LMFDB - Embedded newform 2400.4.a.bw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.23109.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 37x + 10 $ x^3 - x^2 - 37*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.268842$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +27.1562 q^{7} +9.00000 q^{9} -32.2315 q^{11} -42.0808 q^{13} -88.5547 q^{17} +36.6247 q^{19} +81.4685 q^{21} +140.646 q^{23} +27.0000 q^{27} -164.115 q^{29} -80.7863 q^{31} -96.6946 q^{33} -126.522 q^{37} -126.242 q^{39} +285.562 q^{41} +249.271 q^{43} -481.479 q^{47} +394.457 q^{49} -265.664 q^{51} -560.418 q^{53} +109.874 q^{57} +308.486 q^{59} -680.542 q^{61} +244.406 q^{63} -501.172 q^{67} +421.939 q^{69} -259.521 q^{71} -732.857 q^{73} -875.285 q^{77} +1033.49 q^{79} +81.0000 q^{81} -855.387 q^{83} -492.345 q^{87} -1128.33 q^{89} -1142.75 q^{91} -242.359 q^{93} -593.287 q^{97} -290.084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} + 32 q^{7} + 27 q^{9} - 48 q^{11} - 76 q^{13} - 16 q^{17} - 88 q^{19} + 96 q^{21} + 20 q^{23} + 81 q^{27} + 58 q^{29} + 56 q^{31} - 144 q^{33} - 436 q^{37} - 228 q^{39} + 362 q^{41} + 148 q^{43}+ \cdots - 432 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 + 32 * q^7 + 27 * q^9 - 48 * q^11 - 76 * q^13 - 16 * q^17 - 88 * q^19 + 96 * q^21 + 20 * q^23 + 81 * q^27 + 58 * q^29 + 56 * q^31 - 144 * q^33 - 436 * q^37 - 228 * q^39 + 362 * q^41 + 148 * q^43 + 52 * q^47 - 109 * q^49 - 48 * q^51 - 940 * q^53 - 264 * q^57 - 816 * q^59 - 842 * q^61 + 288 * q^63 - 780 * q^67 + 60 * q^69 - 304 * q^71 - 512 * q^73 - 1536 * q^77 + 440 * q^79 + 243 * q^81 - 1212 * q^83 + 174 * q^87 - 590 * q^89 - 944 * q^91 + 168 * q^93 - 1752 * q^97 - 432 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27.1562	1.46630	0.733148	−	0.680070i	$-0.238050\pi$
$7$	27.1562	1.46630	0.733148	+	0.680070i	$0.238050\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−32.2315	−0.883470	−0.441735	−	0.897146i	$-0.645637\pi$
$11$	−32.2315	−0.883470	−0.441735	+	0.897146i	$0.645637\pi$
$12$	0	0
$13$	−42.0808	−0.897778	−0.448889	−	0.893588i	$-0.648180\pi$
$13$	−42.0808	−0.897778	−0.448889	+	0.893588i	$0.648180\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−88.5547	−1.26339	−0.631696	−	0.775216i	$-0.717641\pi$
$17$	−88.5547	−1.26339	−0.631696	+	0.775216i	$0.717641\pi$
$18$	0	0
$19$	36.6247	0.442225	0.221112	−	0.975248i	$-0.429031\pi$
$19$	36.6247	0.442225	0.221112	+	0.975248i	$0.429031\pi$
$20$	0	0
$21$	81.4685	0.846566
$22$	0	0
$23$	140.646	1.27508	0.637539	−	0.770418i	$-0.279952\pi$
$23$	140.646	1.27508	0.637539	+	0.770418i	$0.279952\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−164.115	−1.05087	−0.525437	−	0.850832i	$-0.676098\pi$
$29$	−164.115	−1.05087	−0.525437	+	0.850832i	$0.676098\pi$
$30$	0	0
$31$	−80.7863	−0.468053	−0.234026	−	0.972230i	$-0.575190\pi$
$31$	−80.7863	−0.468053	−0.234026	+	0.972230i	$0.575190\pi$
$32$	0	0
$33$	−96.6946	−0.510072
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−126.522	−0.562165	−0.281083	−	0.959684i	$-0.590693\pi$
$37$	−126.522	−0.562165	−0.281083	+	0.959684i	$0.590693\pi$
$38$	0	0
$39$	−126.242	−0.518332
$40$	0	0
$41$	285.562	1.08774	0.543869	−	0.839170i	$-0.316959\pi$
$41$	285.562	1.08774	0.543869	+	0.839170i	$0.316959\pi$
$42$	0	0
$43$	249.271	0.884034	0.442017	−	0.897007i	$-0.354263\pi$
$43$	249.271	0.884034	0.442017	+	0.897007i	$0.354263\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−481.479	−1.49428	−0.747138	−	0.664669i	$-0.768572\pi$
$47$	−481.479	−1.49428	−0.747138	+	0.664669i	$0.768572\pi$
$48$	0	0
$49$	394.457	1.15002
$50$	0	0
$51$	−265.664	−0.729420
$52$	0	0
$53$	−560.418	−1.45244	−0.726220	−	0.687462i	$-0.758725\pi$
$53$	−560.418	−1.45244	−0.726220	+	0.687462i	$0.758725\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	109.874	0.255319
$58$	0	0
$59$	308.486	0.680703	0.340351	−	0.940298i	$-0.389454\pi$
$59$	308.486	0.680703	0.340351	+	0.940298i	$0.389454\pi$
$60$	0	0
$61$	−680.542	−1.42843	−0.714217	−	0.699924i	$-0.753217\pi$
$61$	−680.542	−1.42843	−0.714217	+	0.699924i	$0.753217\pi$
$62$	0	0
$63$	244.406	0.488765
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−501.172	−0.913849	−0.456924	−	0.889506i	$-0.651049\pi$
$67$	−501.172	−0.913849	−0.456924	+	0.889506i	$0.651049\pi$
$68$	0	0
$69$	421.939	0.736167
$70$	0	0
$71$	−259.521	−0.433796	−0.216898	−	0.976194i	$-0.569594\pi$
$71$	−259.521	−0.433796	−0.216898	+	0.976194i	$0.569594\pi$
$72$	0	0
$73$	−732.857	−1.17499	−0.587496	−	0.809227i	$-0.699886\pi$
$73$	−732.857	−1.17499	−0.587496	+	0.809227i	$0.699886\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−875.285	−1.29543
$78$	0	0
$79$	1033.49	1.47185	0.735927	−	0.677061i	$-0.236747\pi$
