LMFDB - Embedded newform 2300.4.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,4,Mod(1,2300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2300.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,3,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.704393013$
Analytic rank:	$0$
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} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 $ x^5 - 2x^4 - 80x^3 + 121x^2 + 1212x + 1044
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.04116$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04116 q^{3} -21.6113 q^{7} -22.8337 q^{9} -30.3495 q^{11} -69.6786 q^{13} +43.4506 q^{17} +14.3892 q^{19} -44.1120 q^{21} +23.0000 q^{23} -101.718 q^{27} -158.105 q^{29} -46.4782 q^{31} -61.9480 q^{33} -39.6849 q^{37} -142.225 q^{39} -201.771 q^{41} -17.8927 q^{43} +0.702668 q^{47} +124.048 q^{49} +88.6894 q^{51} +661.161 q^{53} +29.3705 q^{57} -732.679 q^{59} -875.997 q^{61} +493.465 q^{63} -746.803 q^{67} +46.9466 q^{69} -275.246 q^{71} +499.633 q^{73} +655.892 q^{77} -797.970 q^{79} +408.887 q^{81} +1150.10 q^{83} -322.716 q^{87} +1166.89 q^{89} +1505.85 q^{91} -94.8693 q^{93} -403.209 q^{97} +692.991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9} + 7 q^{11} - 5 q^{13} + 24 q^{17} - 13 q^{19} + 115 q^{23} + 204 q^{27} - 253 q^{29} - 98 q^{31} + 473 q^{33} + 435 q^{37} - 410 q^{39} - 774 q^{41} + 498 q^{43} + 572 q^{47}+ \cdots + 111 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9 + 7 * q^11 - 5 * q^13 + 24 * q^17 - 13 * q^19 + 115 * q^23 + 204 * q^27 - 253 * q^29 - 98 * q^31 + 473 * q^33 + 435 * q^37 - 410 * q^39 - 774 * q^41 + 498 * q^43 + 572 * q^47 - 683 * q^49 - 657 * q^51 + 665 * q^53 + 932 * q^57 - 763 * q^59 + 337 * q^61 + 1527 * q^63 + 305 * q^67 + 69 * q^69 - 1504 * q^71 + 1304 * q^73 - 182 * q^77 + 626 * q^79 - 959 * q^81 + 1703 * q^83 + 1354 * q^87 + 646 * q^89 - 767 * q^91 + 452 * q^93 + 233 * q^97 + 111 * q^99

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
$3$	2.04116	0.392821	0.196410	−	0.980522i	$-0.437072\pi$
$3$	2.04116	0.392821	0.196410	+	0.980522i	$0.437072\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−21.6113	−1.16690	−0.583450	−	0.812149i	$-0.698298\pi$
$7$	−21.6113	−1.16690	−0.583450	+	0.812149i	$0.698298\pi$
$8$	0	0
$9$	−22.8337	−0.845692
$10$	0	0
$11$	−30.3495	−0.831884	−0.415942	−	0.909391i	$-0.636548\pi$
$11$	−30.3495	−0.831884	−0.415942	+	0.909391i	$0.636548\pi$
$12$	0	0
$13$	−69.6786	−1.48657	−0.743284	−	0.668976i	$-0.766733\pi$
$13$	−69.6786	−1.48657	−0.743284	+	0.668976i	$0.766733\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	43.4506	0.619901	0.309950	−	0.950753i	$-0.399688\pi$
$17$	43.4506	0.619901	0.309950	+	0.950753i	$0.399688\pi$
$18$	0	0
$19$	14.3892	0.173742	0.0868710	−	0.996220i	$-0.472313\pi$
$19$	14.3892	0.173742	0.0868710	+	0.996220i	$0.472313\pi$
$20$	0	0
$21$	−44.1120	−0.458382
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−101.718	−0.725026
$28$	0	0
$29$	−158.105	−1.01239	−0.506195	−	0.862419i	$-0.668948\pi$
$29$	−158.105	−1.01239	−0.506195	+	0.862419i	$0.668948\pi$
$30$	0	0
$31$	−46.4782	−0.269282	−0.134641	−	0.990894i	$-0.542988\pi$
$31$	−46.4782	−0.269282	−0.134641	+	0.990894i	$0.542988\pi$
$32$	0	0
$33$	−61.9480	−0.326781
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−39.6849	−0.176328	−0.0881642	−	0.996106i	$-0.528100\pi$
$37$	−39.6849	−0.176328	−0.0881642	+	0.996106i	$0.528100\pi$
$38$	0	0
$39$	−142.225	−0.583954
$40$	0	0
$41$	−201.771	−0.768571	−0.384285	−	0.923214i	$-0.625552\pi$
$41$	−201.771	−0.768571	−0.384285	+	0.923214i	$0.625552\pi$
$42$	0	0
$43$	−17.8927	−0.0634562	−0.0317281	−	0.999497i	$-0.510101\pi$
$43$	−17.8927	−0.0634562	−0.0317281	+	0.999497i	$0.510101\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.702668	0.00218074	0.00109037	−	0.999999i	$-0.499653\pi$
$47$	0.702668	0.00218074	0.00109037	+	0.999999i	$0.499653\pi$
$48$	0	0
$49$	124.048	0.361655
$50$	0	0
$51$	88.6894	0.243510
$52$	0	0
$53$	661.161	1.71354	0.856769	−	0.515701i	$-0.172468\pi$
$53$	661.161	1.71354	0.856769	+	0.515701i	$0.172468\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	29.3705	0.0682495
$58$	0	0
$59$	−732.679	−1.61672	−0.808361	−	0.588686i	$-0.799645\pi$
$59$	−732.679	−1.61672	−0.808361	+	0.588686i	$0.799645\pi$
$60$	0	0
$61$	−875.997	−1.83869	−0.919344	−	0.393454i	$-0.871280\pi$
$61$	−875.997	−1.83869	−0.919344	+	0.393454i	$0.871280\pi$
$62$	0	0
$63$	493.465	0.986838
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−746.803	−1.36174	−0.680870	−	0.732404i	$-0.738398\pi$
$67$	−746.803	−1.36174	−0.680870	+	0.732404i	$0.738398\pi$
