LMFDB - Embedded newform 1476.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1476,4,Mod(1,1476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1476, 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("1476.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1476 = 2^{2} \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1476.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:	$87.0868191685$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.4344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x + 4 $ x^3 - x^2 - 16*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.66433$ of defining polynomial
Character	$\chi$	$=$	1476.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.09161 q^{5} -16.7758 q^{7} +O(q^{10})$	$q-3.09161 q^{5} -16.7758 q^{7} +1.93531 q^{11} -9.34270 q^{13} +64.5496 q^{17} +34.8771 q^{19} +68.9699 q^{23} -115.442 q^{25} +176.964 q^{29} -119.125 q^{31} +51.8642 q^{35} +72.6883 q^{37} +41.0000 q^{41} +347.660 q^{43} +42.0857 q^{47} -61.5717 q^{49} -346.827 q^{53} -5.98322 q^{55} -72.1421 q^{59} +75.9712 q^{61} +28.8839 q^{65} +46.3660 q^{67} -744.503 q^{71} -651.947 q^{73} -32.4664 q^{77} +202.844 q^{79} -978.917 q^{83} -199.562 q^{85} -438.743 q^{89} +156.731 q^{91} -107.826 q^{95} -1266.55 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 10 q^{5}+O(q^{10}) $ 3 * q + 10 * q^5	$ 3 q + 10 q^{5} + 42 q^{11} - 76 q^{13} + 78 q^{17} - 162 q^{19} + 48 q^{23} - 163 q^{25} + 88 q^{29} - 16 q^{31} - 56 q^{35} - 406 q^{37} + 123 q^{41} - 236 q^{43} - 528 q^{47} - 201 q^{49} + 92 q^{53} + 580 q^{55} - 712 q^{59} - 394 q^{61} - 280 q^{65} + 446 q^{67} - 804 q^{71} + 298 q^{73} - 300 q^{77} + 936 q^{79} - 796 q^{83} - 812 q^{85} + 314 q^{89} - 640 q^{91} - 1444 q^{95} - 302 q^{97}+O(q^{100}) $ 3 * q + 10 * q^5 + 42 * q^11 - 76 * q^13 + 78 * q^17 - 162 * q^19 + 48 * q^23 - 163 * q^25 + 88 * q^29 - 16 * q^31 - 56 * q^35 - 406 * q^37 + 123 * q^41 - 236 * q^43 - 528 * q^47 - 201 * q^49 + 92 * q^53 + 580 * q^55 - 712 * q^59 - 394 * q^61 - 280 * q^65 + 446 * q^67 - 804 * q^71 + 298 * q^73 - 300 * q^77 + 936 * q^79 - 796 * q^83 - 812 * q^85 + 314 * q^89 - 640 * q^91 - 1444 * q^95 - 302 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.09161	−0.276522	−0.138261	−	0.990396i	$-0.544151\pi$
$5$	−3.09161	−0.276522	−0.138261	+	0.990396i	$0.544151\pi$
$6$	0	0
$7$	−16.7758	−0.905809	−0.452905	−	0.891559i	$-0.649612\pi$
$7$	−16.7758	−0.905809	−0.452905	+	0.891559i	$0.649612\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.93531	0.0530471	0.0265235	−	0.999648i	$-0.491556\pi$
$11$	1.93531	0.0530471	0.0265235	+	0.999648i	$0.491556\pi$
$12$	0	0
$13$	−9.34270	−0.199323	−0.0996615	−	0.995021i	$-0.531776\pi$
$13$	−9.34270	−0.199323	−0.0996615	+	0.995021i	$0.531776\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	64.5496	0.920917	0.460458	−	0.887681i	$-0.347685\pi$
$17$	64.5496	0.920917	0.460458	+	0.887681i	$0.347685\pi$
$18$	0	0
$19$	34.8771	0.421124	0.210562	−	0.977580i	$-0.432471\pi$
$19$	34.8771	0.421124	0.210562	+	0.977580i	$0.432471\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	68.9699	0.625270	0.312635	−	0.949873i	$-0.398788\pi$
$23$	68.9699	0.625270	0.312635	+	0.949873i	$0.398788\pi$
$24$	0	0
$25$	−115.442	−0.923536
$26$	0	0
$27$	0	0
$28$	0	0
$29$	176.964	1.13315	0.566574	−	0.824011i	$-0.308268\pi$
$29$	176.964	1.13315	0.566574	+	0.824011i	$0.308268\pi$
$30$	0	0
$31$	−119.125	−0.690177	−0.345088	−	0.938570i	$-0.612151\pi$
$31$	−119.125	−0.690177	−0.345088	+	0.938570i	$0.612151\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	51.8642	0.250476
$36$	0	0
$37$	72.6883	0.322970	0.161485	−	0.986875i	$-0.448372\pi$
$37$	72.6883	0.322970	0.161485	+	0.986875i	$0.448372\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	347.660	1.23297	0.616484	−	0.787368i	$-0.288557\pi$
$43$	347.660	1.23297	0.616484	+	0.787368i	$0.288557\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	42.0857	0.130613	0.0653067	−	0.997865i	$-0.479197\pi$
$47$	42.0857	0.130613	0.0653067	+	0.997865i	$0.479197\pi$
$48$	0	0
$49$	−61.5717	−0.179509
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−346.827	−0.898875	−0.449437	−	0.893312i	$-0.648376\pi$
$53$	−346.827	−0.898875	−0.449437	+	0.893312i	$0.648376\pi$
$54$	0	0
$55$	−5.98322	−0.0146687
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−72.1421	−0.159188	−0.0795941	−	0.996827i	$-0.525362\pi$
$59$	−72.1421	−0.159188	−0.0795941	+	0.996827i	$0.525362\pi$
$60$	0	0
$61$	75.9712	0.159461	0.0797305	−	0.996816i	$-0.474594\pi$
$61$	75.9712	0.159461	0.0797305	+	0.996816i	$0.474594\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	28.8839	0.0551171