$79$	1033.49	1.47185	0.735927	+	0.677061i	$0.236747\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−855.387	−1.13122	−0.565608	−	0.824674i	$-0.691359\pi$
$83$	−855.387	−1.13122	−0.565608	+	0.824674i	$0.691359\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−492.345	−0.606723
$88$	0	0
$89$	−1128.33	−1.34386	−0.671928	−	0.740617i	$-0.734533\pi$
$89$	−1128.33	−1.34386	−0.671928	+	0.740617i	$0.734533\pi$
$90$	0	0
$91$	−1142.75	−1.31641
$92$	0	0
$93$	−242.359	−0.270230
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−593.287	−0.621022	−0.310511	−	0.950570i	$-0.600500\pi$
$97$	−593.287	−0.621022	−0.310511	+	0.950570i	$0.600500\pi$
$98$	0	0
$99$	−290.084	−0.294490
$100$	0	0
$101$	−149.980	−0.147758	−0.0738789	−	0.997267i	$-0.523538\pi$
$101$	−149.980	−0.147758	−0.0738789	+	0.997267i	$0.523538\pi$
$102$	0	0
$103$	1497.86	1.43290	0.716450	−	0.697638i	$-0.245766\pi$
$103$	1497.86	1.43290	0.716450	+	0.697638i	$0.245766\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−358.126	−0.323564	−0.161782	−	0.986827i	$-0.551724\pi$
$107$	−358.126	−0.323564	−0.161782	+	0.986827i	$0.551724\pi$
$108$	0	0
$109$	−483.140	−0.424555	−0.212277	−	0.977209i	$-0.568088\pi$
$109$	−483.140	−0.424555	−0.212277	+	0.977209i	$0.568088\pi$
$110$	0	0
$111$	−379.566	−0.324566
$112$	0	0
$113$	1717.42	1.42975	0.714873	−	0.699254i	$-0.246484\pi$
$113$	1717.42	1.42975	0.714873	+	0.699254i	$0.246484\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−378.727	−0.299259
$118$	0	0
$119$	−2404.81	−1.85251
$120$	0	0
$121$	−292.128	−0.219480
$122$	0	0
$123$	856.685	0.628006
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.7036	0.00817738	0.00408869	−	0.999992i	$-0.498699\pi$
$127$	11.7036	0.00817738	0.00408869	+	0.999992i	$0.498699\pi$
$128$	0	0
$129$	747.813	0.510398
$130$	0	0
$131$	−2766.92	−1.84540	−0.922699	−	0.385521i	$-0.874022\pi$
$131$	−2766.92	−1.84540	−0.922699	+	0.385521i	$0.874022\pi$
$132$	0	0
$133$	994.586	0.648432
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−352.620	−0.219900	−0.109950	−	0.993937i	$-0.535069\pi$
$137$	−352.620	−0.219900	−0.109950	+	0.993937i	$0.535069\pi$
$138$	0	0
$139$	2898.12	1.76846	0.884228	−	0.467055i	$-0.154685\pi$
$139$	2898.12	1.76846	0.884228	+	0.467055i	$0.154685\pi$
$140$	0	0
$141$	−1444.44	−0.862720
$142$	0	0
$143$	1356.33	0.793160
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1183.37	0.663965
$148$	0	0
$149$	233.290	0.128267	0.0641337	−	0.997941i	$-0.479572\pi$
$149$	233.290	0.128267	0.0641337	+	0.997941i	$0.479572\pi$
$150$	0	0
$151$	3092.49	1.66664	0.833322	−	0.552788i	$-0.186436\pi$
$151$	3092.49	1.66664	0.833322	+	0.552788i	$0.186436\pi$
$152$	0	0
$153$	−796.993	−0.421131
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2482.32	1.26185	0.630925	−	0.775843i	$-0.282675\pi$
$157$	2482.32	1.26185	0.630925	+	0.775843i	$0.282675\pi$
$158$	0	0
$159$	−1681.25	−0.838567
$160$	0	0
$161$	3819.42	1.86964
$162$	0	0
$163$	−1938.77	−0.931633	−0.465817	−	0.884881i	$-0.654239\pi$
$163$	−1938.77	−0.931633	−0.465817	+	0.884881i	$0.654239\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1449.63	0.671712	0.335856	−	0.941913i	$-0.390974\pi$
$167$	1449.63	0.671712	0.335856	+	0.941913i	$0.390974\pi$
$168$	0	0
$169$	−426.206	−0.193995
$170$	0	0
$171$	329.622	0.147408
$172$	0	0
$173$	3114.34	1.36866	0.684331	−	0.729171i	$-0.260094\pi$
$173$	3114.34	1.36866	0.684331	+	0.729171i	$0.260094\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	925.458	0.393004
$178$	0	0
$179$	−1023.88	−0.427532	−0.213766	−	0.976885i	$-0.568573\pi$
$179$	−1023.88	−0.427532	−0.213766	+	0.976885i	$0.568573\pi$
$180$	0	0
$181$	939.635	0.385870	0.192935	−	0.981212i	$-0.438199\pi$
$181$	939.635	0.385870	0.192935	+	0.981212i	$0.438199\pi$
$182$	0	0
$183$	−2041.63	−0.824707
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2854.25	1.11617
$188$	0	0
$189$	733.217	0.282189
$190$	0	0
$191$	−2864.56	−1.08519	−0.542597	−	0.839993i	$-0.682559\pi$
$191$	−2864.56	−1.08519	−0.542597	+	0.839993i	$0.682559\pi$
$192$	0	0
$193$	1287.85	0.480320	0.240160	−	0.970733i	$-0.422800\pi$
$193$	1287.85	0.480320	0.240160	+	0.970733i	$0.422800\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4357.21	−1.57583	−0.787915	−	0.615784i	$-0.788840\pi$
$197$	−4357.21	−1.57583	−0.787915	+	0.615784i	$0.788840\pi$
$198$	0	0
$199$	−3416.54	−1.21704	−0.608522	−	0.793537i	$-0.708237\pi$
$199$	−3416.54	−1.21704	−0.608522	+	0.793537i	$0.708237\pi$
$200$	0	0
$201$	−1503.51	−0.527611
$202$	0	0
$203$	−4456.73	−1.54089
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1265.82	0.425026
$208$	0	0
$209$	−1180.47	−0.390693
$210$	0	0