$68$	0	0
$69$	46.9466	0.0819087
$70$	0	0
$71$	−275.246	−0.460079	−0.230040	−	0.973181i	$-0.573886\pi$
$71$	−275.246	−0.460079	−0.230040	+	0.973181i	$0.573886\pi$
$72$	0	0
$73$	499.633	0.801063	0.400532	−	0.916283i	$-0.368825\pi$
$73$	499.633	0.801063	0.400532	+	0.916283i	$0.368825\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	655.892	0.970725
$78$	0	0
$79$	−797.970	−1.13644	−0.568219	−	0.822877i	$-0.692367\pi$
$79$	−797.970	−1.13644	−0.568219	+	0.822877i	$0.692367\pi$
$80$	0	0
$81$	408.887	0.560887
$82$	0	0
$83$	1150.10	1.52097	0.760483	−	0.649358i	$-0.224962\pi$
$83$	1150.10	1.52097	0.760483	+	0.649358i	$0.224962\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−322.716	−0.397687
$88$	0	0
$89$	1166.89	1.38978	0.694890	−	0.719116i	$-0.255453\pi$
$89$	1166.89	1.38978	0.694890	+	0.719116i	$0.255453\pi$
$90$	0	0
$91$	1505.85	1.73468
$92$	0	0
$93$	−94.8693	−0.105779
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−403.209	−0.422058	−0.211029	−	0.977480i	$-0.567682\pi$
$97$	−403.209	−0.422058	−0.211029	+	0.977480i	$0.567682\pi$
$98$	0	0
$99$	692.991	0.703517
$100$	0	0
$101$	484.319	0.477144	0.238572	−	0.971125i	$-0.423321\pi$
$101$	484.319	0.477144	0.238572	+	0.971125i	$0.423321\pi$
$102$	0	0
$103$	2064.69	1.97514	0.987572	−	0.157167i	$-0.0502361\pi$
$103$	2064.69	1.97514	0.987572	+	0.157167i	$0.0502361\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1436.43	1.29780	0.648902	−	0.760872i	$-0.275228\pi$
$107$	1436.43	1.29780	0.648902	+	0.760872i	$0.275228\pi$
$108$	0	0
$109$	1870.62	1.64379	0.821895	−	0.569639i	$-0.192917\pi$
$109$	1870.62	1.64379	0.821895	+	0.569639i	$0.192917\pi$
$110$	0	0
$111$	−81.0030	−0.0692654
$112$	0	0
$113$	−1606.67	−1.33755	−0.668775	−	0.743465i	$-0.733181\pi$
$113$	−1606.67	−1.33755	−0.668775	+	0.743465i	$0.733181\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1591.02	1.25718
$118$	0	0
$119$	−939.023	−0.723362
$120$	0	0
$121$	−409.908	−0.307970
$122$	0	0
$123$	−411.847	−0.301910
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2031.66	1.41953	0.709766	−	0.704438i	$-0.248801\pi$
$127$	2031.66	1.41953	0.709766	+	0.704438i	$0.248801\pi$
$128$	0	0
$129$	−36.5219	−0.0249269
$130$	0	0
$131$	−434.691	−0.289917	−0.144958	−	0.989438i	$-0.546305\pi$
$131$	−434.691	−0.289917	−0.144958	+	0.989438i	$0.546305\pi$
$132$	0	0
$133$	−310.968	−0.202740
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−939.729	−0.586033	−0.293016	−	0.956107i	$-0.594659\pi$
$137$	−939.729	−0.586033	−0.293016	+	0.956107i	$0.594659\pi$
$138$	0	0
$139$	−124.609	−0.0760376	−0.0380188	−	0.999277i	$-0.512105\pi$
$139$	−124.609	−0.0760376	−0.0380188	+	0.999277i	$0.512105\pi$
$140$	0	0
$141$	1.43426	0.000856639 0
$142$	0	0
$143$	2114.71	1.23665
$144$	0	0
$145$	0	0
$146$	0	0
$147$	253.201	0.142066
$148$	0	0
$149$	1375.14	0.756079	0.378039	−	0.925789i	$-0.376598\pi$
$149$	1375.14	0.756079	0.378039	+	0.925789i	$0.376598\pi$
$150$	0	0
$151$	−1196.51	−0.644838	−0.322419	−	0.946597i	$-0.604496\pi$
$151$	−1196.51	−0.644838	−0.322419	+	0.946597i	$0.604496\pi$
$152$	0	0
$153$	−992.137	−0.524245
$154$	0	0
$155$	0	0
$156$	0	0
$157$	212.340	0.107940	0.0539699	−	0.998543i	$-0.482812\pi$
$157$	212.340	0.107940	0.0539699	+	0.998543i	$0.482812\pi$
$158$	0	0
$159$	1349.53	0.673113
$160$	0	0
$161$	−497.060	−0.243315
$162$	0	0
$163$	−1390.16	−0.668013	−0.334006	−	0.942571i	$-0.608401\pi$
$163$	−1390.16	−0.668013	−0.334006	+	0.942571i	$0.608401\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−362.349	−0.167901	−0.0839503	−	0.996470i	$-0.526754\pi$
$167$	−362.349	−0.167901	−0.0839503	+	0.996470i	$0.526754\pi$
$168$	0	0
$169$	2658.11	1.20988
$170$	0	0
$171$	−328.557	−0.146932
$172$	0	0
$173$	−986.596	−0.433581	−0.216790	−	0.976218i	$-0.569559\pi$
$173$	−986.596	−0.433581	−0.216790	+	0.976218i	$0.569559\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1495.51	−0.635082
$178$	0	0
$179$	3977.44	1.66083	0.830413	−	0.557149i	$-0.188105\pi$
$179$	3977.44	1.66083	0.830413	+	0.557149i	$0.188105\pi$
$180$	0	0
$181$	2512.70	1.03186	0.515932	−	0.856630i	$-0.327446\pi$
$181$	2512.70	1.03186	0.515932	+	0.856630i	$0.327446\pi$
$182$	0	0
$183$	−1788.05	−0.722275
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1318.70	−0.515685
$188$	0	0
$189$	2198.26	0.846032
$190$	0	0
$191$	1977.49	0.749142	0.374571	−	0.927198i	$-0.377790\pi$
$191$	1977.49	0.749142	0.374571	+	0.927198i	$0.377790\pi$
$192$	0	0
$193$	1241.30	0.462957	0.231478	−	0.972840i	$-0.425644\pi$
$193$	1241.30	0.462957	0.231478	+	0.972840i	$0.425644\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1969.91	−0.712436	−0.356218	−	0.934403i	$-0.615934\pi$
