LMFDB - Embedded newform 1472.4.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, 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("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.167313.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 16x^{2} + 4x + 24 $ x^4 - x^3 - 16x^2 + 4x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.38136$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.30570 q^{3} +19.3975 q^{5} +22.5880 q^{7} +41.9846 q^{9} -42.8067 q^{11} +61.5287 q^{13} +161.110 q^{15} +73.0895 q^{17} -32.1955 q^{19} +187.609 q^{21} +23.0000 q^{23} +251.264 q^{25} +124.458 q^{27} -82.1985 q^{29} -294.493 q^{31} -355.539 q^{33} +438.151 q^{35} +97.8289 q^{37} +511.038 q^{39} -420.501 q^{41} -97.8130 q^{43} +814.398 q^{45} -78.5106 q^{47} +167.217 q^{49} +607.059 q^{51} +501.744 q^{53} -830.344 q^{55} -267.406 q^{57} -790.472 q^{59} -193.563 q^{61} +948.348 q^{63} +1193.50 q^{65} +562.027 q^{67} +191.031 q^{69} +835.854 q^{71} +548.055 q^{73} +2086.92 q^{75} -966.917 q^{77} -164.766 q^{79} -99.8765 q^{81} +811.712 q^{83} +1417.76 q^{85} -682.716 q^{87} -1066.59 q^{89} +1389.81 q^{91} -2445.97 q^{93} -624.512 q^{95} -1503.73 q^{97} -1797.22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 20 q^{5} + 10 q^{7} + 23 q^{9} - 30 q^{11} + 153 q^{13} + 136 q^{15} - 68 q^{17} - 120 q^{19} + 426 q^{21} + 92 q^{23} - 76 q^{25} + 43 q^{27} + 315 q^{29} - 249 q^{31} - 504 q^{33} + 224 q^{35}+ \cdots - 1470 q^{99}+O(q^{100}) $ 4 * q + q^3 + 20 * q^5 + 10 * q^7 + 23 * q^9 - 30 * q^11 + 153 * q^13 + 136 * q^15 - 68 * q^17 - 120 * q^19 + 426 * q^21 + 92 * q^23 - 76 * q^25 + 43 * q^27 + 315 * q^29 - 249 * q^31 - 504 * q^33 + 224 * q^35 + 348 * q^37 + 61 * q^39 - 929 * q^41 + 50 * q^43 + 1028 * q^45 - 205 * q^47 + 456 * q^49 + 890 * q^51 + 578 * q^53 - 1128 * q^55 + 366 * q^57 - 664 * q^59 + 182 * q^61 - 624 * q^63 + 848 * q^65 + 1078 * q^67 + 23 * q^69 + 739 * q^71 + 921 * q^73 + 2883 * q^75 - 528 * q^77 - 1754 * q^79 + 788 * q^81 + 1020 * q^83 + 1072 * q^85 - 2121 * q^87 - 430 * q^89 + 1334 * q^91 - 1035 * q^93 - 744 * q^95 + 48 * q^97 - 1470 * 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
$3$	8.30570	1.59843	0.799216	−	0.601044i	$-0.205248\pi$
$3$	8.30570	1.59843	0.799216	+	0.601044i	$0.205248\pi$
$4$	0	0
$5$	19.3975	1.73497	0.867484	−	0.497465i	$-0.165736\pi$
$5$	19.3975	1.73497	0.867484	+	0.497465i	$0.165736\pi$
$6$	0	0
$7$	22.5880	1.21964	0.609818	−	0.792541i	$-0.291242\pi$
$7$	22.5880	1.21964	0.609818	+	0.792541i	$0.291242\pi$
$8$	0	0
$9$	41.9846	1.55499
$10$	0	0
$11$	−42.8067	−1.17334	−0.586668	−	0.809827i	$-0.699561\pi$
$11$	−42.8067	−1.17334	−0.586668	+	0.809827i	$0.699561\pi$
$12$	0	0
$13$	61.5287	1.31269	0.656345	−	0.754461i	$-0.272101\pi$
$13$	61.5287	1.31269	0.656345	+	0.754461i	$0.272101\pi$
$14$	0	0
$15$	161.110	2.77323
$16$	0	0
$17$	73.0895	1.04275	0.521377	−	0.853327i	$-0.325419\pi$