$211$	−4359.57	−1.42240	−0.711198	−	0.702992i	$-0.751847\pi$
$211$	−4359.57	−1.42240	−0.711198	+	0.702992i	$0.751847\pi$
$212$	0	0
$213$	−778.564	−0.250452
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2193.85	−0.686304
$218$	0	0
$219$	−2198.57	−0.678382
$220$	0	0
$221$	3726.45	1.13425
$222$	0	0
$223$	−5532.56	−1.66138	−0.830689	−	0.556737i	$-0.812053\pi$
$223$	−5532.56	−1.66138	−0.830689	+	0.556737i	$0.812053\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.4368	0.00480595	0.00240298	−	0.999997i	$-0.499235\pi$
$227$	16.4368	0.00480595	0.00240298	+	0.999997i	$0.499235\pi$
$228$	0	0
$229$	5427.86	1.56630	0.783151	−	0.621832i	$-0.213611\pi$
$229$	5427.86	1.56630	0.783151	+	0.621832i	$0.213611\pi$
$230$	0	0
$231$	−2625.85	−0.747916
$232$	0	0
$233$	−3217.35	−0.904616	−0.452308	−	0.891862i	$-0.649399\pi$
$233$	−3217.35	−0.904616	−0.452308	+	0.891862i	$0.649399\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3100.46	0.849776
$238$	0	0
$239$	−3382.68	−0.915512	−0.457756	−	0.889078i	$-0.651347\pi$
$239$	−3382.68	−0.915512	−0.457756	+	0.889078i	$0.651347\pi$
$240$	0	0
$241$	−5799.70	−1.55017	−0.775086	−	0.631856i	$-0.782293\pi$
$241$	−5799.70	−1.55017	−0.775086	+	0.631856i	$0.782293\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1541.20	−0.397020
$248$	0	0
$249$	−2566.16	−0.653108
$250$	0	0
$251$	1728.60	0.434695	0.217347	−	0.976094i	$-0.430260\pi$
$251$	1728.60	0.434695	0.217347	+	0.976094i	$0.430260\pi$
$252$	0	0
$253$	−4533.25	−1.12649
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−165.237	−0.0401059	−0.0200530	−	0.999799i	$-0.506383\pi$
$257$	−165.237	−0.0401059	−0.0200530	+	0.999799i	$0.506383\pi$
$258$	0	0
$259$	−3435.86	−0.824300
$260$	0	0
$261$	−1477.03	−0.350292
$262$	0	0
$263$	5674.62	1.33047	0.665233	−	0.746636i	$-0.268332\pi$
$263$	5674.62	1.33047	0.665233	+	0.746636i	$0.268332\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3385.00	−0.775875
$268$	0	0
$269$	−3783.08	−0.857467	−0.428734	−	0.903431i	$-0.641040\pi$
$269$	−3783.08	−0.857467	−0.428734	+	0.903431i	$0.641040\pi$
$270$	0	0
$271$	−7269.83	−1.62956	−0.814780	−	0.579770i	$-0.803142\pi$
$271$	−7269.83	−1.62956	−0.814780	+	0.579770i	$0.803142\pi$
$272$	0	0
$273$	−3428.26	−0.760028
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6738.92	1.46174	0.730871	−	0.682515i	$-0.239114\pi$
$277$	6738.92	1.46174	0.730871	+	0.682515i	$0.239114\pi$
$278$	0	0
$279$	−727.076	−0.156018
$280$	0	0
$281$	−4042.01	−0.858099	−0.429050	−	0.903281i	$-0.641151\pi$
$281$	−4042.01	−0.858099	−0.429050	+	0.903281i	$0.641151\pi$
$282$	0	0
$283$	4403.37	0.924923	0.462462	−	0.886639i	$-0.346966\pi$
$283$	4403.37	0.924923	0.462462	+	0.886639i	$0.346966\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7754.76	1.59494
$288$	0	0
$289$	2928.94	0.596161
$290$	0	0
$291$	−1779.86	−0.358547
$292$	0	0
$293$	−1089.36	−0.217206	−0.108603	−	0.994085i	$-0.534638\pi$
$293$	−1089.36	−0.217206	−0.108603	+	0.994085i	$0.534638\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−870.251	−0.170024
$298$	0	0
$299$	−5918.51	−1.14474
$300$	0	0
$301$	6769.25	1.29626
$302$	0	0
$303$	−449.939	−0.0853080
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3042.71	−0.565656	−0.282828	−	0.959171i	$-0.591273\pi$
$307$	−3042.71	−0.565656	−0.282828	+	0.959171i	$0.591273\pi$
$308$	0	0
$309$	4493.59	0.827286
$310$	0	0
$311$	−3197.34	−0.582972	−0.291486	−	0.956575i	$-0.594150\pi$
$311$	−3197.34	−0.582972	−0.291486	+	0.956575i	$0.594150\pi$
$312$	0	0
$313$	7938.44	1.43357	0.716785	−	0.697294i	$-0.245613\pi$
$313$	7938.44	1.43357	0.716785	+	0.697294i	$0.245613\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2561.12	−0.453775	−0.226887	−	0.973921i	$-0.572855\pi$
$317$	−2561.12	−0.453775	−0.226887	+	0.973921i	$0.572855\pi$
$318$	0	0
$319$	5289.67	0.928417
$320$	0	0
$321$	−1074.38	−0.186810
$322$	0	0
$323$	−3243.29	−0.558704
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1449.42	−0.245117
$328$	0	0
$329$	−13075.1	−2.19105
$330$	0	0
$331$	−3467.97	−0.575882	−0.287941	−	0.957648i	$-0.592971\pi$
$331$	−3467.97	−0.575882	−0.287941	+	0.957648i	$0.592971\pi$
$332$	0	0
$333$	−1138.70	−0.187388
$334$	0	0
$335$	0	0
$336$	0	0
$337$	727.539	0.117601	0.0588005	−	0.998270i	$-0.481272\pi$
$337$	727.539	0.117601	0.0588005	+	0.998270i	$0.481272\pi$
$338$	0	0
$339$	5152.26	0.825465
$340$	0	0
$341$	2603.87	0.413511
$342$	0	0
$343$	1397.39	0.219976
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10763.2	−1.66512	−0.832561	−	0.553933i	$-0.813126\pi$
$347$	−10763.2	−1.66512	−0.832561	+	0.553933i	$0.813126\pi$
$348$	0	0
$349$	−3664.76	−0.562092	−0.281046	−	0.959694i	$-0.590681\pi$
$349$	−3664.76	−0.562092	−0.281046	+	0.959694i	$0.590681\pi$