$197$	−1969.91	−0.712436	−0.356218	+	0.934403i	$0.615934\pi$
$198$	0	0
$199$	−4053.09	−1.44380	−0.721899	−	0.691998i	$-0.756731\pi$
$199$	−4053.09	−1.44380	−0.721899	+	0.691998i	$0.756731\pi$
$200$	0	0
$201$	−1524.34	−0.534919
$202$	0	0
$203$	3416.85	1.18136
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−525.175	−0.176339
$208$	0	0
$209$	−436.704	−0.144533
$210$	0	0
$211$	374.110	0.122061	0.0610303	−	0.998136i	$-0.480561\pi$
$211$	374.110	0.122061	0.0610303	+	0.998136i	$0.480561\pi$
$212$	0	0
$213$	−561.819	−0.180729
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1004.45	0.314225
$218$	0	0
$219$	1019.83	0.314674
$220$	0	0
$221$	−3027.58	−0.921524
$222$	0	0
$223$	−3226.90	−0.969010	−0.484505	−	0.874788i	$-0.661000\pi$
$223$	−3226.90	−0.969010	−0.484505	+	0.874788i	$0.661000\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3020.61	−0.883194	−0.441597	−	0.897213i	$-0.645588\pi$
$227$	−3020.61	−0.883194	−0.441597	+	0.897213i	$0.645588\pi$
$228$	0	0
$229$	934.761	0.269741	0.134871	−	0.990863i	$-0.456938\pi$
$229$	934.761	0.269741	0.134871	+	0.990863i	$0.456938\pi$
$230$	0	0
$231$	1338.78	0.381321
$232$	0	0
$233$	3916.88	1.10130	0.550651	−	0.834736i	$-0.314380\pi$
$233$	3916.88	1.10130	0.550651	+	0.834736i	$0.314380\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1628.78	−0.446416
$238$	0	0
$239$	3200.06	0.866086	0.433043	−	0.901373i	$-0.357440\pi$
$239$	3200.06	0.866086	0.433043	+	0.901373i	$0.357440\pi$
$240$	0	0
$241$	−3774.24	−1.00880	−0.504399	−	0.863471i	$-0.668286\pi$
$241$	−3774.24	−1.00880	−0.504399	+	0.863471i	$0.668286\pi$
$242$	0	0
$243$	3581.00	0.945354
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1002.62	−0.258279
$248$	0	0
$249$	2347.54	0.597467
$250$	0	0
$251$	3342.01	0.840420	0.420210	−	0.907427i	$-0.361956\pi$
$251$	3342.01	0.840420	0.420210	+	0.907427i	$0.361956\pi$
$252$	0	0
$253$	−698.039	−0.173460
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2899.20	0.703685	0.351842	−	0.936059i	$-0.385555\pi$
$257$	2899.20	0.703685	0.351842	+	0.936059i	$0.385555\pi$
$258$	0	0
$259$	857.641	0.205758
$260$	0	0
$261$	3610.11	0.856170
$262$	0	0
$263$	6001.83	1.40718	0.703591	−	0.710605i	$-0.251579\pi$
$263$	6001.83	1.40718	0.703591	+	0.710605i	$0.251579\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2381.81	0.545934
$268$	0	0
$269$	8308.29	1.88314	0.941571	−	0.336813i	$-0.109349\pi$
$269$	8308.29	1.88314	0.941571	+	0.336813i	$0.109349\pi$
$270$	0	0
$271$	−2395.28	−0.536911	−0.268456	−	0.963292i	$-0.586513\pi$
$271$	−2395.28	−0.536911	−0.268456	+	0.963292i	$0.586513\pi$
$272$	0	0
$273$	3073.66	0.681416
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7466.49	−1.61956	−0.809780	−	0.586733i	$-0.800414\pi$
$277$	−7466.49	−1.61956	−0.809780	+	0.586733i	$0.800414\pi$
$278$	0	0
$279$	1061.27	0.227729
$280$	0	0
$281$	−5570.87	−1.18267	−0.591335	−	0.806426i	$-0.701399\pi$
$281$	−5570.87	−1.18267	−0.591335	+	0.806426i	$0.701399\pi$
$282$	0	0
$283$	5273.48	1.10769	0.553845	−	0.832620i	$-0.313160\pi$
$283$	5273.48	1.10769	0.553845	+	0.832620i	$0.313160\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4360.54	0.896845
$288$	0	0
$289$	−3025.05	−0.615723
$290$	0	0
$291$	−823.012	−0.165793
$292$	0	0
$293$	511.514	0.101990	0.0509948	−	0.998699i	$-0.483761\pi$
$293$	511.514	0.101990	0.0509948	+	0.998699i	$0.483761\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3087.10	0.603137
$298$	0	0
$299$	−1602.61	−0.309971
$300$	0	0
$301$	386.685	0.0740471
$302$	0	0
$303$	988.570	0.187432
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4157.46	−0.772894	−0.386447	−	0.922312i	$-0.626298\pi$
$307$	−4157.46	−0.772894	−0.386447	+	0.922312i	$0.626298\pi$
$308$	0	0
$309$	4214.35	0.775877
$310$	0	0
$311$	−3408.45	−0.621465	−0.310733	−	0.950497i	$-0.600574\pi$
$311$	−3408.45	−0.621465	−0.310733	+	0.950497i	$0.600574\pi$
$312$	0	0
$313$	−4516.35	−0.815589	−0.407794	−	0.913074i	$-0.633702\pi$
$313$	−4516.35	−0.815589	−0.407794	+	0.913074i	$0.633702\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2693.62	−0.477251	−0.238626	−	0.971112i	$-0.576697\pi$
$317$	−2693.62	−0.477251	−0.238626	+	0.971112i	$0.576697\pi$
$318$	0	0
$319$	4798.40	0.842190
$320$	0	0
$321$	2931.98	0.509804
$322$	0	0
$323$	625.217	0.107703
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3818.23	0.645715
$328$	0	0
$329$	−15.1856	−0.00254470
$330$	0	0
$331$	−11024.4	−1.83068	−0.915338	−	0.402687i	$-0.868076\pi$
$331$	−11024.4	−1.83068	−0.915338	+	0.402687i	$0.868076\pi$
$332$	0	0
$333$	906.152	0.149120
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5218.80	−0.843579	−0.421790	−	0.906694i	$-0.638598\pi$