$17$	73.0895	1.04275	0.521377	+	0.853327i	$0.325419\pi$
$18$	0	0
$19$	−32.1955	−0.388744	−0.194372	−	0.980928i	$-0.562267\pi$
$19$	−32.1955	−0.388744	−0.194372	+	0.980928i	$0.562267\pi$
$20$	0	0
$21$	187.609	1.94951
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	251.264	2.01011
$26$	0	0
$27$	124.458	0.887107
$28$	0	0
$29$	−82.1985	−0.526340	−0.263170	−	0.964749i	$-0.584768\pi$
$29$	−82.1985	−0.526340	−0.263170	+	0.964749i	$0.584768\pi$
$30$	0	0
$31$	−294.493	−1.70621	−0.853105	−	0.521739i	$-0.825283\pi$
$31$	−294.493	−1.70621	−0.853105	+	0.521739i	$0.825283\pi$
$32$	0	0
$33$	−355.539	−1.87550
$34$	0	0
$35$	438.151	2.11603
$36$	0	0
$37$	97.8289	0.434675	0.217337	−	0.976097i	$-0.430263\pi$
$37$	97.8289	0.434675	0.217337	+	0.976097i	$0.430263\pi$
$38$	0	0
$39$	511.038	2.09825
$40$	0	0
$41$	−420.501	−1.60174	−0.800868	−	0.598841i	$-0.795628\pi$
$41$	−420.501	−1.60174	−0.800868	+	0.598841i	$0.795628\pi$
$42$	0	0
$43$	−97.8130	−0.346892	−0.173446	−	0.984843i	$-0.555490\pi$
$43$	−97.8130	−0.346892	−0.173446	+	0.984843i	$0.555490\pi$
$44$	0	0
$45$	814.398	2.69785
$46$	0	0
$47$	−78.5106	−0.243659	−0.121829	−	0.992551i	$-0.538876\pi$
$47$	−78.5106	−0.243659	−0.121829	+	0.992551i	$0.538876\pi$
$48$	0	0
$49$	167.217	0.487514
$50$	0	0
$51$	607.059	1.66677
$52$	0	0
$53$	501.744	1.30037	0.650187	−	0.759774i	$-0.274690\pi$
$53$	501.744	1.30037	0.650187	+	0.759774i	$0.274690\pi$
$54$	0	0
$55$	−830.344	−2.03570
$56$	0	0
$57$	−267.406	−0.621382
$58$	0	0
$59$	−790.472	−1.74425	−0.872125	−	0.489283i	$-0.837258\pi$
$59$	−790.472	−1.74425	−0.872125	+	0.489283i	$0.837258\pi$
$60$	0	0
$61$	−193.563	−0.406283	−0.203141	−	0.979149i	$-0.565115\pi$
$61$	−193.563	−0.406283	−0.203141	+	0.979149i	$0.565115\pi$
$62$	0	0
$63$	948.348	1.89652
$64$	0	0
$65$	1193.50	2.27748
$66$	0	0
$67$	562.027	1.02481	0.512407	−	0.858743i	$-0.328754\pi$
$67$	562.027	1.02481	0.512407	+	0.858743i	$0.328754\pi$
$68$	0	0
$69$	191.031	0.333296
$70$	0	0
$71$	835.854	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$71$	835.854	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$72$	0	0
$73$	548.055	0.878699	0.439349	−	0.898316i	$-0.355209\pi$
$73$	548.055	0.878699	0.439349	+	0.898316i	$0.355209\pi$
$74$	0	0
$75$	2086.92	3.21303
$76$	0	0
$77$	−966.917	−1.43104
$78$	0	0
$79$	−164.766	−0.234654	−0.117327	−	0.993093i	$-0.537433\pi$
$79$	−164.766	−0.234654	−0.117327	+	0.993093i	$0.537433\pi$
$80$	0	0
$81$	−99.8765	−0.137005
$82$	0	0
$83$	811.712	1.07346	0.536729	−	0.843755i	$-0.319660\pi$
$83$	811.712	1.07346	0.536729	+	0.843755i	$0.319660\pi$
$84$	0	0
$85$	1417.76	1.80914
$86$	0	0
$87$	−682.716	−0.841319
$88$	0	0
$89$	−1066.59	−1.27031	−0.635157	−	0.772383i	$-0.719065\pi$
$89$	−1066.59	−1.27031	−0.635157	+	0.772383i	$0.719065\pi$
$90$	0	0
$91$	1389.81	1.60101
$92$	0	0
$93$	−2445.97	−2.72726
$94$	0	0
$95$	−624.512	−0.674459
$96$	0	0