$66$	0	0
$67$	46.3660	0.0845448	0.0422724	−	0.999106i	$-0.486540\pi$
$67$	46.3660	0.0845448	0.0422724	+	0.999106i	$0.486540\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−744.503	−1.24445	−0.622227	−	0.782837i	$-0.713772\pi$
$71$	−744.503	−1.24445	−0.622227	+	0.782837i	$0.713772\pi$
$72$	0	0
$73$	−651.947	−1.04527	−0.522634	−	0.852557i	$-0.675051\pi$
$73$	−651.947	−1.04527	−0.522634	+	0.852557i	$0.675051\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−32.4664	−0.0480505
$78$	0	0
$79$	202.844	0.288883	0.144441	−	0.989513i	$-0.453861\pi$
$79$	202.844	0.288883	0.144441	+	0.989513i	$0.453861\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−978.917	−1.29458	−0.647290	−	0.762244i	$-0.724098\pi$
$83$	−978.917	−1.29458	−0.647290	+	0.762244i	$0.724098\pi$
$84$	0	0
$85$	−199.562	−0.254653
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−438.743	−0.522546	−0.261273	−	0.965265i	$-0.584142\pi$
$89$	−438.743	−0.522546	−0.261273	+	0.965265i	$0.584142\pi$
$90$	0	0
$91$	156.731	0.180549
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−107.826	−0.116450
$96$	0	0
$97$	−1266.55	−1.32576	−0.662879	−	0.748726i	$-0.730666\pi$
$97$	−1266.55	−1.32576	−0.662879	+	0.748726i	$0.730666\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	311.979	0.307357	0.153679	−	0.988121i	$-0.450888\pi$
$101$	311.979	0.307357	0.153679	+	0.988121i	$0.450888\pi$
$102$	0	0
$103$	1308.21	1.25147	0.625737	−	0.780034i	$-0.284798\pi$
$103$	1308.21	1.25147	0.625737	+	0.780034i	$0.284798\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−493.109	−0.445520	−0.222760	−	0.974873i	$-0.571507\pi$
$107$	−493.109	−0.445520	−0.222760	+	0.974873i	$0.571507\pi$
$108$	0	0
$109$	−557.037	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$109$	−557.037	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1368.37	1.13916	0.569581	−	0.821935i	$-0.307105\pi$
$113$	1368.37	1.13916	0.569581	+	0.821935i	$0.307105\pi$
$114$	0	0
$115$	−213.228	−0.172901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1082.87	−0.834175
$120$	0	0
$121$	−1327.25	−0.997186
$122$	0	0
$123$	0	0
$124$	0	0
$125$	743.352	0.531899
$126$	0	0
$127$	−1946.13	−1.35977	−0.679886	−	0.733318i	$-0.737971\pi$
$127$	−1946.13	−1.35977	−0.679886	+	0.733318i	$0.737971\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1586.75	1.05828	0.529141	−	0.848534i	$-0.322514\pi$
$131$	1586.75	1.05828	0.529141	+	0.848534i	$0.322514\pi$
$132$	0	0
$133$	−585.092	−0.381458
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1505.33	−0.938753	−0.469377	−	0.882998i	$-0.655521\pi$
$137$	−1505.33	−0.938753	−0.469377	+	0.882998i	$0.655521\pi$
$138$	0	0
$139$	−1202.23	−0.733613	−0.366807	−	0.930297i	$-0.619549\pi$
$139$	−1202.23	−0.733613	−0.366807	+	0.930297i	$0.619549\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−18.0810	−0.0105735
$144$	0	0
$145$	−547.101	−0.313340
$146$	0	0
$147$	0	0
$148$	0	0
$149$	469.222	0.257988	0.128994	−	0.991645i	$-0.458825\pi$
$149$	469.222	0.257988	0.128994	+	0.991645i	$0.458825\pi$
$150$	0	0
$151$	266.088	0.143403	0.0717017	−	0.997426i	$-0.477157\pi$
$151$	266.088	0.143403	0.0717017	+	0.997426i	$0.477157\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	368.288	0.190849
$156$	0	0
$157$	−1661.21	−0.844454	−0.422227	−	0.906490i	$-0.638751\pi$
$157$	−1661.21	−0.844454	−0.422227	+	0.906490i	$0.638751\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1157.03	−0.566376
$162$	0	0
$163$	−874.695	−0.420315	−0.210158	−	0.977668i	$-0.567398\pi$
$163$	−874.695	−0.420315	−0.210158	+	0.977668i	$0.567398\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−718.153	−0.332768	−0.166384	−	0.986061i	$-0.553209\pi$
$167$	−718.153	−0.332768	−0.166384	+	0.986061i	$0.553209\pi$
$168$	0	0
$169$	−2109.71	−0.960270
$170$	0	0
$171$	0	0
$172$	0	0
$173$	642.840	0.282510	0.141255	−	0.989973i	$-0.454886\pi$
$173$	642.840	0.282510	0.141255	+	0.989973i	$0.454886\pi$
$174$	0	0
$175$	1936.63	0.836547
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2692.31	1.12421	0.562103	−	0.827067i	$-0.309992\pi$
$179$	2692.31	1.12421	0.562103	+	0.827067i	$0.309992\pi$
$180$	0	0
$181$	−2090.95	−0.858669	−0.429335	−	0.903145i	$-0.641252\pi$
$181$	−2090.95	−0.858669	−0.429335	+	0.903145i	$0.641252\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−224.724	−0.0893082
$186$	0	0
$187$	124.924	0.0488520
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1759.86	−0.666695	−0.333348	−	0.942804i	$-0.608178\pi$
$191$	−1759.86	−0.666695	−0.333348	+	0.942804i	$0.608178\pi$
$192$	0	0
$193$	462.509	0.172498	0.0862490	−	0.996274i	$-0.472512\pi$