$337$	−5218.80	−0.843579	−0.421790	+	0.906694i	$0.638598\pi$
$338$	0	0
$339$	−3279.47	−0.525417
$340$	0	0
$341$	1410.59	0.224011
$342$	0	0
$343$	4731.84	0.744885
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−93.8512	−0.0145193	−0.00725965	−	0.999974i	$-0.502311\pi$
$347$	−93.8512	−0.0145193	−0.00725965	+	0.999974i	$0.502311\pi$
$348$	0	0
$349$	4540.27	0.696376	0.348188	−	0.937425i	$-0.386797\pi$
$349$	4540.27	0.696376	0.348188	+	0.937425i	$0.386797\pi$
$350$	0	0
$351$	7087.59	1.07780
$352$	0	0
$353$	−4287.12	−0.646403	−0.323202	−	0.946330i	$-0.604759\pi$
$353$	−4287.12	−0.646403	−0.323202	+	0.946330i	$0.604759\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1916.69	−0.284151
$358$	0	0
$359$	−176.418	−0.0259359	−0.0129680	−	0.999916i	$-0.504128\pi$
$359$	−176.418	−0.0259359	−0.0129680	+	0.999916i	$0.504128\pi$
$360$	0	0
$361$	−6651.95	−0.969814
$362$	0	0
$363$	−836.686	−0.120977
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7745.87	1.10172	0.550860	−	0.834598i	$-0.314300\pi$
$367$	7745.87	1.10172	0.550860	+	0.834598i	$0.314300\pi$
$368$	0	0
$369$	4607.18	0.649974
$370$	0	0
$371$	−14288.5	−1.99953
$372$	0	0
$373$	8268.02	1.14773	0.573863	−	0.818951i	$-0.305444\pi$
$373$	8268.02	1.14773	0.573863	+	0.818951i	$0.305444\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11016.5	1.50499
$378$	0	0
$379$	7229.18	0.979783	0.489892	−	0.871783i	$-0.337036\pi$
$379$	7229.18	0.979783	0.489892	+	0.871783i	$0.337036\pi$
$380$	0	0
$381$	4146.93	0.557621
$382$	0	0
$383$	519.390	0.0692940	0.0346470	−	0.999400i	$-0.488969\pi$
$383$	519.390	0.0692940	0.0346470	+	0.999400i	$0.488969\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	408.557	0.0536644
$388$	0	0
$389$	−10746.4	−1.40068	−0.700341	−	0.713809i	$-0.746969\pi$
$389$	−10746.4	−1.40068	−0.700341	+	0.713809i	$0.746969\pi$
$390$	0	0
$391$	999.363	0.129258
$392$	0	0
$393$	−887.271	−0.113885
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11738.8	1.48401	0.742004	−	0.670395i	$-0.233875\pi$
$397$	11738.8	1.48401	0.742004	+	0.670395i	$0.233875\pi$
$398$	0	0
$399$	−634.734	−0.0796403
$400$	0	0
$401$	−94.4786	−0.0117657	−0.00588284	−	0.999983i	$-0.501873\pi$
$401$	−94.4786	−0.0117657	−0.00588284	+	0.999983i	$0.501873\pi$
$402$	0	0
$403$	3238.54	0.400306
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1204.42	0.146685
$408$	0	0
$409$	−13712.5	−1.65780	−0.828901	−	0.559396i	$-0.811033\pi$
$409$	−13712.5	−1.65780	−0.828901	+	0.559396i	$0.811033\pi$
$410$	0	0
$411$	−1918.13	−0.230206
$412$	0	0
$413$	15834.1	1.88655
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−254.347	−0.0298691
$418$	0	0
$419$	4392.60	0.512154	0.256077	−	0.966656i	$-0.417570\pi$
$419$	4392.60	0.512154	0.256077	+	0.966656i	$0.417570\pi$
$420$	0	0
$421$	112.287	0.0129989	0.00649943	−	0.999979i	$-0.497931\pi$
$421$	112.287	0.0129989	0.00649943	+	0.999979i	$0.497931\pi$
$422$	0	0
$423$	−16.0445	−0.00184423
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18931.4	2.14556
$428$	0	0
$429$	4316.46	0.485782
$430$	0	0
$431$	−7035.30	−0.786261	−0.393131	−	0.919483i	$-0.628608\pi$
$431$	−7035.30	−0.786261	−0.393131	+	0.919483i	$0.628608\pi$
$432$	0	0
$433$	−12046.0	−1.33693	−0.668467	−	0.743742i	$-0.733049\pi$
$433$	−12046.0	−1.33693	−0.668467	+	0.743742i	$0.733049\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	330.951	0.0362277
$438$	0	0
$439$	−2089.02	−0.227115	−0.113557	−	0.993531i	$-0.536225\pi$
$439$	−2089.02	−0.227115	−0.113557	+	0.993531i	$0.536225\pi$
$440$	0	0
$441$	−2832.47	−0.305849
$442$	0	0
$443$	7436.42	0.797551	0.398776	−	0.917049i	$-0.369435\pi$
$443$	7436.42	0.797551	0.398776	+	0.917049i	$0.369435\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2806.87	0.297003
$448$	0	0
$449$	1796.33	0.188806	0.0944032	−	0.995534i	$-0.469906\pi$
$449$	1796.33	0.188806	0.0944032	+	0.995534i	$0.469906\pi$
$450$	0	0
$451$	6123.66	0.639361
$452$	0	0
$453$	−2442.26	−0.253306
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3232.73	0.330899	0.165450	−	0.986218i	$-0.447093\pi$
$457$	3232.73	0.330899	0.165450	+	0.986218i	$0.447093\pi$
$458$	0	0
$459$	−4419.72	−0.449444
$460$	0	0
$461$	−10750.2	−1.08609	−0.543044	−	0.839704i	$-0.682728\pi$
$461$	−10750.2	−1.08609	−0.543044	+	0.839704i	$0.682728\pi$
$462$	0	0
$463$	−6598.81	−0.662360	−0.331180	−	0.943568i	$-0.607447\pi$
$463$	−6598.81	−0.662360	−0.331180	+	0.943568i	$0.607447\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8153.38	−0.807909	−0.403954	−	0.914779i	$-0.632365\pi$
$467$	−8153.38	−0.807909	−0.403954	+	0.914779i	$0.632365\pi$
$468$	0	0
$469$	16139.4	1.58901
$470$	0	0