$97$	−1503.73	−1.57403	−0.787016	−	0.616933i	$-0.788375\pi$
$97$	−1503.73	−1.57403	−0.787016	+	0.616933i	$0.788375\pi$
$98$	0	0
$99$	−1797.22	−1.82452
$100$	0	0
$101$	−1522.89	−1.50032	−0.750162	−	0.661254i	$-0.770025\pi$
$101$	−1522.89	−1.50032	−0.750162	+	0.661254i	$0.770025\pi$
$102$	0	0
$103$	−680.287	−0.650783	−0.325392	−	0.945579i	$-0.605496\pi$
$103$	−680.287	−0.650783	−0.325392	+	0.945579i	$0.605496\pi$
$104$	0	0
$105$	3639.15	3.38233
$106$	0	0
$107$	−936.924	−0.846503	−0.423252	−	0.906012i	$-0.639111\pi$
$107$	−936.924	−0.846503	−0.423252	+	0.906012i	$0.639111\pi$
$108$	0	0
$109$	428.377	0.376432	0.188216	−	0.982128i	$-0.439730\pi$
$109$	428.377	0.376432	0.188216	+	0.982128i	$0.439730\pi$
$110$	0	0
$111$	812.537	0.694798
$112$	0	0
$113$	−858.464	−0.714668	−0.357334	−	0.933977i	$-0.616314\pi$
$113$	−858.464	−0.714668	−0.357334	+	0.933977i	$0.616314\pi$
$114$	0	0
$115$	446.143	0.361766
$116$	0	0
$117$	2583.26	2.04122
$118$	0	0
$119$	1650.94	1.27178
$120$	0	0
$121$	501.412	0.376718
$122$	0	0
$123$	−3492.55	−2.56027
$124$	0	0
$125$	2449.21	1.75251
$126$	0	0
$127$	−446.125	−0.311710	−0.155855	−	0.987780i	$-0.549813\pi$
$127$	−446.125	−0.311710	−0.155855	+	0.987780i	$0.549813\pi$
$128$	0	0
$129$	−812.405	−0.554483
$130$	0	0
$131$	−1405.24	−0.937226	−0.468613	−	0.883403i	$-0.655246\pi$
$131$	−1405.24	−0.937226	−0.468613	+	0.883403i	$0.655246\pi$
$132$	0	0
$133$	−727.231	−0.474127
$134$	0	0
$135$	2414.17	1.53910
$136$	0	0
$137$	2000.41	1.24749	0.623747	−	0.781626i	$-0.285609\pi$
$137$	2000.41	1.24749	0.623747	+	0.781626i	$0.285609\pi$
$138$	0	0
$139$	1659.33	1.01254	0.506269	−	0.862375i	$-0.331024\pi$
$139$	1659.33	1.01254	0.506269	+	0.862375i	$0.331024\pi$
$140$	0	0
$141$	−652.086	−0.389472
$142$	0	0
$143$	−2633.84	−1.54023
$144$	0	0
$145$	−1594.45	−0.913184
$146$	0	0
$147$	1388.86	0.779258
$148$	0	0
$149$	−1068.09	−0.587257	−0.293628	−	0.955920i	$-0.594863\pi$
$149$	−1068.09	−0.587257	−0.293628	+	0.955920i	$0.594863\pi$
$150$	0	0
$151$	−2902.06	−1.56402	−0.782008	−	0.623269i	$-0.785804\pi$
$151$	−2902.06	−1.56402	−0.782008	+	0.623269i	$0.785804\pi$
$152$	0	0
$153$	3068.63	1.62147
$154$	0	0
$155$	−5712.44	−2.96022
$156$	0	0
$157$	3244.69	1.64939	0.824694	−	0.565578i	$-0.191347\pi$
$157$	3244.69	1.64939	0.824694	+	0.565578i	$0.191347\pi$
$158$	0	0
$159$	4167.33	2.07856
$160$	0	0
$161$	519.524	0.254312
$162$	0	0
$163$	1690.87	0.812509	0.406255	−	0.913760i	$-0.366835\pi$
$163$	1690.87	0.812509	0.406255	+	0.913760i	$0.366835\pi$
$164$	0	0
$165$	−6896.58	−3.25393
$166$	0	0
$167$	−848.474	−0.393155	−0.196578	−	0.980488i	$-0.562983\pi$
$167$	−848.474	−0.393155	−0.196578	+	0.980488i	$0.562983\pi$
$168$	0	0
$169$	1588.78	0.723157
$170$	0	0
$171$	−1351.71	−0.604492
$172$	0	0
$173$	2586.68	1.13677	0.568387	−	0.822762i	$-0.307568\pi$
$173$	2586.68	1.13677	0.568387	+	0.822762i	$0.307568\pi$
$174$	0	0
$175$	5675.55	2.45161
$176$	0	0