$193$	462.509	0.172498	0.0862490	+	0.996274i	$0.472512\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2809.02	−1.01591	−0.507956	−	0.861383i	$-0.669599\pi$
$197$	−2809.02	−1.01591	−0.507956	+	0.861383i	$0.669599\pi$
$198$	0	0
$199$	855.296	0.304675	0.152338	−	0.988329i	$-0.451320\pi$
$199$	855.296	0.304675	0.152338	+	0.988329i	$0.451320\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2968.71	−1.02642
$204$	0	0
$205$	−126.756	−0.0431854
$206$	0	0
$207$	0	0
$208$	0	0
$209$	67.4980	0.0223394
$210$	0	0
$211$	−2946.64	−0.961398	−0.480699	−	0.876886i	$-0.659617\pi$
$211$	−2946.64	−0.961398	−0.480699	+	0.876886i	$0.659617\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1074.83	−0.340942
$216$	0	0
$217$	1998.42	0.625169
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−603.068	−0.183560
$222$	0	0
$223$	432.711	0.129939	0.0649697	−	0.997887i	$-0.479305\pi$
$223$	432.711	0.129939	0.0649697	+	0.997887i	$0.479305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−680.586	−0.198996	−0.0994980	−	0.995038i	$-0.531724\pi$
$227$	−680.586	−0.198996	−0.0994980	+	0.995038i	$0.531724\pi$
$228$	0	0
$229$	2929.61	0.845390	0.422695	−	0.906272i	$-0.361084\pi$
$229$	2929.61	0.845390	0.422695	+	0.906272i	$0.361084\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1668.81	−0.469216	−0.234608	−	0.972090i	$-0.575381\pi$
$233$	−1668.81	−0.469216	−0.234608	+	0.972090i	$0.575381\pi$
$234$	0	0
$235$	−130.112	−0.0361174
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−112.178	−0.0303606	−0.0151803	−	0.999885i	$-0.504832\pi$
$239$	−112.178	−0.0303606	−0.0151803	+	0.999885i	$0.504832\pi$
$240$	0	0
$241$	535.567	0.143149	0.0715745	−	0.997435i	$-0.477198\pi$
$241$	535.567	0.143149	0.0715745	+	0.997435i	$0.477198\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	190.356	0.0496382
$246$	0	0
$247$	−325.846	−0.0839397
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1885.51	0.474153	0.237076	−	0.971491i	$-0.423811\pi$
$251$	1885.51	0.474153	0.237076	+	0.971491i	$0.423811\pi$
$252$	0	0
$253$	133.478	0.0331688
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3189.00	0.774025	0.387013	−	0.922074i	$-0.373507\pi$
$257$	3189.00	0.774025	0.387013	+	0.922074i	$0.373507\pi$
$258$	0	0
$259$	−1219.41	−0.292549
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1185.01	−0.277836	−0.138918	−	0.990304i	$-0.544363\pi$
$263$	−1185.01	−0.277836	−0.138918	+	0.990304i	$0.544363\pi$
$264$	0	0
$265$	1072.25	0.248558
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3105.32	−0.703848	−0.351924	−	0.936029i	$-0.614472\pi$
$269$	−3105.32	−0.703848	−0.351924	+	0.936029i	$0.614472\pi$
$270$	0	0
$271$	−4613.88	−1.03422	−0.517110	−	0.855919i	$-0.672992\pi$
$271$	−4613.88	−1.03422	−0.517110	+	0.855919i	$0.672992\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−223.416	−0.0489909
$276$	0	0
$277$	1477.87	0.320566	0.160283	−	0.987071i	$-0.448759\pi$
$277$	1477.87	0.320566	0.160283	+	0.987071i	$0.448759\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5233.42	−1.11103	−0.555516	−	0.831506i	$-0.687479\pi$
$281$	−5233.42	−1.11103	−0.555516	+	0.831506i	$0.687479\pi$
$282$	0	0
$283$	−5249.12	−1.10257	−0.551286	−	0.834316i	$-0.685863\pi$
$283$	−5249.12	−1.10257	−0.551286	+	0.834316i	$0.685863\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−687.809	−0.141464
$288$	0	0
$289$	−746.344	−0.151912
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6331.92	−1.26251	−0.631254	−	0.775576i	$-0.717459\pi$
$293$	−6331.92	−1.26251	−0.631254	+	0.775576i	$0.717459\pi$
$294$	0	0
$295$	223.035	0.0440190
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−644.365	−0.124631
$300$	0	0
$301$	−5832.28	−1.11683
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−234.873	−0.0440944
$306$	0	0
$307$	−8972.43	−1.66802	−0.834012	−	0.551746i	$-0.813962\pi$
$307$	−8972.43	−1.66802	−0.834012	+	0.551746i	$0.813962\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9243.37	1.68535	0.842674	−	0.538424i	$-0.180980\pi$
$311$	9243.37	1.68535	0.842674	+	0.538424i	$0.180980\pi$
$312$	0	0
$313$	−1538.02	−0.277745	−0.138872	−	0.990310i	$-0.544348\pi$
$313$	−1538.02	−0.277745	−0.138872	+	0.990310i	$0.544348\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2315.48	0.410253	0.205127	−	0.978735i	$-0.434239\pi$
$317$	2315.48	0.410253	0.205127	+	0.978735i	$0.434239\pi$
$318$	0	0
$319$	342.479	0.0601102
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2251.30	0.387820
$324$	0	0
$325$	1078.54	0.184082
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−706.022	−0.118311
$330$	0	0
$331$	−7719.37	−1.28186	−0.640929	−	0.767600i	$-0.721451\pi$