$471$	433.418	0.0424010
$472$	0	0
$473$	543.036	0.0527882
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15096.8	−1.44913
$478$	0	0
$479$	15056.7	1.43624	0.718120	−	0.695919i	$-0.245003\pi$
$479$	15056.7	1.43624	0.718120	+	0.695919i	$0.245003\pi$
$480$	0	0
$481$	2765.19	0.262124
$482$	0	0
$483$	−1014.58	−0.0955793
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2098.90	−0.195298	−0.0976491	−	0.995221i	$-0.531132\pi$
$487$	−2098.90	−0.195298	−0.0976491	+	0.995221i	$0.531132\pi$
$488$	0	0
$489$	−2837.54	−0.262409
$490$	0	0
$491$	−2245.71	−0.206410	−0.103205	−	0.994660i	$-0.532910\pi$
$491$	−2245.71	−0.206410	−0.103205	+	0.994660i	$0.532910\pi$
$492$	0	0
$493$	−6869.74	−0.627581
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5948.41	0.536866
$498$	0	0
$499$	3216.32	0.288542	0.144271	−	0.989538i	$-0.453916\pi$
$499$	3216.32	0.288542	0.144271	+	0.989538i	$0.453916\pi$
$500$	0	0
$501$	−739.611	−0.0659548
$502$	0	0
$503$	15459.5	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$503$	15459.5	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5425.62	0.475267
$508$	0	0
$509$	12581.8	1.09564	0.547818	−	0.836597i	$-0.315459\pi$
$509$	12581.8	1.09564	0.547818	+	0.836597i	$0.315459\pi$
$510$	0	0
$511$	−10797.7	−0.934761
$512$	0	0
$513$	−1463.64	−0.125967
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−21.3256	−0.00181412
$518$	0	0
$519$	−2013.80	−0.170320
$520$	0	0
$521$	2681.76	0.225509	0.112754	−	0.993623i	$-0.464033\pi$
$521$	2681.76	0.225509	0.112754	+	0.993623i	$0.464033\pi$
$522$	0	0
$523$	9009.09	0.753232	0.376616	−	0.926370i	$-0.377088\pi$
$523$	9009.09	0.753232	0.376616	+	0.926370i	$0.377088\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2019.50	−0.166928
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	16729.8	1.36725
$532$	0	0
$533$	14059.2	1.14253
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8118.57	0.652406
$538$	0	0
$539$	−3764.78	−0.300855
$540$	0	0
$541$	22789.6	1.81109	0.905546	−	0.424249i	$-0.139462\pi$
$541$	22789.6	1.81109	0.905546	+	0.424249i	$0.139462\pi$
$542$	0	0
$543$	5128.80	0.405337
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20709.0	−1.61874	−0.809371	−	0.587297i	$-0.800192\pi$
$547$	−20709.0	−1.61874	−0.809371	+	0.587297i	$0.800192\pi$
$548$	0	0
$549$	20002.3	1.55496
$550$	0	0
$551$	−2274.99	−0.175895
$552$	0	0
$553$	17245.2	1.32611
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−586.443	−0.0446111	−0.0223055	−	0.999751i	$-0.507101\pi$
$557$	−586.443	−0.0446111	−0.0223055	+	0.999751i	$0.507101\pi$
$558$	0	0
$559$	1246.74	0.0943320
$560$	0	0
$561$	−2691.68	−0.202572
$562$	0	0
$563$	4988.10	0.373399	0.186699	−	0.982417i	$-0.440221\pi$
$563$	4988.10	0.373399	0.186699	+	0.982417i	$0.440221\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−8836.57	−0.654499
$568$	0	0
$569$	5548.19	0.408773	0.204387	−	0.978890i	$-0.434480\pi$
$569$	5548.19	0.408773	0.204387	+	0.978890i	$0.434480\pi$
$570$	0	0
$571$	−2684.93	−0.196779	−0.0983893	−	0.995148i	$-0.531369\pi$
$571$	−2684.93	−0.196779	−0.0983893	+	0.995148i	$0.531369\pi$
$572$	0	0
$573$	4036.36	0.294278
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1462.86	−0.105546	−0.0527728	−	0.998607i	$-0.516806\pi$
$577$	−1462.86	−0.105546	−0.0527728	+	0.998607i	$0.516806\pi$
$578$	0	0
$579$	2533.68	0.181859
$580$	0	0
$581$	−24855.2	−1.77481
$582$	0	0
$583$	−20065.9	−1.42546
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11020.7	0.774912	0.387456	−	0.921888i	$-0.373354\pi$
$587$	11020.7	0.774912	0.387456	+	0.921888i	$0.373354\pi$
$588$	0	0
$589$	−668.782	−0.0467856
$590$	0	0
$591$	−4020.88	−0.279860
$592$	0	0
$593$	−7341.16	−0.508373	−0.254186	−	0.967155i	$-0.581808\pi$
$593$	−7341.16	−0.508373	−0.254186	+	0.967155i	$0.581808\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8272.98	−0.567154
$598$	0	0
$599$	−12642.5	−0.862366	−0.431183	−	0.902264i	$-0.641904\pi$
$599$	−12642.5	−0.862366	−0.431183	+	0.902264i	$0.641904\pi$
$600$	0	0
$601$	7426.60	0.504056	0.252028	−	0.967720i	$-0.418903\pi$
$601$	7426.60	0.504056	0.252028	+	0.967720i	$0.418903\pi$
$602$	0	0
$603$	17052.3	1.15161
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7506.30	0.501929	0.250965	−	0.967996i	$-0.419252\pi$
$607$	7506.30	0.501929	0.250965	+	0.967996i	$0.419252\pi$
$608$	0	0
$609$	6974.31	0.464061
$610$	0	0
$611$	−48.9610	−0.00324182
$612$	0	0
$613$	−23467.7	−1.54625	−0.773124	−	0.634254i	$-0.781307\pi$
$613$	−23467.7	−1.54625	−0.773124	+	0.634254i	$0.781307\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1436.56	−0.0937337	−0.0468669	−	0.998901i	$-0.514924\pi$