$177$	−6565.43	−2.78807
$178$	0	0
$179$	−577.163	−0.241001	−0.120501	−	0.992713i	$-0.538450\pi$
$179$	−577.163	−0.241001	−0.120501	+	0.992713i	$0.538450\pi$
$180$	0	0
$181$	1889.29	0.775857	0.387929	−	0.921689i	$-0.373191\pi$
$181$	1889.29	0.775857	0.387929	+	0.921689i	$0.373191\pi$
$182$	0	0
$183$	−1607.68	−0.649415
$184$	0	0
$185$	1897.64	0.754147
$186$	0	0
$187$	−3128.72	−1.22350
$188$	0	0
$189$	2811.25	1.08195
$190$	0	0
$191$	−4725.47	−1.79017	−0.895086	−	0.445893i	$-0.852886\pi$
$191$	−4725.47	−1.79017	−0.895086	+	0.445893i	$0.852886\pi$
$192$	0	0
$193$	279.011	0.104060	0.0520301	−	0.998646i	$-0.483431\pi$
$193$	279.011	0.104060	0.0520301	+	0.998646i	$0.483431\pi$
$194$	0	0
$195$	9912.88	3.64039
$196$	0	0
$197$	5223.37	1.88909	0.944543	−	0.328389i	$-0.106506\pi$
$197$	5223.37	1.88909	0.944543	+	0.328389i	$0.106506\pi$
$198$	0	0
$199$	2512.67	0.895069	0.447535	−	0.894267i	$-0.352302\pi$
$199$	2512.67	0.895069	0.447535	+	0.894267i	$0.352302\pi$
$200$	0	0
$201$	4668.03	1.63810
$202$	0	0
$203$	−1856.70	−0.641944
$204$	0	0
$205$	−8156.67	−2.77896
$206$	0	0
$207$	965.646	0.324237
$208$	0	0
$209$	1378.18	0.456128
$210$	0	0
$211$	−1071.94	−0.349740	−0.174870	−	0.984591i	$-0.555951\pi$
$211$	−1071.94	−0.349740	−0.174870	+	0.984591i	$0.555951\pi$
$212$	0	0
$213$	6942.35	2.23325
$214$	0	0
$215$	−1897.33	−0.601846
$216$	0	0
$217$	−6652.01	−2.08096
$218$	0	0
$219$	4551.98	1.40454
$220$	0	0
$221$	4497.10	1.36881
$222$	0	0
$223$	1768.84	0.531168	0.265584	−	0.964088i	$-0.414435\pi$
$223$	1768.84	0.531168	0.265584	+	0.964088i	$0.414435\pi$
$224$	0	0
$225$	10549.2	3.12570
$226$	0	0
$227$	50.1307	0.0146577	0.00732883	−	0.999973i	$-0.497667\pi$
$227$	50.1307	0.0146577	0.00732883	+	0.999973i	$0.497667\pi$
$228$	0	0
$229$	−2711.84	−0.782548	−0.391274	−	0.920274i	$-0.627966\pi$
$229$	−2711.84	−0.782548	−0.391274	+	0.920274i	$0.627966\pi$
$230$	0	0
$231$	−8030.92	−2.28743
$232$	0	0
$233$	−1058.20	−0.297533	−0.148766	−	0.988872i	$-0.547530\pi$
$233$	−1058.20	−0.297533	−0.148766	+	0.988872i	$0.547530\pi$
$234$	0	0
$235$	−1522.91	−0.422740
$236$	0	0
$237$	−1368.50	−0.375079
$238$	0	0
$239$	3234.31	0.875355	0.437678	−	0.899132i	$-0.355801\pi$
$239$	3234.31	0.875355	0.437678	+	0.899132i	$0.355801\pi$
$240$	0	0
$241$	−600.544	−0.160516	−0.0802581	−	0.996774i	$-0.525574\pi$
$241$	−600.544	−0.160516	−0.0802581	+	0.996774i	$0.525574\pi$
$242$	0	0
$243$	−4189.90	−1.10610
$244$	0	0
$245$	3243.60	0.845821
$246$	0	0
$247$	−1980.94	−0.510301
$248$	0	0
$249$	6741.84	1.71585
$250$	0	0
$251$	−1766.38	−0.444196	−0.222098	−	0.975024i	$-0.571290\pi$
$251$	−1766.38	−0.444196	−0.222098	+	0.975024i	$0.571290\pi$
$252$	0	0
$253$	−984.554	−0.244658
$254$	0	0
$255$	11775.4	2.89179
$256$	0	0
$257$	−2536.75	−0.615713	−0.307856	−	0.951433i	$-0.599612\pi$
$257$	−2536.75	−0.615713	−0.307856	+	0.951433i	$0.599612\pi$
$258$	0	0
$259$	2209.76	0.530145
$260$	0	0