$331$	−7719.37	−1.28186	−0.640929	+	0.767600i	$0.721451\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−143.345	−0.0233785
$336$	0	0
$337$	−10486.5	−1.69507	−0.847534	−	0.530741i	$-0.821914\pi$
$337$	−10486.5	−1.69507	−0.847534	+	0.530741i	$0.821914\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−230.544	−0.0366119
$342$	0	0
$343$	6787.02	1.06841
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3601.48	0.557168	0.278584	−	0.960412i	$-0.410135\pi$
$347$	3601.48	0.557168	0.278584	+	0.960412i	$0.410135\pi$
$348$	0	0
$349$	1161.86	0.178203	0.0891016	−	0.996023i	$-0.471600\pi$
$349$	1161.86	0.178203	0.0891016	+	0.996023i	$0.471600\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1526.41	−0.230148	−0.115074	−	0.993357i	$-0.536711\pi$
$353$	−1526.41	−0.230148	−0.115074	+	0.993357i	$0.536711\pi$
$354$	0	0
$355$	2301.71	0.344119
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5901.23	−0.867562	−0.433781	−	0.901018i	$-0.642821\pi$
$359$	−5901.23	−0.867562	−0.433781	+	0.901018i	$0.642821\pi$
$360$	0	0
$361$	−5642.59	−0.822655
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2015.56	0.289039
$366$	0	0
$367$	3027.41	0.430598	0.215299	−	0.976548i	$-0.430927\pi$
$367$	3027.41	0.430598	0.215299	+	0.976548i	$0.430927\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5818.31	0.814209
$372$	0	0
$373$	3810.61	0.528970	0.264485	−	0.964390i	$-0.414798\pi$
$373$	3810.61	0.528970	0.264485	+	0.964390i	$0.414798\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1653.32	−0.225862
$378$	0	0
$379$	−4465.84	−0.605264	−0.302632	−	0.953107i	$-0.597865\pi$
$379$	−4465.84	−0.605264	−0.302632	+	0.953107i	$0.597865\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1122.94	−0.149816	−0.0749082	−	0.997190i	$-0.523866\pi$
$383$	−1122.94	−0.149816	−0.0749082	+	0.997190i	$0.523866\pi$
$384$	0	0
$385$	100.373	0.0132870
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5013.37	0.653440	0.326720	−	0.945121i	$-0.394057\pi$
$389$	5013.37	0.653440	0.326720	+	0.945121i	$0.394057\pi$
$390$	0	0
$391$	4451.98	0.575822
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−627.114	−0.0798824
$396$	0	0
$397$	3627.07	0.458533	0.229266	−	0.973364i	$-0.426367\pi$
$397$	3627.07	0.458533	0.229266	+	0.973364i	$0.426367\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	148.294	0.0184674	0.00923371	−	0.999957i	$-0.497061\pi$
$401$	148.294	0.0184674	0.00923371	+	0.999957i	$0.497061\pi$
$402$	0	0
$403$	1112.95	0.137568
$404$	0	0
$405$	0	0
$406$	0	0
$407$	140.674	0.0171326
$408$	0	0
$409$	−10080.6	−1.21872	−0.609359	−	0.792894i	$-0.708573\pi$
$409$	−10080.6	−1.21872	−0.609359	+	0.792894i	$0.708573\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1210.24	0.144194
$414$	0	0
$415$	3026.43	0.357979
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3428.59	0.399756	0.199878	−	0.979821i	$-0.435945\pi$
$419$	3428.59	0.399756	0.199878	+	0.979821i	$0.435945\pi$
$420$	0	0
$421$	−2790.34	−0.323023	−0.161511	−	0.986871i	$-0.551637\pi$
$421$	−2790.34	−0.323023	−0.161511	+	0.986871i	$0.551637\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7451.74	−0.850500
$426$	0	0
$427$	−1274.48	−0.144441
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13121.0	1.46639	0.733195	−	0.680018i	$-0.238028\pi$
$431$	13121.0	1.46639	0.733195	+	0.680018i	$0.238028\pi$
$432$	0	0
$433$	15752.5	1.74830	0.874152	−	0.485652i	$-0.161418\pi$
$433$	15752.5	1.74830	0.874152	+	0.485652i	$0.161418\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2405.47	0.263316
$438$	0	0
$439$	9831.84	1.06890	0.534451	−	0.845199i	$-0.320518\pi$
$439$	9831.84	1.06890	0.534451	+	0.845199i	$0.320518\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5337.28	−0.572419	−0.286210	−	0.958167i	$-0.592395\pi$
$443$	−5337.28	−0.572419	−0.286210	+	0.958167i	$0.592395\pi$
$444$	0	0
$445$	1356.42	0.144495
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5991.97	−0.629796	−0.314898	−	0.949126i	$-0.601970\pi$
$449$	−5991.97	−0.629796	−0.314898	+	0.949126i	$0.601970\pi$
$450$	0	0
$451$	79.3477	0.00828456
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−484.552	−0.0499256
$456$	0	0
$457$	7516.44	0.769374	0.384687	−	0.923047i	$-0.374309\pi$
$457$	7516.44	0.769374	0.384687	+	0.923047i	$0.374309\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18851.1	−1.90452	−0.952258	−	0.305293i	$-0.901246\pi$
$461$	−18851.1	−1.90452	−0.952258	+	0.305293i	$0.901246\pi$
$462$	0	0
$463$	16138.9	1.61995	0.809977	−	0.586462i	$-0.199480\pi$
$463$	16138.9	1.61995	0.809977	+	0.586462i	$0.199480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8759.74	−0.867993	−0.433996	−	0.900915i	$-0.642897\pi$