$617$	−1436.56	−0.0937337	−0.0468669	+	0.998901i	$0.514924\pi$
$618$	0	0
$619$	12664.4	0.822332	0.411166	−	0.911561i	$-0.365122\pi$
$619$	12664.4	0.822332	0.411166	+	0.911561i	$0.365122\pi$
$620$	0	0
$621$	−2339.52	−0.151178
$622$	0	0
$623$	−25218.1	−1.62173
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−891.380	−0.0567756
$628$	0	0
$629$	−1724.33	−0.109306
$630$	0	0
$631$	5383.68	0.339653	0.169826	−	0.985474i	$-0.445679\pi$
$631$	5383.68	0.339653	0.169826	+	0.985474i	$0.445679\pi$
$632$	0	0
$633$	763.616	0.0479479
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8643.47	−0.537625
$638$	0	0
$639$	6284.87	0.389085
$640$	0	0
$641$	31245.1	1.92529	0.962643	−	0.270775i	$-0.0872799\pi$
$641$	31245.1	1.92529	0.962643	+	0.270775i	$0.0872799\pi$
$642$	0	0
$643$	−22381.8	−1.37271	−0.686354	−	0.727268i	$-0.740790\pi$
$643$	−22381.8	−1.37271	−0.686354	+	0.727268i	$0.740790\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13062.8	−0.793741	−0.396870	−	0.917875i	$-0.629904\pi$
$647$	−13062.8	−0.793741	−0.396870	+	0.917875i	$0.629904\pi$
$648$	0	0
$649$	22236.4	1.34493
$650$	0	0
$651$	2050.25	0.123434
$652$	0	0
$653$	17486.1	1.04791	0.523955	−	0.851746i	$-0.324456\pi$
$653$	17486.1	1.04791	0.523955	+	0.851746i	$0.324456\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−11408.5	−0.677453
$658$	0	0
$659$	−29707.1	−1.75603	−0.878016	−	0.478631i	$-0.841133\pi$
$659$	−29707.1	−1.75603	−0.878016	+	0.478631i	$0.841133\pi$
$660$	0	0
$661$	−9905.74	−0.582888	−0.291444	−	0.956588i	$-0.594136\pi$
$661$	−9905.74	−0.582888	−0.291444	+	0.956588i	$0.594136\pi$
$662$	0	0
$663$	−6179.76	−0.361994
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3636.41	−0.211098
$668$	0	0
$669$	−6586.61	−0.380647
$670$	0	0
$671$	26586.1	1.52957
$672$	0	0
$673$	7803.48	0.446957	0.223478	−	0.974709i	$-0.428259\pi$
$673$	7803.48	0.446957	0.223478	+	0.974709i	$0.428259\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29359.1	−1.66671	−0.833354	−	0.552740i	$-0.813582\pi$
$677$	−29359.1	−1.66671	−0.833354	+	0.552740i	$0.813582\pi$
$678$	0	0
$679$	8713.86	0.492500
$680$	0	0
$681$	−6165.54	−0.346937
$682$	0	0
$683$	26055.8	1.45973	0.729866	−	0.683591i	$-0.239583\pi$
$683$	26055.8	1.45973	0.729866	+	0.683591i	$0.239583\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1907.99	0.105960
$688$	0	0
$689$	−46068.8	−2.54729
$690$	0	0
$691$	15166.4	0.834959	0.417480	−	0.908686i	$-0.362914\pi$
$691$	15166.4	0.834959	0.417480	+	0.908686i	$0.362914\pi$
$692$	0	0
$693$	−14976.4	−0.820934
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8767.08	−0.476437
$698$	0	0
$699$	7994.95	0.432614
$700$	0	0
$701$	−26527.8	−1.42930	−0.714650	−	0.699482i	$-0.753414\pi$
$701$	−26527.8	−1.42930	−0.714650	+	0.699482i	$0.753414\pi$
$702$	0	0
$703$	−571.032	−0.0306357
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10466.7	−0.556779
$708$	0	0
$709$	21268.1	1.12657	0.563287	−	0.826261i	$-0.309536\pi$
$709$	21268.1	1.12657	0.563287	+	0.826261i	$0.309536\pi$
$710$	0	0
$711$	18220.6	0.961077
$712$	0	0
$713$	−1069.00	−0.0561491
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6531.82	0.340216
$718$	0	0
$719$	−27640.4	−1.43368	−0.716838	−	0.697240i	$-0.754411\pi$
$719$	−27640.4	−1.43368	−0.716838	+	0.697240i	$0.754411\pi$
$720$	0	0
$721$	−44620.6	−2.30480
$722$	0	0
$723$	−7703.81	−0.396276
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10860.2	0.554036	0.277018	−	0.960865i	$-0.410654\pi$
$727$	10860.2	0.554036	0.277018	+	0.960865i	$0.410654\pi$
$728$	0	0
$729$	−3730.57	−0.189533
$730$	0	0
$731$	−777.450	−0.0393366
$732$	0	0
$733$	2575.74	0.129791	0.0648956	−	0.997892i	$-0.479329\pi$
$733$	2575.74	0.129791	0.0648956	+	0.997892i	$0.479329\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22665.1	1.13281
$738$	0	0
$739$	−25649.2	−1.27675	−0.638377	−	0.769724i	$-0.720394\pi$
$739$	−25649.2	−1.27675	−0.638377	+	0.769724i	$0.720394\pi$
$740$	0	0
$741$	−2046.50	−0.101457
$742$	0	0
$743$	−3767.84	−0.186041	−0.0930207	−	0.995664i	$-0.529652\pi$
$743$	−3767.84	−0.186041	−0.0930207	+	0.995664i	$0.529652\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−26261.1	−1.28627
$748$	0	0
$749$	−31043.1	−1.51441
$750$	0	0
$751$	16030.6	0.778913	0.389456	−	0.921045i	$-0.372663\pi$
$751$	16030.6	0.778913	0.389456	+	0.921045i	$0.372663\pi$
$752$	0	0
$753$	6821.55	0.330134
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3408.80	0.163666	0.0818329	−	0.996646i	$-0.473923\pi$
$757$	3408.80	0.163666	0.0818329	+	0.996646i	$0.473923\pi$
$758$	0	0
$759$	−1424.81	−0.0681385
$760$	0	0
$761$	28208.1	1.34368	0.671841	−	0.740696i	$-0.265504\pi$
$761$	28208.1	1.34368	0.671841	+	0.740696i	$0.265504\pi$