$261$	−3451.07	−0.818452
$262$	0	0
$263$	−2229.51	−0.522727	−0.261363	−	0.965240i	$-0.584172\pi$
$263$	−2229.51	−0.522727	−0.261363	+	0.965240i	$0.584172\pi$
$264$	0	0
$265$	9732.60	2.25611
$266$	0	0
$267$	−8858.75	−2.03051
$268$	0	0
$269$	7158.72	1.62258	0.811292	−	0.584642i	$-0.198765\pi$
$269$	7158.72	1.62258	0.811292	+	0.584642i	$0.198765\pi$
$270$	0	0
$271$	−1145.71	−0.256816	−0.128408	−	0.991721i	$-0.540987\pi$
$271$	−1145.71	−0.256816	−0.128408	+	0.991721i	$0.540987\pi$
$272$	0	0
$273$	11543.3	2.55910
$274$	0	0
$275$	−10755.8	−2.35854
$276$	0	0
$277$	7186.12	1.55874	0.779372	−	0.626561i	$-0.215538\pi$
$277$	7186.12	1.55874	0.779372	+	0.626561i	$0.215538\pi$
$278$	0	0
$279$	−12364.2	−2.65313
$280$	0	0
$281$	−570.342	−0.121081	−0.0605405	−	0.998166i	$-0.519282\pi$
$281$	−570.342	−0.121081	−0.0605405	+	0.998166i	$0.519282\pi$
$282$	0	0
$283$	4196.38	0.881446	0.440723	−	0.897643i	$-0.354722\pi$
$283$	4196.38	0.881446	0.440723	+	0.897643i	$0.354722\pi$
$284$	0	0
$285$	−5187.01	−1.07808
$286$	0	0
$287$	−9498.27	−1.95354
$288$	0	0
$289$	429.072	0.0873341
$290$	0	0
$291$	−12489.6	−2.51598
$292$	0	0
$293$	906.601	0.180765	0.0903826	−	0.995907i	$-0.471191\pi$
$293$	906.601	0.180765	0.0903826	+	0.995907i	$0.471191\pi$
$294$	0	0
$295$	−15333.2	−3.02622
$296$	0	0
$297$	−5327.62	−1.04088
$298$	0	0
$299$	1415.16	0.273715
$300$	0	0
$301$	−2209.40	−0.423082
$302$	0	0
$303$	−12648.6	−2.39817
$304$	0	0
$305$	−3754.65	−0.704887
$306$	0	0
$307$	−61.8809	−0.0115040	−0.00575200	−	0.999983i	$-0.501831\pi$
$307$	−61.8809	−0.0115040	−0.00575200	+	0.999983i	$0.501831\pi$
$308$	0	0
$309$	−5650.26	−1.04023
$310$	0	0
$311$	2435.15	0.444003	0.222001	−	0.975046i	$-0.428741\pi$
$311$	2435.15	0.444003	0.222001	+	0.975046i	$0.428741\pi$
$312$	0	0
$313$	2983.10	0.538705	0.269352	−	0.963042i	$-0.413190\pi$
$313$	2983.10	0.538705	0.269352	+	0.963042i	$0.413190\pi$
$314$	0	0
$315$	18395.6	3.29040
$316$	0	0
$317$	−7897.30	−1.39923	−0.699616	−	0.714519i	$-0.746646\pi$
$317$	−7897.30	−1.39923	−0.699616	+	0.714519i	$0.746646\pi$
$318$	0	0
$319$	3518.64	0.617574
$320$	0	0
$321$	−7781.81	−1.35308
$322$	0	0
$323$	−2353.15	−0.405364
$324$	0	0
$325$	15459.9	2.63866
$326$	0	0
$327$	3557.97	0.601701
$328$	0	0
$329$	−1773.40	−0.297175
$330$	0	0
$331$	7166.54	1.19006	0.595028	−	0.803705i	$-0.297141\pi$
$331$	7166.54	1.19006	0.595028	+	0.803705i	$0.297141\pi$
$332$	0	0
$333$	4107.31	0.675913
$334$	0	0
$335$	10901.9	1.77802
$336$	0	0
$337$	10628.7	1.71806	0.859028	−	0.511929i	$-0.171069\pi$
$337$	10628.7	1.71806	0.859028	+	0.511929i	$0.171069\pi$
$338$	0	0
$339$	−7130.15	−1.14235
$340$	0	0
$341$	12606.3	2.00196
$342$	0	0
$343$	−3970.58	−0.625047
$344$	0	0
$345$	3705.53	0.578258
$346$	0	0
$347$	5709.84	0.883344	0.441672	−	0.897177i	$-0.354386\pi$
$347$	5709.84	0.883344	0.441672	+	0.897177i	$0.354386\pi$
$348$	0	0
$349$	−6740.22	−1.03380	−0.516899	−	0.856046i	$-0.672914\pi$