$350$	0	0
$351$	−1136.18	−0.172777
$352$	0	0
$353$	−1359.24	−0.204943	−0.102472	−	0.994736i	$-0.532675\pi$
$353$	−1359.24	−0.204943	−0.102472	+	0.994736i	$0.532675\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7214.42	−1.06955
$358$	0	0
$359$	7054.37	1.03709	0.518545	−	0.855050i	$-0.326474\pi$
$359$	7054.37	1.03709	0.518545	+	0.855050i	$0.326474\pi$
$360$	0	0
$361$	−5517.63	−0.804437
$362$	0	0
$363$	−876.384	−0.126717
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10592.4	1.50659	0.753297	−	0.657680i	$-0.228462\pi$
$367$	10592.4	1.50659	0.753297	+	0.657680i	$0.228462\pi$
$368$	0	0
$369$	2570.06	0.362579
$370$	0	0
$371$	−15218.8	−2.12971
$372$	0	0
$373$	−5681.19	−0.788635	−0.394318	−	0.918974i	$-0.629019\pi$
$373$	−5681.19	−0.788635	−0.394318	+	0.918974i	$0.629019\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6906.09	0.943452
$378$	0	0
$379$	−14337.1	−1.94314	−0.971568	−	0.236762i	$-0.923914\pi$
$379$	−14337.1	−1.94314	−0.971568	+	0.236762i	$0.923914\pi$
$380$	0	0
$381$	35.1108	0.00472121
$382$	0	0
$383$	7779.08	1.03784	0.518920	−	0.854823i	$-0.326334\pi$
$383$	7779.08	1.03784	0.518920	+	0.854823i	$0.326334\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2243.44	0.294678
$388$	0	0
$389$	9522.97	1.24122	0.620609	−	0.784120i	$-0.286885\pi$
$389$	9522.97	1.24122	0.620609	+	0.784120i	$0.286885\pi$
$390$	0	0
$391$	−12454.9	−1.61092
$392$	0	0
$393$	−8300.77	−1.06544
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−339.790	−0.0429561	−0.0214781	−	0.999769i	$-0.506837\pi$
$397$	−339.790	−0.0429561	−0.0214781	+	0.999769i	$0.506837\pi$
$398$	0	0
$399$	2983.76	0.374373
$400$	0	0
$401$	3401.85	0.423642	0.211821	−	0.977308i	$-0.432061\pi$
$401$	3401.85	0.423642	0.211821	+	0.977308i	$0.432061\pi$
$402$	0	0
$403$	3399.55	0.420208
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4078.00	0.496656
$408$	0	0
$409$	1463.68	0.176954	0.0884772	−	0.996078i	$-0.471800\pi$
$409$	1463.68	0.176954	0.0884772	+	0.996078i	$0.471800\pi$
$410$	0	0
$411$	−1057.86	−0.126959
$412$	0	0
$413$	8377.30	0.998111
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8694.36	1.02102
$418$	0	0
$419$	7480.14	0.872145	0.436073	−	0.899912i	$-0.356369\pi$
$419$	7480.14	0.872145	0.436073	+	0.899912i	$0.356369\pi$
$420$	0	0
$421$	3071.73	0.355599	0.177799	−	0.984067i	$-0.443102\pi$
$421$	3071.73	0.355599	0.177799	+	0.984067i	$0.443102\pi$
$422$	0	0
$423$	−4333.31	−0.498092
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−18480.9	−2.09451
$428$	0	0
$429$	4068.99	0.457931
$430$	0	0
$431$	−15707.4	−1.75545	−0.877724	−	0.479167i	$-0.840939\pi$
$431$	−15707.4	−1.75545	−0.877724	+	0.479167i	$0.840939\pi$
$432$	0	0
$433$	16201.6	1.79815	0.899075	−	0.437794i	$-0.144240\pi$
$433$	16201.6	1.79815	0.899075	+	0.437794i	$0.144240\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5151.13	0.563871
$438$	0	0
$439$	−10776.4	−1.17160	−0.585798	−	0.810457i	$-0.699219\pi$
$439$	−10776.4	−1.17160	−0.585798	+	0.810457i	$0.699219\pi$
$440$	0	0
$441$	3550.12	0.383341
$442$	0	0
$443$	5597.38	0.600315	0.300158	−	0.953890i	$-0.402961\pi$
$443$	5597.38	0.600315	0.300158	+	0.953890i	$0.402961\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	699.870	0.0740553
$448$	0	0
$449$	9698.36	1.01936	0.509682	−	0.860363i	$-0.329763\pi$
$449$	9698.36	1.01936	0.509682	+	0.860363i	$0.329763\pi$
$450$	0	0
$451$	−9204.09	−0.960984
$452$	0	0
$453$	9277.47	0.962237
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8730.79	0.893674	0.446837	−	0.894615i	$-0.352550\pi$
$457$	8730.79	0.893674	0.446837	+	0.894615i	$0.352550\pi$
$458$	0	0
$459$	−2390.98	−0.243140
$460$	0	0
$461$	−14055.8	−1.42006	−0.710028	−	0.704173i	$-0.751318\pi$
$461$	−14055.8	−1.42006	−0.710028	+	0.704173i	$0.751318\pi$
$462$	0	0
$463$	−1031.28	−0.103515	−0.0517577	−	0.998660i	$-0.516482\pi$
$463$	−1031.28	−0.103515	−0.0517577	+	0.998660i	$0.516482\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18394.5	−1.82269	−0.911346	−	0.411642i	$-0.864955\pi$
$467$	−18394.5	−1.82269	−0.911346	+	0.411642i	$0.864955\pi$
$468$	0	0
$469$	−13609.9	−1.33997
$470$	0	0
$471$	7446.96	0.728530
$472$	0	0
$473$	−8034.39	−0.781018
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5043.76	−0.484147
$478$	0	0
$479$	8341.07	0.795643	0.397822	−	0.917463i	$-0.369766\pi$
$479$	8341.07	0.795643	0.397822	+	0.917463i	$0.369766\pi$
$480$	0	0
$481$	5324.15	0.504700
$482$	0	0
$483$	11458.3	1.07944
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1440.98	0.134080	0.0670401	−	0.997750i	$-0.478644\pi$
$487$	1440.98	0.134080	0.0670401	+	0.997750i	$0.478644\pi$
$488$	0	0
$489$	−5816.31	−0.537879
$490$	0	0