$762$	0	0
$763$	−40426.6	−1.91814
$764$	0	0
$765$	0	0
$766$	0	0
$767$	51052.1	2.40337
$768$	0	0
$769$	26349.3	1.23560	0.617802	−	0.786334i	$-0.288023\pi$
$769$	26349.3	1.23560	0.617802	+	0.786334i	$0.288023\pi$
$770$	0	0
$771$	5917.71	0.276422
$772$	0	0
$773$	−29491.9	−1.37225	−0.686126	−	0.727482i	$-0.740690\pi$
$773$	−29491.9	−1.37225	−0.686126	+	0.727482i	$0.740690\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1750.58	0.0808258
$778$	0	0
$779$	−2903.32	−0.133533
$780$	0	0
$781$	8353.56	0.382732
$782$	0	0
$783$	16082.1	0.734009
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−18485.5	−0.837276	−0.418638	−	0.908153i	$-0.637492\pi$
$787$	−18485.5	−0.837276	−0.418638	+	0.908153i	$0.637492\pi$
$788$	0	0
$789$	12250.7	0.552770
$790$	0	0
$791$	34722.3	1.56079
$792$	0	0
$793$	61038.3	2.73333
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36684.3	−1.63039	−0.815197	−	0.579184i	$-0.803371\pi$
$797$	−36684.3	−1.63039	−0.815197	+	0.579184i	$0.803371\pi$
$798$	0	0
$799$	30.5313	0.00135184
$800$	0	0
$801$	−26644.5	−1.17533
$802$	0	0
$803$	−15163.6	−0.666391
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16958.5	0.739737
$808$	0	0
$809$	216.567	0.00941173	0.00470586	−	0.999989i	$-0.498502\pi$
$809$	216.567	0.00941173	0.00470586	+	0.999989i	$0.498502\pi$
$810$	0	0
$811$	25897.5	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$811$	25897.5	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$812$	0	0
$813$	−4889.14	−0.210910
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−257.462	−0.0110250
$818$	0	0
$819$	−34384.0	−1.46700
$820$	0	0
$821$	12735.6	0.541384	0.270692	−	0.962666i	$-0.412748\pi$
$821$	12735.6	0.541384	0.270692	+	0.962666i	$0.412748\pi$
$822$	0	0
$823$	−25798.2	−1.09267	−0.546336	−	0.837566i	$-0.683978\pi$
$823$	−25798.2	−1.09267	−0.546336	+	0.837566i	$0.683978\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21295.2	−0.895413	−0.447706	−	0.894181i	$-0.647759\pi$
$827$	−21295.2	−0.895413	−0.447706	+	0.894181i	$0.647759\pi$
$828$	0	0
$829$	−43456.4	−1.82063	−0.910315	−	0.413915i	$-0.864161\pi$
$829$	−43456.4	−1.82063	−0.910315	+	0.413915i	$0.864161\pi$
$830$	0	0
$831$	−15240.3	−0.636197
$832$	0	0
$833$	5389.94	0.224190
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4727.68	0.195236
$838$	0	0
$839$	14245.5	0.586184	0.293092	−	0.956084i	$-0.405316\pi$
$839$	14245.5	0.586184	0.293092	+	0.956084i	$0.405316\pi$
$840$	0	0
$841$	608.090	0.0249330
$842$	0	0
$843$	−11371.0	−0.464577
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8858.64	0.359370
$848$	0	0
$849$	10764.0	0.435123
$850$	0	0
$851$	−912.752	−0.0367670
$852$	0	0
$853$	−1967.69	−0.0789830	−0.0394915	−	0.999220i	$-0.512574\pi$
$853$	−1967.69	−0.0789830	−0.0394915	+	0.999220i	$0.512574\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32176.3	1.28252	0.641262	−	0.767322i	$-0.278411\pi$
$857$	32176.3	1.28252	0.641262	+	0.767322i	$0.278411\pi$
$858$	0	0
$859$	−1273.26	−0.0505741	−0.0252871	−	0.999680i	$-0.508050\pi$
$859$	−1273.26	−0.0505741	−0.0252871	+	0.999680i	$0.508050\pi$
$860$	0	0
$861$	8900.54	0.352299
$862$	0	0
$863$	−69.6881	−0.00274880	−0.00137440	−	0.999999i	$-0.500437\pi$
$863$	−69.6881	−0.00274880	−0.00137440	+	0.999999i	$0.500437\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−6174.59	−0.241869
$868$	0	0
$869$	24218.0	0.945384
$870$	0	0
$871$	52036.3	2.02432
$872$	0	0
$873$	9206.75	0.356931
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20284.8	0.781038	0.390519	−	0.920595i	$-0.372296\pi$
$877$	20284.8	0.781038	0.390519	+	0.920595i	$0.372296\pi$
$878$	0	0
$879$	1044.08	0.0400636
$880$	0	0
$881$	−29637.5	−1.13339	−0.566693	−	0.823929i	$-0.691778\pi$
$881$	−29637.5	−1.13339	−0.566693	+	0.823929i	$0.691778\pi$
$882$	0	0
$883$	16749.3	0.638344	0.319172	−	0.947697i	$-0.396595\pi$
$883$	16749.3	0.638344	0.319172	+	0.947697i	$0.396595\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22904.8	0.867042	0.433521	−	0.901143i	$-0.357271\pi$
$887$	22904.8	0.867042	0.433521	+	0.901143i	$0.357271\pi$
$888$	0	0
$889$	−43906.7	−1.65645
$890$	0	0
$891$	−12409.5	−0.466593
$892$	0	0
$893$	10.1108	0.000378886 0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3271.17	−0.121763
$898$	0	0
$899$	7348.42	0.272618
$900$	0	0
$901$	28727.8	1.06222
$902$	0	0
$903$	789.285	0.0290872
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20869.7	0.764020	0.382010	−	0.924158i	$-0.375232\pi$
$907$	20869.7	0.764020	0.382010	+	0.924158i	$0.375232\pi$
$908$	0	0
$909$	−11058.8	−0.403517
$910$	0	0
$911$	−42361.0	−1.54060	−0.770298	−	0.637684i	$-0.779892\pi$