$349$	−6740.22	−1.03380	−0.516899	+	0.856046i	$0.672914\pi$
$350$	0	0
$351$	7657.72	1.16450
$352$	0	0
$353$	117.762	0.0177559	0.00887795	−	0.999961i	$-0.497174\pi$
$353$	117.762	0.0177559	0.00887795	+	0.999961i	$0.497174\pi$
$354$	0	0
$355$	16213.5	2.42401
$356$	0	0
$357$	13712.2	2.03285
$358$	0	0
$359$	7386.91	1.08598	0.542989	−	0.839740i	$-0.317292\pi$
$359$	7386.91	1.08598	0.542989	+	0.839740i	$0.317292\pi$
$360$	0	0
$361$	−5822.45	−0.848878
$362$	0	0
$363$	4164.57	0.602158
$364$	0	0
$365$	10630.9	1.52451
$366$	0	0
$367$	−5281.05	−0.751141	−0.375570	−	0.926794i	$-0.622553\pi$
$367$	−5281.05	−0.751141	−0.375570	+	0.926794i	$0.622553\pi$
$368$	0	0
$369$	−17654.6	−2.49068
$370$	0	0
$371$	11333.4	1.58598
$372$	0	0
$373$	7997.34	1.11015	0.555076	−	0.831800i	$-0.312689\pi$
$373$	7997.34	1.11015	0.555076	+	0.831800i	$0.312689\pi$
$374$	0	0
$375$	20342.4	2.80128
$376$	0	0
$377$	−5057.56	−0.690922
$378$	0	0
$379$	6863.39	0.930208	0.465104	−	0.885256i	$-0.346017\pi$
$379$	6863.39	0.930208	0.465104	+	0.885256i	$0.346017\pi$
$380$	0	0
$381$	−3705.38	−0.498247
$382$	0	0
$383$	12526.2	1.67117	0.835584	−	0.549362i	$-0.185129\pi$
$383$	12526.2	1.67117	0.835584	+	0.549362i	$0.185129\pi$
$384$	0	0
$385$	−18755.8	−2.48282
$386$	0	0
$387$	−4106.64	−0.539412
$388$	0	0
$389$	8304.89	1.08245	0.541227	−	0.840876i	$-0.317960\pi$
$389$	8304.89	1.08245	0.541227	+	0.840876i	$0.317960\pi$
$390$	0	0
$391$	1681.06	0.217429
$392$	0	0
$393$	−11671.5	−1.49809
$394$	0	0
$395$	−3196.06	−0.407117
$396$	0	0
$397$	−3118.81	−0.394279	−0.197140	−	0.980375i	$-0.563165\pi$
$397$	−3118.81	−0.394279	−0.197140	+	0.980375i	$0.563165\pi$
$398$	0	0
$399$	−6040.16	−0.757860
$400$	0	0
$401$	6961.17	0.866893	0.433447	−	0.901179i	$-0.357297\pi$
$401$	6961.17	0.866893	0.433447	+	0.901179i	$0.357297\pi$
$402$	0	0
$403$	−18119.8	−2.23973
$404$	0	0
$405$	−1937.36	−0.237699
$406$	0	0
$407$	−4187.73	−0.510020
$408$	0	0
$409$	−11447.5	−1.38397	−0.691983	−	0.721913i	$-0.743263\pi$
$409$	−11447.5	−1.38397	−0.691983	+	0.721913i	$0.743263\pi$
$410$	0	0
$411$	16614.8	1.99404
$412$	0	0
$413$	−17855.2	−2.12735
$414$	0	0
$415$	15745.2	1.86241
$416$	0	0
$417$	13781.9	1.61847
$418$	0	0
$419$	−75.2583	−0.00877472	−0.00438736	−	0.999990i	$-0.501397\pi$
$419$	−75.2583	−0.00877472	−0.00438736	+	0.999990i	$0.501397\pi$
$420$	0	0
$421$	−88.6778	−0.0102658	−0.00513289	−	0.999987i	$-0.501634\pi$
$421$	−88.6778	−0.0102658	−0.00513289	+	0.999987i	$0.501634\pi$
$422$	0	0
$423$	−3296.24	−0.378886
$424$	0	0
$425$	18364.8	2.09605
$426$	0	0
$427$	−4372.21	−0.495517
$428$	0	0
$429$	−21875.9	−2.46195
$430$	0	0
$431$	−4231.39	−0.472898	−0.236449	−	0.971644i	$-0.575984\pi$
$431$	−4231.39	−0.472898	−0.236449	+	0.971644i	$0.575984\pi$
$432$	0	0
$433$	6903.87	0.766233	0.383116	−	0.923700i	$-0.374851\pi$
$433$	6903.87	0.766233	0.383116	+	0.923700i	$0.374851\pi$
$434$	0	0