$491$	865.844	0.0795824	0.0397912	−	0.999208i	$-0.487331\pi$
$491$	865.844	0.0795824	0.0397912	+	0.999208i	$0.487331\pi$
$492$	0	0
$493$	14533.1	1.32767
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7047.60	−0.636073
$498$	0	0
$499$	−274.122	−0.0245920	−0.0122960	−	0.999924i	$-0.503914\pi$
$499$	−274.122	−0.0245920	−0.0122960	+	0.999924i	$0.503914\pi$
$500$	0	0
$501$	4348.89	0.387813
$502$	0	0
$503$	6215.83	0.550994	0.275497	−	0.961302i	$-0.411158\pi$
$503$	6215.83	0.550994	0.275497	+	0.961302i	$0.411158\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1278.62	−0.112003
$508$	0	0
$509$	−1932.27	−0.168264	−0.0841319	−	0.996455i	$-0.526812\pi$
$509$	−1932.27	−0.168264	−0.0841319	+	0.996455i	$0.526812\pi$
$510$	0	0
$511$	−19901.6	−1.72289
$512$	0	0
$513$	988.866	0.0851062
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15518.8	1.32015
$518$	0	0
$519$	9343.01	0.790198
$520$	0	0
$521$	13231.7	1.11265	0.556327	−	0.830964i	$-0.312210\pi$
$521$	13231.7	1.11265	0.556327	+	0.830964i	$0.312210\pi$
$522$	0	0
$523$	−1832.84	−0.153240	−0.0766201	−	0.997060i	$-0.524413\pi$
$523$	−1832.84	−0.153240	−0.0766201	+	0.997060i	$0.524413\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7154.01	0.591335
$528$	0	0
$529$	7614.41	0.625824
$530$	0	0
$531$	2776.37	0.226901
$532$	0	0
$533$	−12016.7	−0.976547
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3071.63	−0.246835
$538$	0	0
$539$	−12714.0	−1.01601
$540$	0	0
$541$	−837.229	−0.0665347	−0.0332673	−	0.999446i	$-0.510591\pi$
$541$	−837.229	−0.0665347	−0.0332673	+	0.999446i	$0.510591\pi$
$542$	0	0
$543$	2818.90	0.222782
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2432.47	−0.190137	−0.0950685	−	0.995471i	$-0.530307\pi$
$547$	−2432.47	−0.190137	−0.0950685	+	0.995471i	$0.530307\pi$
$548$	0	0
$549$	−6124.88	−0.476145
$550$	0	0
$551$	−6010.65	−0.464723
$552$	0	0
$553$	28065.6	2.15817
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9938.31	0.756014	0.378007	−	0.925803i	$-0.376610\pi$
$557$	9938.31	0.756014	0.378007	+	0.925803i	$0.376610\pi$
$558$	0	0
$559$	−10489.5	−0.793667
$560$	0	0
$561$	8562.76	0.644421
$562$	0	0
$563$	−21469.3	−1.60715	−0.803575	−	0.595204i	$-0.797071\pi$
$563$	−21469.3	−1.60715	−0.803575	+	0.595204i	$0.797071\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2199.65	0.162922
$568$	0	0
$569$	14161.5	1.04337	0.521686	−	0.853137i	$-0.325303\pi$
$569$	14161.5	1.04337	0.521686	+	0.853137i	$0.325303\pi$
$570$	0	0
$571$	10634.3	0.779390	0.389695	−	0.920944i	$-0.372580\pi$
$571$	10634.3	0.779390	0.389695	+	0.920944i	$0.372580\pi$
$572$	0	0
$573$	−8593.67	−0.626537
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−9661.60	−0.697084	−0.348542	−	0.937293i	$-0.613323\pi$
$577$	−9661.60	−0.697084	−0.348542	+	0.937293i	$0.613323\pi$
$578$	0	0
$579$	3863.56	0.277313
$580$	0	0
$581$	−23229.0	−1.65870
$582$	0	0
$583$	18063.1	1.28319
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4614.00	0.324430	0.162215	−	0.986755i	$-0.448136\pi$
$587$	4614.00	0.324430	0.162215	+	0.986755i	$0.448136\pi$
$588$	0	0
$589$	−2958.77	−0.206985
$590$	0	0
$591$	−13071.6	−0.909806
$592$	0	0
$593$	−19062.9	−1.32010	−0.660050	−	0.751222i	$-0.729465\pi$
$593$	−19062.9	−1.32010	−0.660050	+	0.751222i	$0.729465\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10249.6	−0.702661
$598$	0	0
$599$	−15491.4	−1.05670	−0.528349	−	0.849027i	$-0.677189\pi$
$599$	−15491.4	−1.05670	−0.528349	+	0.849027i	$0.677189\pi$
$600$	0	0
$601$	−22419.5	−1.52165	−0.760825	−	0.648957i	$-0.775205\pi$
$601$	−22419.5	−1.52165	−0.760825	+	0.648957i	$0.775205\pi$
$602$	0	0
$603$	−4510.54	−0.304616
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21836.9	1.46019	0.730093	−	0.683348i	$-0.239477\pi$
$607$	21836.9	1.46019	0.730093	+	0.683348i	$0.239477\pi$
$608$	0	0
$609$	−13370.2	−0.889635
$610$	0	0
$611$	20261.0	1.34153
$612$	0	0
$613$	18540.6	1.22161	0.610805	−	0.791781i	$-0.290846\pi$
$613$	18540.6	1.22161	0.610805	+	0.791781i	$0.290846\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10562.3	−0.689178	−0.344589	−	0.938754i	$-0.611982\pi$
$617$	−10562.3	−0.689178	−0.344589	+	0.938754i	$0.611982\pi$
$618$	0	0
$619$	−2041.84	−0.132582	−0.0662911	−	0.997800i	$-0.521117\pi$
$619$	−2041.84	−0.132582	−0.0662911	+	0.997800i	$0.521117\pi$
$620$	0	0
$621$	3797.45	0.245389
$622$	0	0
$623$	−30641.2	−1.97049
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3541.41	−0.225567
$628$	0	0
$629$	11204.1	0.710235
$630$	0	0
$631$	5344.21	0.337162	0.168581	−	0.985688i	$-0.446081\pi$
$631$	5344.21	0.337162	0.168581	+	0.985688i	$0.446081\pi$
$632$	0	0
$633$	−13078.7	−0.821220
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−16599.1	−1.03246
