$878$	0	0
$879$	−42561.5	−1.63318
$880$	0	0
$881$	−29887.5	−1.14295	−0.571473	−	0.820621i	$-0.693628\pi$
$881$	−29887.5	−1.14295	−0.571473	+	0.820621i	$0.693628\pi$
$882$	0	0
$883$	−34321.4	−1.30805	−0.654024	−	0.756474i	$-0.726920\pi$
$883$	−34321.4	−1.30805	−0.654024	+	0.756474i	$0.726920\pi$
$884$	0	0
$885$	1609.63	0.0611379
$886$	0	0
$887$	−29971.4	−1.13455	−0.567273	−	0.823530i	$-0.692001\pi$
$887$	−29971.4	−1.13455	−0.567273	+	0.823530i	$0.692001\pi$
$888$	0	0
$889$	−19827.9	−0.748039
$890$	0	0
$891$	5567.21	0.209325
$892$	0	0
$893$	−19707.0	−0.738486
$894$	0	0
$895$	−500.691	−0.0186997
$896$	0	0
$897$	132656.	4.93784
$898$	0	0
$899$	24879.4	0.922995
$900$	0	0
$901$	−12039.0	−0.445148
$902$	0	0
$903$	−16650.1	−0.613601
$904$	0	0
$905$	−5037.69	−0.185037
$906$	0	0
$907$	53353.2	1.95321	0.976607	−	0.215032i	$-0.0689857\pi$
$907$	53353.2	1.95321	0.976607	+	0.215032i	$0.0689857\pi$
$908$	0	0
$909$	13671.6	0.498853
$910$	0	0
$911$	36185.6	1.31601	0.658004	−	0.753014i	$-0.271401\pi$
$911$	36185.6	1.31601	0.658004	+	0.753014i	$0.271401\pi$
$912$	0	0
$913$	1188.13	0.0430681
$914$	0	0
$915$	8489.04	0.306709
$916$	0	0
$917$	9950.78	0.358346
$918$	0	0
$919$	−38631.3	−1.38665	−0.693324	−	0.720626i	$-0.743854\pi$
$919$	−38631.3	−1.38665	−0.693324	+	0.720626i	$0.743854\pi$
$920$	0	0
$921$	14092.1	0.504179
$922$	0	0
$923$	−31978.0	−1.14038
$924$	0	0
$925$	4540.86	0.161408
$926$	0	0
$927$	−66397.5	−2.35251
$928$	0	0
$929$	25067.5	0.885294	0.442647	−	0.896696i	$-0.354040\pi$
$929$	25067.5	0.885294	0.442647	+	0.896696i	$0.354040\pi$
$930$	0	0
$931$	−11733.5	−0.413052
$932$	0	0
$933$	2020.03	0.0708818
$934$	0	0
$935$	220.853	0.00772476
$936$	0	0
$937$	−5069.83	−0.176760	−0.0883801	−	0.996087i	$-0.528169\pi$
$937$	−5069.83	−0.176760	−0.0883801	+	0.996087i	$0.528169\pi$
$938$	0	0
$939$	83999.1	2.91928
$940$	0	0
$941$	−6417.99	−0.222338	−0.111169	−	0.993801i	$-0.535460\pi$
$941$	−6417.99	−0.222338	−0.111169	+	0.993801i	$0.535460\pi$
$942$	0	0
$943$	69243.6	2.39118
$944$	0	0
$945$	6394.95	0.220135
$946$	0	0
$947$	−11504.2	−0.394760	−0.197380	−	0.980327i	$-0.563243\pi$
$947$	−11504.2	−0.394760	−0.197380	+	0.980327i	$0.563243\pi$
$948$	0	0
$949$	−33369.9	−1.14145
$950$	0	0
$951$	−22767.5	−0.776327
$952$	0	0
$953$	−39729.1	−1.35042	−0.675212	−	0.737624i	$-0.735948\pi$
$953$	−39729.1	−1.35042	−0.675212	+	0.737624i	$0.735948\pi$
$954$	0	0
$955$	−260.466	−0.00882564
$956$	0	0
$957$	−9112.02	−0.307785
$958$	0	0
$959$	−26739.3	−0.900372
$960$	0	0
$961$	−20307.6	−0.681667
$962$	0	0
$963$	−59161.8	−1.97971
$964$	0	0
$965$	5327.83	0.177730
$966$	0	0
$967$	54102.8	1.79920	0.899602	−	0.436711i	$-0.143857\pi$
$967$	54102.8	1.79920	0.899602	+	0.436711i	$0.143857\pi$
$968$	0	0
$969$	−23656.1	−0.784256
$970$	0	0
$971$	11991.7	0.396325	0.198162	−	0.980169i	$-0.436503\pi$
$971$	11991.7	0.396325	0.198162	+	0.980169i	$0.436503\pi$
$972$	0	0
$973$	25707.3	0.847006
$974$	0	0
$975$	−82374.4	−2.70573
$976$	0	0
$977$	−30943.4	−1.01327	−0.506636	−	0.862160i	$-0.669111\pi$
$977$	−30943.4	−1.01327	−0.506636	+	0.862160i	$0.669111\pi$
$978$	0	0
$979$	−2914.82	−0.0951562
$980$	0	0
$981$	52183.2	1.69835
$982$	0	0
$983$	−25930.5	−0.841359	−0.420679	−	0.907209i	$-0.638208\pi$
$983$	−25930.5	−0.841359	−0.420679	+	0.907209i	$0.638208\pi$
$984$	0	0
$985$	3074.02	0.0994379
$986$	0	0
$987$	−36662.2	−1.18234
$988$	0	0
$989$	−27434.6	−0.882073