$435$	−13243.0	−1.45966
$436$	0	0
$437$	−740.495	−0.0810588
$438$	0	0
$439$	5134.77	0.558244	0.279122	−	0.960256i	$-0.409957\pi$
$439$	5134.77	0.558244	0.279122	+	0.960256i	$0.409957\pi$
$440$	0	0
$441$	7020.56	0.758078
$442$	0	0
$443$	−3454.62	−0.370505	−0.185253	−	0.982691i	$-0.559310\pi$
$443$	−3454.62	−0.370505	−0.185253	+	0.982691i	$0.559310\pi$
$444$	0	0
$445$	−20689.2	−2.20396
$446$	0	0
$447$	−8871.22	−0.938690
$448$	0	0
$449$	11358.9	1.19389	0.596947	−	0.802281i	$-0.296380\pi$
$449$	11358.9	1.19389	0.596947	+	0.802281i	$0.296380\pi$
$450$	0	0
$451$	18000.2	1.87937
$452$	0	0
$453$	−24103.6	−2.49997
$454$	0	0
$455$	26958.9	2.77769
$456$	0	0
$457$	−1736.42	−0.177738	−0.0888689	−	0.996043i	$-0.528325\pi$
$457$	−1736.42	−0.177738	−0.0888689	+	0.996043i	$0.528325\pi$
$458$	0	0
$459$	9096.55	0.925034
$460$	0	0
$461$	−4748.97	−0.479787	−0.239893	−	0.970799i	$-0.577113\pi$
$461$	−4748.97	−0.479787	−0.239893	+	0.970799i	$0.577113\pi$
$462$	0	0
$463$	−4160.49	−0.417612	−0.208806	−	0.977957i	$-0.566958\pi$
$463$	−4160.49	−0.417612	−0.208806	+	0.977957i	$0.566958\pi$
$464$	0	0
$465$	−47445.8	−4.73171
$466$	0	0
$467$	5468.27	0.541844	0.270922	−	0.962601i	$-0.412671\pi$
$467$	5468.27	0.541844	0.270922	+	0.962601i	$0.412671\pi$
$468$	0	0
$469$	12695.1	1.24990
$470$	0	0
$471$	26949.4	2.63644
$472$	0	0
$473$	4187.05	0.407021
$474$	0	0
$475$	−8089.56	−0.781420
$476$	0	0
$477$	21065.5	2.02206
$478$	0	0
$479$	3514.63	0.335256	0.167628	−	0.985850i	$-0.446389\pi$
$479$	3514.63	0.335256	0.167628	+	0.985850i	$0.446389\pi$
$480$	0	0
$481$	6019.28	0.570594
$482$	0	0
$483$	4315.01	0.406500
$484$	0	0
$485$	−29168.7	−2.73089
$486$	0	0
$487$	1494.52	0.139062	0.0695309	−	0.997580i	$-0.477850\pi$
$487$	1494.52	0.139062	0.0695309	+	0.997580i	$0.477850\pi$
$488$	0	0
$489$	14043.8	1.29874
$490$	0	0
$491$	−3658.63	−0.336276	−0.168138	−	0.985763i	$-0.553775\pi$
$491$	−3658.63	−0.336276	−0.168138	+	0.985763i	$0.553775\pi$
$492$	0	0
$493$	−6007.84	−0.548843
$494$	0	0
$495$	−34861.7	−3.16549
$496$	0	0
$497$	18880.3	1.70402
$498$	0	0
$499$	−10572.4	−0.948469	−0.474235	−	0.880398i	$-0.657275\pi$
$499$	−10572.4	−0.948469	−0.474235	+	0.880398i	$0.657275\pi$
$500$	0	0
$501$	−7047.17	−0.628432
$502$	0	0
$503$	−17973.1	−1.59320	−0.796602	−	0.604504i	$-0.793372\pi$
$503$	−17973.1	−1.59320	−0.796602	+	0.604504i	$0.793372\pi$
$504$	0	0
$505$	−29540.2	−2.60301
$506$	0	0
$507$	13195.9	1.15592
$508$	0	0
$509$	1687.64	0.146961	0.0734807	−	0.997297i	$-0.476589\pi$
$509$	1687.64	0.146961	0.0734807	+	0.997297i	$0.476589\pi$
$510$	0	0
$511$	12379.5	1.07169
$512$	0	0
$513$	−4006.97	−0.344858
$514$	0	0
$515$	−13195.9	−1.12909
$516$	0	0
$517$	3360.78	0.285893
$518$	0	0
$519$	21484.2	1.81706
$520$	0	0
$521$	9466.62	0.796046	0.398023	−	0.917375i	$-0.369696\pi$
$521$	9466.62	0.796046	0.398023	+	0.917375i	$0.369696\pi$
$522$	0	0
$523$	8894.18	0.743624	0.371812	−	0.928308i	$-0.378737\pi$