$638$	0	0
$639$	−2335.69	−0.144599
$640$	0	0
$641$	11052.4	0.681036	0.340518	−	0.940238i	$-0.389398\pi$
$641$	11052.4	0.681036	0.340518	+	0.940238i	$0.389398\pi$
$642$	0	0
$643$	−18549.5	−1.13767	−0.568833	−	0.822453i	$-0.692605\pi$
$643$	−18549.5	−1.13767	−0.568833	+	0.822453i	$0.692605\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24532.8	1.49070	0.745349	−	0.666674i	$-0.232283\pi$
$647$	24532.8	1.49070	0.745349	+	0.666674i	$0.232283\pi$
$648$	0	0
$649$	−9942.98	−0.601381
$650$	0	0
$651$	−6581.54	−0.396238
$652$	0	0
$653$	−17293.7	−1.03638	−0.518190	−	0.855266i	$-0.673394\pi$
$653$	−17293.7	−1.03638	−0.518190	+	0.855266i	$0.673394\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6595.72	−0.391664
$658$	0	0
$659$	3937.73	0.232765	0.116383	−	0.993204i	$-0.462870\pi$
$659$	3937.73	0.232765	0.116383	+	0.993204i	$0.462870\pi$
$660$	0	0
$661$	2429.27	0.142946	0.0714732	−	0.997443i	$-0.477230\pi$
$661$	2429.27	0.142946	0.0714732	+	0.997443i	$0.477230\pi$
$662$	0	0
$663$	11179.4	0.654857
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23082.2	−1.33995
$668$	0	0
$669$	−16597.7	−0.959197
$670$	0	0
$671$	21934.9	1.26198
$672$	0	0
$673$	−15823.0	−0.906287	−0.453144	−	0.891438i	$-0.649697\pi$
$673$	−15823.0	−0.906287	−0.453144	+	0.891438i	$0.649697\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5762.14	−0.327115	−0.163558	−	0.986534i	$-0.552297\pi$
$677$	−5762.14	−0.327115	−0.163558	+	0.986534i	$0.552297\pi$
$678$	0	0
$679$	−16111.4	−0.910602
$680$	0	0
$681$	49.3105	0.00277472
$682$	0	0
$683$	−3348.28	−0.187582	−0.0937909	−	0.995592i	$-0.529899\pi$
$683$	−3348.28	−0.187582	−0.0937909	+	0.995592i	$0.529899\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16283.6	0.904305
$688$	0	0
$689$	23582.8	1.30397
$690$	0	0
$691$	−4671.24	−0.257167	−0.128584	−	0.991699i	$-0.541043\pi$
$691$	−4671.24	−0.257167	−0.128584	+	0.991699i	$0.541043\pi$
$692$	0	0
$693$	−7877.56	−0.431809
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−25287.8	−1.37424
$698$	0	0
$699$	−9652.05	−0.522280
$700$	0	0
$701$	−13479.4	−0.726262	−0.363131	−	0.931738i	$-0.618292\pi$
$701$	−13479.4	−0.726262	−0.363131	+	0.931738i	$0.618292\pi$
$702$	0	0
$703$	−4633.83	−0.248604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4072.87	−0.216657
$708$	0	0
$709$	−16781.2	−0.888903	−0.444451	−	0.895803i	$-0.646601\pi$
$709$	−16781.2	−0.888903	−0.444451	+	0.895803i	$0.646601\pi$
$710$	0	0
$711$	9301.39	0.490618
$712$	0	0
$713$	−11362.3	−0.596804
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10148.0	−0.528571
$718$	0	0
$719$	9221.15	0.478290	0.239145	−	0.970984i	$-0.423133\pi$
$719$	9221.15	0.478290	0.239145	+	0.970984i	$0.423133\pi$
$720$	0	0
$721$	40676.2	2.10106
$722$	0	0
$723$	−17399.1	−0.894992
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26693.8	1.36179	0.680893	−	0.732383i	$-0.261592\pi$
$727$	26693.8	1.36179	0.680893	+	0.732383i	$0.261592\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−22074.1	−1.11688
$732$	0	0
$733$	−7298.53	−0.367773	−0.183886	−	0.982947i	$-0.558868\pi$
$733$	−7298.53	−0.367773	−0.183886	+	0.982947i	$0.558868\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16153.5	0.807358
$738$	0	0
$739$	12789.8	0.636644	0.318322	−	0.947983i	$-0.396881\pi$
$739$	12789.8	0.636644	0.318322	+	0.947983i	$0.396881\pi$
$740$	0	0
$741$	−4623.59	−0.229220
$742$	0	0
$743$	16034.2	0.791708	0.395854	−	0.918313i	$-0.370449\pi$
$743$	16034.2	0.791708	0.395854	+	0.918313i	$0.370449\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7698.49	−0.377072
$748$	0	0
$749$	−9725.33	−0.474440
$750$	0	0
$751$	31294.8	1.52059	0.760294	−	0.649579i	$-0.225055\pi$
$751$	31294.8	1.52059	0.760294	+	0.649579i	$0.225055\pi$
$752$	0	0
$753$	5185.80	0.250971
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21880.8	1.05056	0.525279	−	0.850930i	$-0.323961\pi$
$757$	21880.8	1.05056	0.525279	+	0.850930i	$0.323961\pi$
$758$	0	0
$759$	−13599.7	−0.650381
$760$	0	0
$761$	−25262.9	−1.20339	−0.601695	−	0.798726i	$-0.705508\pi$
$761$	−25262.9	−1.20339	−0.601695	+	0.798726i	$0.705508\pi$
$762$	0	0
$763$	−13120.2	−0.622522
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12981.3	−0.611120
$768$	0	0
$769$	11559.9	0.542081	0.271040	−	0.962568i	$-0.412632\pi$
$769$	11559.9	0.542081	0.271040	+	0.962568i	$0.412632\pi$
$770$	0	0
$771$	−495.712	−0.0231552
$772$	0	0
$773$	−29021.8	−1.35038	−0.675188	−	0.737646i	$-0.735937\pi$
$773$	−29021.8	−1.35038	−0.675188	+	0.737646i	$0.735937\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10307.6	−0.475910
$778$	0	0
$779$	10458.6	0.481025
$780$	0	0
$781$	8364.77	0.383246
$782$	0	0
$783$	−4431.10	−0.202241