$467$	−8759.74	−0.867993	−0.433996	+	0.900915i	$0.642897\pi$
$468$	0	0
$469$	−777.827	−0.0765815
$470$	0	0
$471$	0	0
$472$	0	0
$473$	672.829	0.0654053
$474$	0	0
$475$	−4026.28	−0.388923
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10121.7	−0.965494	−0.482747	−	0.875760i	$-0.660361\pi$
$479$	−10121.7	−0.965494	−0.482747	+	0.875760i	$0.660361\pi$
$480$	0	0
$481$	−679.105	−0.0643753
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3915.67	0.366601
$486$	0	0
$487$	5253.64	0.488840	0.244420	−	0.969669i	$-0.421402\pi$
$487$	5253.64	0.488840	0.244420	+	0.969669i	$0.421402\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15116.3	1.38939	0.694693	−	0.719306i	$-0.255540\pi$
$491$	15116.3	1.38939	0.694693	+	0.719306i	$0.255540\pi$
$492$	0	0
$493$	11422.9	1.04354
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12489.6	1.12724
$498$	0	0
$499$	−11514.5	−1.03299	−0.516494	−	0.856291i	$-0.672763\pi$
$499$	−11514.5	−1.03299	−0.516494	+	0.856291i	$0.672763\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17266.9	−1.53061	−0.765303	−	0.643670i	$-0.777411\pi$
$503$	−17266.9	−1.53061	−0.765303	+	0.643670i	$0.777411\pi$
$504$	0	0
$505$	−964.517	−0.0849910
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19770.1	1.72160	0.860798	−	0.508947i	$-0.169965\pi$
$509$	19770.1	1.72160	0.860798	+	0.508947i	$0.169965\pi$
$510$	0	0
$511$	10936.9	0.946814
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4044.47	−0.346060
$516$	0	0
$517$	81.4489	0.00692866
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1671.29	0.140538	0.0702692	−	0.997528i	$-0.477614\pi$
$521$	1671.29	0.140538	0.0702692	+	0.997528i	$0.477614\pi$
$522$	0	0
$523$	14765.1	1.23448	0.617238	−	0.786776i	$-0.288252\pi$
$523$	14765.1	1.23448	0.617238	+	0.786776i	$0.288252\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7689.48	−0.635596
$528$	0	0
$529$	−7410.15	−0.609037
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−383.051	−0.0311290
$534$	0	0
$535$	1524.50	0.123196
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−119.160	−0.00952245
$540$	0	0
$541$	−3776.95	−0.300155	−0.150078	−	0.988674i	$-0.547952\pi$
$541$	−3776.95	−0.300155	−0.150078	+	0.988674i	$0.547952\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1722.14	0.135355
$546$	0	0
$547$	7131.53	0.557445	0.278722	−	0.960372i	$-0.410089\pi$
$547$	7131.53	0.557445	0.278722	+	0.960372i	$0.410089\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6171.98	0.477196
$552$	0	0
$553$	−3402.88	−0.261673
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6615.63	0.503256	0.251628	−	0.967824i	$-0.419034\pi$
$557$	6615.63	0.503256	0.251628	+	0.967824i	$0.419034\pi$
$558$	0	0
$559$	−3248.08	−0.245759
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16297.2	−1.21998	−0.609988	−	0.792411i	$-0.708826\pi$
$563$	−16297.2	−1.21998	−0.609988	+	0.792411i	$0.708826\pi$
$564$	0	0
$565$	−4230.46	−0.315003
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7111.08	−0.523923	−0.261961	−	0.965078i	$-0.584369\pi$
$569$	−7111.08	−0.523923	−0.261961	+	0.965078i	$0.584369\pi$
$570$	0	0
$571$	19924.3	1.46025	0.730127	−	0.683312i	$-0.239461\pi$
$571$	19924.3	1.46025	0.730127	+	0.683312i	$0.239461\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7962.02	−0.577459
$576$	0	0
$577$	−3025.53	−0.218292	−0.109146	−	0.994026i	$-0.534812\pi$
$577$	−3025.53	−0.218292	−0.109146	+	0.994026i	$0.534812\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16422.1	1.17264
$582$	0	0
$583$	−671.218	−0.0476827
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26960.4	1.89570	0.947848	−	0.318724i	$-0.103254\pi$
$587$	26960.4	1.89570	0.947848	+	0.318724i	$0.103254\pi$
$588$	0	0
$589$	−4154.74	−0.290650
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3013.03	−0.208651	−0.104326	−	0.994543i	$-0.533268\pi$
$593$	−3013.03	−0.208651	−0.104326	+	0.994543i	$0.533268\pi$
$594$	0	0
$595$	3347.82	0.230667
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4068.37	0.277511	0.138756	−	0.990327i	$-0.455690\pi$
$599$	4068.37	0.277511	0.138756	+	0.990327i	$0.455690\pi$
$600$	0	0
$601$	−21942.6	−1.48928	−0.744641	−	0.667465i	$-0.767380\pi$
$601$	−21942.6	−1.48928	−0.744641	+	0.667465i	$0.767380\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4103.35	0.275744
$606$	0	0
$607$	13318.5	0.890579	0.445289	−	0.895387i	$-0.353101\pi$
$607$	13318.5	0.890579	0.445289	+	0.895387i	$0.353101\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−393.194	−0.0260343
$612$	0	0
$613$	−12536.4	−0.826003	−0.413001	−	0.910730i	$-0.635519\pi$