$911$	−42361.0	−1.54060	−0.770298	+	0.637684i	$0.779892\pi$
$912$	0	0
$913$	−34905.0	−1.26527
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9394.22	0.338304
$918$	0	0
$919$	14136.3	0.507413	0.253707	−	0.967281i	$-0.418350\pi$
$919$	14136.3	0.507413	0.253707	+	0.967281i	$0.418350\pi$
$920$	0	0
$921$	−8486.01	−0.303609
$922$	0	0
$923$	19178.7	0.683939
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−47144.5	−1.67036
$928$	0	0
$929$	−7897.49	−0.278911	−0.139455	−	0.990228i	$-0.544535\pi$
$929$	−7897.49	−0.278911	−0.139455	+	0.990228i	$0.544535\pi$
$930$	0	0
$931$	1784.94	0.0628347
$932$	0	0
$933$	−6957.19	−0.244124
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−21499.1	−0.749567	−0.374783	−	0.927112i	$-0.622283\pi$
$937$	−21499.1	−0.749567	−0.374783	+	0.927112i	$0.622283\pi$
$938$	0	0
$939$	−9218.58	−0.320380
$940$	0	0
$941$	30258.7	1.04825	0.524125	−	0.851641i	$-0.324392\pi$
$941$	30258.7	1.04825	0.524125	+	0.851641i	$0.324392\pi$
$942$	0	0
$943$	−4640.74	−0.160258
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12769.1	0.438163	0.219082	−	0.975707i	$-0.429694\pi$
$947$	12769.1	0.438163	0.219082	+	0.975707i	$0.429694\pi$
$948$	0	0
$949$	−34813.8	−1.19083
$950$	0	0
$951$	−5498.09	−0.187474
$952$	0	0
$953$	−28174.8	−0.957683	−0.478841	−	0.877901i	$-0.658943\pi$
$953$	−28174.8	−0.957683	−0.478841	+	0.877901i	$0.658943\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9794.28	0.330830
$958$	0	0
$959$	20308.8	0.683841
$960$	0	0
$961$	−27630.8	−0.927487
$962$	0	0
$963$	−32799.0	−1.09754
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2696.72	0.0896800	0.0448400	−	0.998994i	$-0.485722\pi$
$967$	2696.72	0.0896800	0.0448400	+	0.998994i	$0.485722\pi$
$968$	0	0
$969$	1276.17	0.0423079
$970$	0	0
$971$	−29192.7	−0.964817	−0.482408	−	0.875946i	$-0.660238\pi$
$971$	−29192.7	−0.964817	−0.482408	+	0.875946i	$0.660238\pi$
$972$	0	0
$973$	2692.97	0.0887283
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1150.88	0.0376867	0.0188433	−	0.999822i	$-0.494002\pi$
$977$	1150.88	0.0376867	0.0188433	+	0.999822i	$0.494002\pi$
$978$	0	0
$979$	−35414.6	−1.15613
$980$	0	0
$981$	−42713.2	−1.39014
$982$	0	0
$983$	9924.92	0.322030	0.161015	−	0.986952i	$-0.448523\pi$
$983$	9924.92	0.322030	0.161015	+	0.986952i	$0.448523\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−30.9961	−0.000999612 0
$988$	0	0
$989$	−411.533	−0.0132315
$990$	0	0
$991$	32446.8	1.04007	0.520034	−	0.854146i	$-0.325919\pi$
$991$	32446.8	1.04007	0.520034	+	0.854146i	$0.325919\pi$
$992$	0	0
$993$	−22502.4	−0.719127
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−49280.6	−1.56543	−0.782715	−	0.622381i	$-0.786166\pi$
$997$	−49280.6	−1.56543	−0.782715	+	0.622381i	$0.786166\pi$
$998$	0	0
$999$	4036.68	0.127843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.4.a.c.1.3		5
5.2	odd	4		2300.4.c.d.1749.5		10
5.3	odd	4		2300.4.c.d.1749.6		10
5.4	even	2		460.4.a.b.1.3	✓	5
20.19	odd	2		1840.4.a.o.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.3	✓	5	5.4	even	2
1840.4.a.o.1.3		5	20.19	odd	2
2300.4.a.c.1.3		5	1.1	even	1	trivial
2300.4.c.d.1749.5		10	5.2	odd	4
2300.4.c.d.1749.6		10	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q+2.04116 q^{3} -21.6113 q^{7} -22.8337 q^{9} -30.3495 q^{11} -69.6786 q^{13} +43.4506 q^{17} +14.3892 q^{19} -44.1120 q^{21} +23.0000 q^{23} -101.718 q^{27} -158.105 q^{29} -46.4782 q^{31} -61.9480 q^{33} -39.6849 q^{37} -142.225 q^{39} -201.771 q^{41} -17.8927 q^{43} +0.702668 q^{47} +124.048 q^{49} +88.6894 q^{51} +661.161 q^{53} +29.3705 q^{57} -732.679 q^{59} -875.997 q^{61} +493.465 q^{63} -746.803 q^{67} +46.9466 q^{69} -275.246 q^{71} +499.633 q^{73} +655.892 q^{77} -797.970 q^{79} +408.887 q^{81} +1150.10 q^{83} -322.716 q^{87} +1166.89 q^{89} +1505.85 q^{91} -94.8693 q^{93} -403.209 q^{97} +692.991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9} + 7 q^{11} - 5 q^{13} + 24 q^{17} - 13 q^{19} + 115 q^{23} + 204 q^{27} - 253 q^{29} - 98 q^{31} + 473 q^{33} + 435 q^{37} - 410 q^{39} - 774 q^{41} + 498 q^{43} + 572 q^{47}+ \cdots + 111 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9 + 7 * q^11 - 5 * q^13 + 24 * q^17 - 13 * q^19 + 115 * q^23 + 204 * q^27 - 253 * q^29 - 98 * q^31 + 473 * q^33 + 435 * q^37 - 410 * q^39 - 774 * q^41 + 498 * q^43 + 572 * q^47 - 683 * q^49 - 657 * q^51 + 665 * q^53 + 932 * q^57 - 763 * q^59 + 337 * q^61 + 1527 * q^63 + 305 * q^67 + 69 * q^69 - 1504 * q^71 + 1304 * q^73 - 182 * q^77 + 626 * q^79 - 959 * q^81 + 1703 * q^83 + 1354 * q^87 + 646 * q^89 - 767 * q^91 + 452 * q^93 + 233 * q^97 + 111 * q^99

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