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36531.1	1.65463	0.827315	−	0.561739i	$-0.189867\pi$
$787$	36531.1	1.65463	0.827315	+	0.561739i	$0.189867\pi$
$788$	0	0
$789$	17023.9	0.768144
$790$	0	0
$791$	46638.6	2.09643
$792$	0	0
$793$	28637.8	1.28242
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9428.35	−0.419033	−0.209516	−	0.977805i	$-0.567189\pi$
$797$	−9428.35	−0.419033	−0.209516	+	0.977805i	$0.567189\pi$
$798$	0	0
$799$	42637.3	1.88786
$800$	0	0
$801$	−10155.0	−0.447952
$802$	0	0
$803$	23621.1	1.03807
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11349.3	−0.495059
$808$	0	0
$809$	16756.0	0.728195	0.364097	−	0.931361i	$-0.381378\pi$
$809$	16756.0	0.728195	0.364097	+	0.931361i	$0.381378\pi$
$810$	0	0
$811$	20683.4	0.895552	0.447776	−	0.894146i	$-0.352216\pi$
$811$	20683.4	0.895552	0.447776	+	0.894146i	$0.352216\pi$
$812$	0	0
$813$	−21809.5	−0.940827
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9129.47	0.390942
$818$	0	0
$819$	−10284.8	−0.438803
$820$	0	0
$821$	−26056.8	−1.10766	−0.553829	−	0.832630i	$-0.686834\pi$
$821$	−26056.8	−1.10766	−0.553829	+	0.832630i	$0.686834\pi$
$822$	0	0
$823$	3095.88	0.131125	0.0655623	−	0.997848i	$-0.479116\pi$
$823$	3095.88	0.131125	0.0655623	+	0.997848i	$0.479116\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4468.94	−0.187908	−0.0939542	−	0.995577i	$-0.529951\pi$
$827$	−4468.94	−0.187908	−0.0939542	+	0.995577i	$0.529951\pi$
$828$	0	0
$829$	4928.52	0.206483	0.103242	−	0.994656i	$-0.467079\pi$
$829$	4928.52	0.206483	0.103242	+	0.994656i	$0.467079\pi$
$830$	0	0
$831$	20216.8	0.843937
$832$	0	0
$833$	−34931.1	−1.45293
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2181.23	−0.0900768
$838$	0	0
$839$	−26858.6	−1.10520	−0.552600	−	0.833447i	$-0.686364\pi$
$839$	−26858.6	−1.10520	−0.552600	+	0.833447i	$0.686364\pi$
$840$	0	0
$841$	2544.70	0.104338
$842$	0	0
$843$	−12126.0	−0.495424
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7933.08	−0.321823
$848$	0	0
$849$	13210.1	0.534005
$850$	0	0
$851$	−17794.9	−0.716805
$852$	0	0
$853$	48489.4	1.94636	0.973181	−	0.230043i	$-0.0738866\pi$
$853$	48489.4	1.94636	0.973181	+	0.230043i	$0.0738866\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20828.8	0.830221	0.415110	−	0.909771i	$-0.363743\pi$
$857$	20828.8	0.830221	0.415110	+	0.909771i	$0.363743\pi$
$858$	0	0
$859$	20744.2	0.823963	0.411982	−	0.911192i	$-0.364837\pi$
$859$	20744.2	0.823963	0.411982	+	0.911192i	$0.364837\pi$
$860$	0	0
$861$	23264.3	0.920842
$862$	0	0
$863$	−1172.37	−0.0462431	−0.0231215	−	0.999733i	$-0.507360\pi$
$863$	−1172.37	−0.0462431	−0.0231215	+	0.999733i	$0.507360\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8786.82	0.344194
$868$	0	0
$869$	−33310.9	−1.30034
$870$	0	0
$871$	21089.7	0.820433
$872$	0	0
$873$	−5339.58	−0.207007
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14600.4	0.562166	0.281083	−	0.959683i	$-0.409306\pi$
$877$	14600.4	0.562166	0.281083	+	0.959683i	$0.409306\pi$
$878$	0	0
$879$	−3268.09	−0.125404
$880$	0	0
$881$	2582.23	0.0987486	0.0493743	−	0.998780i	$-0.484277\pi$
$881$	2582.23	0.0987486	0.0493743	+	0.998780i	$0.484277\pi$
$882$	0	0
$883$	−8079.59	−0.307927	−0.153964	−	0.988077i	$-0.549204\pi$
$883$	−8079.59	−0.307927	−0.153964	+	0.988077i	$0.549204\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12509.2	0.473527	0.236764	−	0.971567i	$-0.423913\pi$
$887$	12509.2	0.473527	0.236764	+	0.971567i	$0.423913\pi$
$888$	0	0
$889$	317.825	0.0119905
$890$	0	0
$891$	−2610.75	−0.0981634
$892$	0	0
$893$	−17634.0	−0.660806
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−17755.5	−0.660914
$898$	0	0
$899$	13258.2	0.491865
$900$	0	0
$901$	49627.7	1.83500
$902$	0	0
$903$	20307.7	0.748393
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−45428.9	−1.66311	−0.831556	−	0.555440i	$-0.812550\pi$
$907$	−45428.9	−1.66311	−0.831556	+	0.555440i	$0.812550\pi$
$908$	0	0
$909$	−1349.82	−0.0492526
$910$	0	0
$911$	−32437.4	−1.17969	−0.589847	−	0.807515i	$-0.700812\pi$
$911$	−32437.4	−1.17969	−0.589847	+	0.807515i	$0.700812\pi$
$912$	0	0
$913$	27570.4	0.999396
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−75139.0	−2.70590
$918$	0	0
$919$	31513.2	1.13115	0.565573	−	0.824698i	$-0.308655\pi$
$919$	31513.2	1.13115	0.565573	+	0.824698i	$0.308655\pi$
$920$	0	0
$921$	−9128.12	−0.326582
$922$	0	0
$923$	10920.9	0.389452
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13480.8	0.477634
$928$	0	0
$929$	−22220.7	−0.784756	−0.392378	−	0.919804i	$-0.628347\pi$
$929$	−22220.7	−0.784756	−0.392378	+	0.919804i	$0.628347\pi$
$930$	0	0
$931$	14446.9	0.508568
$932$	0	0
$933$	−9592.01	−0.336579
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25800.7	0.899542	0.449771	−	0.893144i	$-0.351506\pi$