$990$	0	0
$991$	15448.4	0.495192	0.247596	−	0.968863i	$-0.420359\pi$
$991$	15448.4	0.495192	0.247596	+	0.968863i	$0.420359\pi$
$992$	0	0
$993$	−15056.7	−0.481179
$994$	0	0
$995$	−5155.31	−0.164256
$996$	0	0
$997$	4372.65	0.138900	0.0694499	−	0.997585i	$-0.477876\pi$
$997$	4372.65	0.138900	0.0694499	+	0.997585i	$0.477876\pi$
$998$	0	0
$999$	12365.2	0.391609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.4.a.g.1.1		5
4.3	odd	2		37.4.a.b.1.2	✓	5
8.3	odd	2		2368.4.a.m.1.1		5
8.5	even	2		2368.4.a.r.1.5		5
12.11	even	2		333.4.a.f.1.4		5
20.19	odd	2		925.4.a.b.1.4		5
28.27	even	2		1813.4.a.c.1.2		5
148.147	odd	2		1369.4.a.d.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.a.b.1.2	✓	5	4.3	odd	2
333.4.a.f.1.4		5	12.11	even	2
592.4.a.g.1.1		5	1.1	even	1	trivial
925.4.a.b.1.4		5	20.19	odd	2
1369.4.a.d.1.4		5	148.147	odd	2
1813.4.a.c.1.2		5	28.27	even	2
2368.4.a.m.1.1		5	8.3	odd	2
2368.4.a.r.1.5		5	8.5	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q-9.45274 q^{3} +1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} +O(q^{10})\)	\(q-9.45274 q^{3} +1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} +3.77312 q^{11} -71.0065 q^{13} -14.2549 q^{15} +38.8146 q^{17} +64.4749 q^{19} +119.947 q^{21} +197.638 q^{23} -122.726 q^{25} -334.195 q^{27} +255.479 q^{29} +97.3830 q^{31} -35.6664 q^{33} -19.1354 q^{35} -37.0000 q^{37} +671.206 q^{39} +350.356 q^{41} -138.813 q^{43} +94.0313 q^{45} -305.653 q^{47} -181.986 q^{49} -366.904 q^{51} -310.167 q^{53} +5.68993 q^{55} -609.465 q^{57} -112.918 q^{59} -595.518 q^{61} -791.221 q^{63} -107.079 q^{65} -899.470 q^{67} -1868.22 q^{69} +450.353 q^{71} +469.956 q^{73} +1160.10 q^{75} -47.8776 q^{77} -263.304 q^{79} +1475.49 q^{81} +314.892 q^{83} +58.5331 q^{85} -2414.98 q^{87} -772.521 q^{89} +901.010 q^{91} -920.536 q^{93} +97.2293 q^{95} +190.846 q^{97} +235.270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 13 q^{3} + 11 q^{5} - 24 q^{7} + 46 q^{9}+O(q^{10}) \) 5 * q - 13 * q^3 + 11 * q^5 - 24 * q^7 + 46 * q^9	\( 5 q - 13 q^{3} + 11 q^{5} - 24 q^{7} + 46 q^{9} - 61 q^{11} - 37 q^{13} + 116 q^{15} + 130 q^{17} + 22 q^{19} - 44 q^{21} - 73 q^{23} + 26 q^{25} - 472 q^{27} + 271 q^{29} - 363 q^{31} + 198 q^{33} - 604 q^{35} - 185 q^{37} + 65 q^{39} + 381 q^{41} + 408 q^{43} - 704 q^{45} - 276 q^{47} - 949 q^{49} + 38 q^{51} + 156 q^{53} + 843 q^{55} - 1618 q^{57} - 100 q^{59} - 1711 q^{61} - 94 q^{63} - 890 q^{65} - 787 q^{67} - 2335 q^{69} - 1578 q^{71} - 313 q^{73} - 684 q^{75} - 342 q^{77} - 569 q^{79} + 385 q^{81} - 2422 q^{83} - 2210 q^{85} - 2371 q^{87} - 2466 q^{89} + 1678 q^{91} - 1142 q^{93} - 794 q^{95} - 2406 q^{97} - 1746 q^{99}+O(q^{100}) \) 5 * q - 13 * q^3 + 11 * q^5 - 24 * q^7 + 46 * q^9 - 61 * q^11 - 37 * q^13 + 116 * q^15 + 130 * q^17 + 22 * q^19 - 44 * q^21 - 73 * q^23 + 26 * q^25 - 472 * q^27 + 271 * q^29 - 363 * q^31 + 198 * q^33 - 604 * q^35 - 185 * q^37 + 65 * q^39 + 381 * q^41 + 408 * q^43 - 704 * q^45 - 276 * q^47 - 949 * q^49 + 38 * q^51 + 156 * q^53 + 843 * q^55 - 1618 * q^57 - 100 * q^59 - 1711 * q^61 - 94 * q^63 - 890 * q^65 - 787 * q^67 - 2335 * q^69 - 1578 * q^71 - 313 * q^73 - 684 * q^75 - 342 * q^77 - 569 * q^79 + 385 * q^81 - 2422 * q^83 - 2210 * q^85 - 2371 * q^87 - 2466 * q^89 + 1678 * q^91 - 1142 * q^93 - 794 * q^95 - 2406 * q^97 - 1746 * q^99

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