$613$	−12536.4	−0.826003	−0.413001	+	0.910730i	$0.635519\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4927.19	0.321493	0.160747	−	0.986996i	$-0.448610\pi$
$617$	4927.19	0.321493	0.160747	+	0.986996i	$0.448610\pi$
$618$	0	0
$619$	−1358.47	−0.0882092	−0.0441046	−	0.999027i	$-0.514043\pi$
$619$	−1358.47	−0.0882092	−0.0441046	+	0.999027i	$0.514043\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7360.27	0.473327
$624$	0	0
$625$	12132.1	0.776454
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4692.01	0.297429
$630$	0	0
$631$	−19632.6	−1.23861	−0.619305	−	0.785151i	$-0.712585\pi$
$631$	−19632.6	−1.23861	−0.619305	+	0.785151i	$0.712585\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6016.67	0.376006
$636$	0	0
$637$	575.246	0.0357803
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7306.62	0.450225	0.225112	−	0.974333i	$-0.427725\pi$
$641$	7306.62	0.450225	0.225112	+	0.974333i	$0.427725\pi$
$642$	0	0
$643$	20600.7	1.26347	0.631735	−	0.775185i	$-0.282343\pi$
$643$	20600.7	1.26347	0.631735	+	0.775185i	$0.282343\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16361.4	0.994180	0.497090	−	0.867699i	$-0.334402\pi$
$647$	16361.4	0.994180	0.497090	+	0.867699i	$0.334402\pi$
$648$	0	0
$649$	−139.617	−0.00844447
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17566.8	−1.05274	−0.526371	−	0.850255i	$-0.676448\pi$
$653$	−17566.8	−1.05274	−0.526371	+	0.850255i	$0.676448\pi$
$654$	0	0
$655$	−4905.60	−0.292638
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28097.4	1.66088	0.830441	−	0.557107i	$-0.188089\pi$
$659$	28097.4	1.66088	0.830441	+	0.557107i	$0.188089\pi$
$660$	0	0
$661$	−23933.8	−1.40835	−0.704173	−	0.710029i	$-0.748682\pi$
$661$	−23933.8	−1.40835	−0.704173	+	0.710029i	$0.748682\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1808.87	0.105481
$666$	0	0
$667$	12205.2	0.708524
$668$	0	0
$669$	0	0
$670$	0	0
$671$	147.028	0.00845894
$672$	0	0
$673$	−15227.7	−0.872192	−0.436096	−	0.899900i	$-0.643639\pi$
$673$	−15227.7	−0.872192	−0.436096	+	0.899900i	$0.643639\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7648.80	−0.434220	−0.217110	−	0.976147i	$-0.569663\pi$
$677$	−7648.80	−0.434220	−0.217110	+	0.976147i	$0.569663\pi$
$678$	0	0
$679$	21247.4	1.20088
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1012.96	0.0567493	0.0283747	−	0.999597i	$-0.490967\pi$
$683$	1012.96	0.0567493	0.0283747	+	0.999597i	$0.490967\pi$
$684$	0	0
$685$	4653.89	0.259586
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3240.30	0.179166
$690$	0	0
$691$	−3020.23	−0.166274	−0.0831368	−	0.996538i	$-0.526494\pi$
$691$	−3020.23	−0.166274	−0.0831368	+	0.996538i	$0.526494\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3716.84	0.202860
$696$	0	0
$697$	2646.54	0.143823
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28212.6	1.52008	0.760040	−	0.649876i	$-0.225179\pi$
$701$	28212.6	1.52008	0.760040	+	0.649876i	$0.225179\pi$
$702$	0	0
$703$	2535.16	0.136010
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5233.71	−0.278407
$708$	0	0
$709$	36223.7	1.91877	0.959385	−	0.282099i	$-0.0910308\pi$
$709$	36223.7	1.91877	0.959385	+	0.282099i	$0.0910308\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8216.04	−0.431547
$714$	0	0
$715$	55.8994	0.00292380
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20742.0	−1.07586	−0.537931	−	0.842989i	$-0.680794\pi$
$719$	−20742.0	−1.07586	−0.537931	+	0.842989i	$0.680794\pi$
$720$	0	0
$721$	−21946.3	−1.13360
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−20429.0	−1.04650
$726$	0	0
$727$	30593.7	1.56074	0.780370	−	0.625318i	$-0.215031\pi$
$727$	30593.7	1.56074	0.780370	+	0.625318i	$0.215031\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22441.3	1.13546
$732$	0	0
$733$	8330.87	0.419792	0.209896	−	0.977724i	$-0.432687\pi$
$733$	8330.87	0.419792	0.209896	+	0.977724i	$0.432687\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	89.7325	0.00448486
$738$	0	0
$739$	29773.6	1.48206	0.741029	−	0.671473i	$-0.234338\pi$
$739$	29773.6	1.48206	0.741029	+	0.671473i	$0.234338\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21964.1	1.08450	0.542250	−	0.840217i	$-0.317572\pi$
$743$	21964.1	1.08450	0.542250	+	0.840217i	$0.317572\pi$
$744$	0	0
$745$	−1450.65	−0.0713392
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8272.31	0.403556
$750$	0	0
$751$	27105.0	1.31701	0.658506	−	0.752575i	$-0.271189\pi$
$751$	27105.0	1.31701	0.658506	+	0.752575i	$0.271189\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−822.638	−0.0396541
$756$	0	0
$757$	4506.25	0.216357	0.108179	−	0.994131i	$-0.465498\pi$
$757$	4506.25	0.216357	0.108179	+	0.994131i	$0.465498\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24622.6	−1.17289	−0.586446	−	0.809989i	$-0.699473\pi$