$937$	25800.7	0.899542	0.449771	+	0.893144i	$0.351506\pi$
$938$	0	0
$939$	23815.3	0.827672
$940$	0	0
$941$	25758.2	0.892341	0.446170	−	0.894948i	$-0.352788\pi$
$941$	25758.2	0.892341	0.446170	+	0.894948i	$0.352788\pi$
$942$	0	0
$943$	40163.2	1.38695
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17272.7	−0.592701	−0.296350	−	0.955079i	$-0.595770\pi$
$947$	−17272.7	−0.592701	−0.296350	+	0.955079i	$0.595770\pi$
$948$	0	0
$949$	30839.2	1.05488
$950$	0	0
$951$	−7683.35	−0.261987
$952$	0	0
$953$	33704.2	1.14563	0.572816	−	0.819684i	$-0.305851\pi$
$953$	33704.2	1.14563	0.572816	+	0.819684i	$0.305851\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	15869.0	0.536022
$958$	0	0
$959$	−9575.80	−0.322439
$960$	0	0
$961$	−23264.6	−0.780926
$962$	0	0
$963$	−3223.13	−0.107855
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36732.9	−1.22156	−0.610780	−	0.791800i	$-0.709144\pi$
$967$	−36732.9	−1.22156	−0.610780	+	0.791800i	$0.709144\pi$
$968$	0	0
$969$	−9729.86	−0.322568
$970$	0	0
$971$	42831.6	1.41558	0.707792	−	0.706421i	$-0.249691\pi$
$971$	42831.6	1.41558	0.707792	+	0.706421i	$0.249691\pi$
$972$	0	0
$973$	78701.9	2.59308
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17406.1	−0.569981	−0.284991	−	0.958530i	$-0.591991\pi$
$977$	−17406.1	−0.569981	−0.284991	+	0.958530i	$0.591991\pi$
$978$	0	0
$979$	36367.9	1.18726
$980$	0	0
$981$	−4348.26	−0.141518
$982$	0	0
$983$	32571.0	1.05682	0.528410	−	0.848989i	$-0.322788\pi$
$983$	32571.0	1.05682	0.528410	+	0.848989i	$0.322788\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−39225.4	−1.26500
$988$	0	0
$989$	35059.1	1.12721
$990$	0	0
$991$	35747.2	1.14586	0.572930	−	0.819604i	$-0.305807\pi$
$991$	35747.2	1.14586	0.572930	+	0.819604i	$0.305807\pi$
$992$	0	0
$993$	−10403.9	−0.332486
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11858.8	0.376701	0.188351	−	0.982102i	$-0.439686\pi$
$997$	11858.8	0.376701	0.188351	+	0.982102i	$0.439686\pi$
$998$	0	0
$999$	−3416.10	−0.108189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bw.1.3		3
4.3	odd	2		2400.4.a.bf.1.1		3
5.2	odd	4		480.4.f.d.289.1	✓	6
5.3	odd	4		480.4.f.d.289.4	yes	6
5.4	even	2		2400.4.a.be.1.1		3
15.2	even	4		1440.4.f.j.289.5		6
15.8	even	4		1440.4.f.j.289.6		6
20.3	even	4		480.4.f.e.289.1	yes	6
20.7	even	4		480.4.f.e.289.4	yes	6
20.19	odd	2		2400.4.a.bx.1.3		3
40.3	even	4		960.4.f.r.769.6		6
40.13	odd	4		960.4.f.s.769.3		6
40.27	even	4		960.4.f.r.769.3		6
40.37	odd	4		960.4.f.s.769.6		6
60.23	odd	4		1440.4.f.i.289.6		6
60.47	odd	4		1440.4.f.i.289.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.d.289.1	✓	6	5.2	odd	4
480.4.f.d.289.4	yes	6	5.3	odd	4
480.4.f.e.289.1	yes	6	20.3	even	4
480.4.f.e.289.4	yes	6	20.7	even	4
960.4.f.r.769.3		6	40.27	even	4
960.4.f.r.769.6		6	40.3	even	4
960.4.f.s.769.3		6	40.13	odd	4
960.4.f.s.769.6		6	40.37	odd	4
1440.4.f.i.289.5		6	60.47	odd	4
1440.4.f.i.289.6		6	60.23	odd	4
1440.4.f.j.289.5		6	15.2	even	4
1440.4.f.j.289.6		6	15.8	even	4
2400.4.a.be.1.1		3	5.4	even	2
2400.4.a.bf.1.1		3	4.3	odd	2
2400.4.a.bw.1.3		3	1.1	even	1	trivial
2400.4.a.bx.1.3		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +27.1562 q^{7} +9.00000 q^{9} -32.2315 q^{11} -42.0808 q^{13} -88.5547 q^{17} +36.6247 q^{19} +81.4685 q^{21} +140.646 q^{23} +27.0000 q^{27} -164.115 q^{29} -80.7863 q^{31} -96.6946 q^{33} -126.522 q^{37} -126.242 q^{39} +285.562 q^{41} +249.271 q^{43} -481.479 q^{47} +394.457 q^{49} -265.664 q^{51} -560.418 q^{53} +109.874 q^{57} +308.486 q^{59} -680.542 q^{61} +244.406 q^{63} -501.172 q^{67} +421.939 q^{69} -259.521 q^{71} -732.857 q^{73} -875.285 q^{77} +1033.49 q^{79} +81.0000 q^{81} -855.387 q^{83} -492.345 q^{87} -1128.33 q^{89} -1142.75 q^{91} -242.359 q^{93} -593.287 q^{97} -290.084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} + 32 q^{7} + 27 q^{9} - 48 q^{11} - 76 q^{13} - 16 q^{17} - 88 q^{19} + 96 q^{21} + 20 q^{23} + 81 q^{27} + 58 q^{29} + 56 q^{31} - 144 q^{33} - 436 q^{37} - 228 q^{39} + 362 q^{41} + 148 q^{43}+ \cdots - 432 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 + 32 * q^7 + 27 * q^9 - 48 * q^11 - 76 * q^13 - 16 * q^17 - 88 * q^19 + 96 * q^21 + 20 * q^23 + 81 * q^27 + 58 * q^29 + 56 * q^31 - 144 * q^33 - 436 * q^37 - 228 * q^39 + 362 * q^41 + 148 * q^43 + 52 * q^47 - 109 * q^49 - 48 * q^51 - 940 * q^53 - 264 * q^57 - 816 * q^59 - 842 * q^61 + 288 * q^63 - 780 * q^67 + 60 * q^69 - 304 * q^71 - 512 * q^73 - 1536 * q^77 + 440 * q^79 + 243 * q^81 - 1212 * q^83 + 174 * q^87 - 590 * q^89 - 944 * q^91 + 168 * q^93 - 1752 * q^97 - 432 * q^99

Introduction

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