$761$	−24622.6	−1.17289	−0.586446	+	0.809989i	$0.699473\pi$
$762$	0	0
$763$	9344.75	0.443385
$764$	0	0
$765$	0	0
$766$	0	0
$767$	674.002	0.0317299
$768$	0	0
$769$	−8049.55	−0.377470	−0.188735	−	0.982028i	$-0.560439\pi$
$769$	−8049.55	−0.377470	−0.188735	+	0.982028i	$0.560439\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−35523.0	−1.65287	−0.826437	−	0.563029i	$-0.809636\pi$
$773$	−35523.0	−1.65287	−0.826437	+	0.563029i	$0.809636\pi$
$774$	0	0
$775$	13752.0	0.637403
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1429.96	0.0657685
$780$	0	0
$781$	−1440.84	−0.0660147
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5135.82	0.233510
$786$	0	0
$787$	−11079.0	−0.501808	−0.250904	−	0.968012i	$-0.580728\pi$
$787$	−11079.0	−0.501808	−0.250904	+	0.968012i	$0.580728\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−22955.5	−1.03186
$792$	0	0
$793$	−709.776	−0.0317842
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15045.2	0.668666	0.334333	−	0.942455i	$-0.391489\pi$
$797$	15045.2	0.668666	0.334333	+	0.942455i	$0.391489\pi$
$798$	0	0
$799$	2716.62	0.120284
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1261.72	−0.0554485
$804$	0	0
$805$	3577.07	0.156615
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36761.3	−1.59760	−0.798799	−	0.601597i	$-0.794531\pi$
$809$	−36761.3	−1.59760	−0.798799	+	0.601597i	$0.794531\pi$
$810$	0	0
$811$	3589.99	0.155440	0.0777198	−	0.996975i	$-0.475236\pi$
$811$	3589.99	0.155440	0.0777198	+	0.996975i	$0.475236\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2704.21	0.116226
$816$	0	0
$817$	12125.4	0.519232
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38006.9	−1.61565	−0.807825	−	0.589422i	$-0.799356\pi$
$821$	−38006.9	−1.61565	−0.807825	+	0.589422i	$0.799356\pi$
$822$	0	0
$823$	−12560.4	−0.531990	−0.265995	−	0.963974i	$-0.585701\pi$
$823$	−12560.4	−0.531990	−0.265995	+	0.963974i	$0.585701\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16793.8	0.706139	0.353070	−	0.935597i	$-0.385138\pi$
$827$	16793.8	0.706139	0.353070	+	0.935597i	$0.385138\pi$
$828$	0	0
$829$	−23823.0	−0.998078	−0.499039	−	0.866579i	$-0.666314\pi$
$829$	−23823.0	−0.998078	−0.499039	+	0.866579i	$0.666314\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3974.43	−0.165313
$834$	0	0
$835$	2220.24	0.0920177
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15284.0	0.628919	0.314459	−	0.949271i	$-0.398177\pi$
$839$	15284.0	0.628919	0.314459	+	0.949271i	$0.398177\pi$
$840$	0	0
$841$	6927.09	0.284025
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6522.40	0.265536
$846$	0	0
$847$	22265.8	0.903260
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5013.31	0.201944
$852$	0	0
$853$	27575.2	1.10687	0.553433	−	0.832894i	$-0.313318\pi$
$853$	27575.2	1.10687	0.553433	+	0.832894i	$0.313318\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7879.22	0.314060	0.157030	−	0.987594i	$-0.449808\pi$
$857$	7879.22	0.314060	0.157030	+	0.987594i	$0.449808\pi$
$858$	0	0
$859$	38691.1	1.53681	0.768406	−	0.639962i	$-0.221050\pi$
$859$	38691.1	1.53681	0.768406	+	0.639962i	$0.221050\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6102.33	0.240702	0.120351	−	0.992731i	$-0.461598\pi$
$863$	6102.33	0.240702	0.120351	+	0.992731i	$0.461598\pi$
$864$	0	0
$865$	−1987.41	−0.0781201
$866$	0	0
$867$	0	0
$868$	0	0
$869$	392.566	0.0153244
$870$	0	0
$871$	−433.183	−0.0168517
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12470.3	−0.481799
$876$	0	0
$877$	−5749.76	−0.221386	−0.110693	−	0.993855i	$-0.535307\pi$
$877$	−5749.76	−0.221386	−0.110693	+	0.993855i	$0.535307\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35299.1	1.34990	0.674948	−	0.737866i	$-0.264166\pi$
$881$	35299.1	1.34990	0.674948	+	0.737866i	$0.264166\pi$
$882$	0	0
$883$	4058.30	0.154669	0.0773344	−	0.997005i	$-0.475359\pi$
$883$	4058.30	0.154669	0.0773344	+	0.997005i	$0.475359\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8452.80	0.319974	0.159987	−	0.987119i	$-0.448855\pi$
$887$	8452.80	0.319974	0.159987	+	0.987119i	$0.448855\pi$
$888$	0	0
$889$	32647.9	1.23169
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1467.83	0.0550045
$894$	0	0
$895$	−8323.57	−0.310867
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21080.8	−0.782073
$900$	0	0
$901$	−22387.6	−0.827789
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6464.40	0.237441
$906$	0	0
$907$	−41552.3	−1.52119	−0.760597	−	0.649225i	$-0.775094\pi$
$907$	−41552.3	−1.52119	−0.760597	+	0.649225i	$0.775094\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	40526.1	1.47386	0.736932	−	0.675967i	$-0.236274\pi$
$911$	40526.1	1.47386	0.736932	+	0.675967i	$0.236274\pi$
$912$	0	0
$913$	−1894.51	−0.0686737
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26619.0	−0.958601
$918$	0	0
$919$	−40119.2	−1.44006	−0.720028	−	0.693945i	$-0.755871\pi$
$919$	−40119.2	−1.44006	−0.720028	+	0.693945i	$0.755871\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6955.67	0.248048
$924$	0	0
$925$	−8391.29	−0.298274
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7960.63	0.281141	0.140570	−	0.990071i	$-0.455106\pi$
$929$	7960.63	0.281141	0.140570	+	0.990071i	$0.455106\pi$
$930$	0	0
$931$	−2147.44	−0.0755957
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−386.214	−0.0135086
$936$	0	0
$937$	42245.8	1.47290	0.736451	−	0.676491i	$-0.236500\pi$
$937$	42245.8	1.47290	0.736451	+	0.676491i	$0.236500\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13101.3	−0.453868	−0.226934	−	0.973910i	$-0.572870\pi$
$941$	−13101.3	−0.453868	−0.226934	+	0.973910i	$0.572870\pi$
$942$	0	0
$943$	2827.77	0.0976508
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37987.7	1.30352	0.651760	−	0.758425i	$-0.274031\pi$
$947$	37987.7	1.30352	0.651760	+	0.758425i	$0.274031\pi$
$948$	0	0
$949$	6090.94	0.208346
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13686.5	−0.465213	−0.232607	−	0.972571i	$-0.574725\pi$
$953$	−13686.5	−0.465213	−0.232607	+	0.972571i	$0.574725\pi$
$954$	0	0
$955$	5440.79	0.184356
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25253.2	0.850331
$960$	0	0
$961$	−15600.2	−0.523656
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1429.89	−0.0476994
$966$	0	0
$967$	−52395.8	−1.74244	−0.871218	−	0.490896i	$-0.836669\pi$
$967$	−52395.8	−1.74244	−0.871218	+	0.490896i	$0.836669\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−54696.6	−1.80772	−0.903861	−	0.427825i	$-0.859280\pi$
$971$	−54696.6	−1.80772	−0.903861	+	0.427825i	$0.859280\pi$
$972$	0	0
$973$	20168.5	0.664514
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43649.8	1.42936	0.714678	−	0.699454i	$-0.246573\pi$
$977$	43649.8	1.42936	0.714678	+	0.699454i	$0.246573\pi$
$978$	0	0
$979$	−849.103	−0.0277196
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37485.8	1.21629	0.608144	−	0.793827i	$-0.291914\pi$
$983$	37485.8	1.21629	0.608144	+	0.793827i	$0.291914\pi$
$984$	0	0
$985$	8684.40	0.280922
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23978.0	0.770938
$990$	0	0
$991$	−17392.8	−0.557518	−0.278759	−	0.960361i	$-0.589923\pi$
$991$	−17392.8	−0.557518	−0.278759	+	0.960361i	$0.589923\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2644.24	−0.0842492
$996$	0	0
$997$	−17394.6	−0.552550	−0.276275	−	0.961079i	$-0.589100\pi$
$997$	−17394.6	−0.552550	−0.276275	+	0.961079i	$0.589100\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1476.4.a.d.1.1		3
3.2	odd	2		164.4.a.a.1.2	✓	3
12.11	even	2		656.4.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.4.a.a.1.2	✓	3	3.2	odd	2
656.4.a.d.1.2		3	12.11	even	2
1476.4.a.d.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1476.a (trivial)

\(f(q)\)	\(=\)	\(q-3.09161 q^{5} -16.7758 q^{7} +O(q^{10})\)	\(q-3.09161 q^{5} -16.7758 q^{7} +1.93531 q^{11} -9.34270 q^{13} +64.5496 q^{17} +34.8771 q^{19} +68.9699 q^{23} -115.442 q^{25} +176.964 q^{29} -119.125 q^{31} +51.8642 q^{35} +72.6883 q^{37} +41.0000 q^{41} +347.660 q^{43} +42.0857 q^{47} -61.5717 q^{49} -346.827 q^{53} -5.98322 q^{55} -72.1421 q^{59} +75.9712 q^{61} +28.8839 q^{65} +46.3660 q^{67} -744.503 q^{71} -651.947 q^{73} -32.4664 q^{77} +202.844 q^{79} -978.917 q^{83} -199.562 q^{85} -438.743 q^{89} +156.731 q^{91} -107.826 q^{95} -1266.55 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 10 q^{5}+O(q^{10}) \) 3 * q + 10 * q^5	\( 3 q + 10 q^{5} + 42 q^{11} - 76 q^{13} + 78 q^{17} - 162 q^{19} + 48 q^{23} - 163 q^{25} + 88 q^{29} - 16 q^{31} - 56 q^{35} - 406 q^{37} + 123 q^{41} - 236 q^{43} - 528 q^{47} - 201 q^{49} + 92 q^{53} + 580 q^{55} - 712 q^{59} - 394 q^{61} - 280 q^{65} + 446 q^{67} - 804 q^{71} + 298 q^{73} - 300 q^{77} + 936 q^{79} - 796 q^{83} - 812 q^{85} + 314 q^{89} - 640 q^{91} - 1444 q^{95} - 302 q^{97}+O(q^{100}) \) 3 * q + 10 * q^5 + 42 * q^11 - 76 * q^13 + 78 * q^17 - 162 * q^19 + 48 * q^23 - 163 * q^25 + 88 * q^29 - 16 * q^31 - 56 * q^35 - 406 * q^37 + 123 * q^41 - 236 * q^43 - 528 * q^47 - 201 * q^49 + 92 * q^53 + 580 * q^55 - 712 * q^59 - 394 * q^61 - 280 * q^65 + 446 * q^67 - 804 * q^71 + 298 * q^73 - 300 * q^77 + 936 * q^79 - 796 * q^83 - 812 * q^85 + 314 * q^89 - 640 * q^91 - 1444 * q^95 - 302 * q^97

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