$523$	8894.18	0.743624	0.371812	+	0.928308i	$0.378737\pi$
$524$	0	0
$525$	47139.4	3.91873
$526$	0	0
$527$	−21524.3	−1.77916
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−33187.7	−2.71228
$532$	0	0
$533$	−25872.8	−2.10258
$534$	0	0
$535$	−18174.0	−1.46866
$536$	0	0
$537$	−4793.74	−0.385224
$538$	0	0
$539$	−7158.02	−0.572018
$540$	0	0
$541$	−10683.4	−0.849013	−0.424507	−	0.905425i	$-0.639552\pi$
$541$	−10683.4	−0.849013	−0.424507	+	0.905425i	$0.639552\pi$
$542$	0	0
$543$	15691.9	1.24016
$544$	0	0
$545$	8309.45	0.653097
$546$	0	0
$547$	−1992.36	−0.155735	−0.0778675	−	0.996964i	$-0.524811\pi$
$547$	−1992.36	−0.155735	−0.0778675	+	0.996964i	$0.524811\pi$
$548$	0	0
$549$	−8126.68	−0.631764
$550$	0	0
$551$	2646.42	0.204612
$552$	0	0
$553$	−3721.74	−0.286193
$554$	0	0
$555$	15761.2	1.20545
$556$	0	0
$557$	−12302.2	−0.935834	−0.467917	−	0.883772i	$-0.654995\pi$
$557$	−12302.2	−0.935834	−0.467917	+	0.883772i	$0.654995\pi$
$558$	0	0
$559$	−6018.30	−0.455362
$560$	0	0
$561$	−25986.2	−1.95568
$562$	0	0
$563$	2734.61	0.204707	0.102354	−	0.994748i	$-0.467363\pi$
$563$	2734.61	0.204707	0.102354	+	0.994748i	$0.467363\pi$
$564$	0	0
$565$	−16652.1	−1.23993
$566$	0	0
$567$	−2256.01	−0.167096
$568$	0	0
$569$	10236.0	0.754160	0.377080	−	0.926181i	$-0.376928\pi$
$569$	10236.0	0.754160	0.377080	+	0.926181i	$0.376928\pi$
$570$	0	0
$571$	−14018.6	−1.02743	−0.513714	−	0.857961i	$-0.671731\pi$
$571$	−14018.6	−1.02743	−0.513714	+	0.857961i	$0.671731\pi$
$572$	0	0
$573$	−39248.3	−2.86147
$574$	0	0
$575$	5779.08	0.419138
$576$	0	0
$577$	−17229.0	−1.24307	−0.621536	−	0.783385i	$-0.713491\pi$
$577$	−17229.0	−1.24307	−0.621536	+	0.783385i	$0.713491\pi$
$578$	0	0
$579$	2317.38	0.166333
$580$	0	0
$581$	18334.9	1.30923
$582$	0	0
$583$	−21478.0	−1.52578
$584$	0	0
$585$	50108.8	3.54144
$586$	0	0
$587$	1110.47	0.0780821	0.0390411	−	0.999238i	$-0.487570\pi$
$587$	1110.47	0.0780821	0.0390411	+	0.999238i	$0.487570\pi$
$588$	0	0
$589$	9481.34	0.663280
$590$	0	0
$591$	43383.7	3.01957
$592$	0	0
$593$	9073.40	0.628331	0.314165	−	0.949368i	$-0.398275\pi$
$593$	9073.40	0.628331	0.314165	+	0.949368i	$0.398275\pi$
$594$	0	0
$595$	32024.2	2.20650
$596$	0	0
$597$	20869.5	1.43071
$598$	0	0
$599$	7518.48	0.512849	0.256425	−	0.966564i	$-0.417455\pi$
$599$	7518.48	0.512849	0.256425	+	0.966564i	$0.417455\pi$
$600$	0	0
$601$	6950.70	0.471755	0.235878	−	0.971783i	$-0.424204\pi$
$601$	6950.70	0.471755	0.235878	+	0.971783i	$0.424204\pi$
$602$	0	0
$603$	23596.5	1.59357
$604$	0	0
$605$	9726.15	0.653593
$606$	0	0
$607$	−1400.06	−0.0936188	−0.0468094	−	0.998904i	$-0.514905\pi$
$607$	−1400.06	−0.0936188	−0.0468094	+	0.998904i	$0.514905\pi$
$608$	0	0
$609$	−15421.2	−1.02610
$610$	0	0
$611$	−4830.65	−0.319848
$612$	0	0
$613$	16420.8	1.08194	0.540970	−	0.841042i	$-0.318057\pi$
$613$	16420.8	1.08194	0.540970	+	0.841042i	$0.318057\pi$
$614$	0	0
$615$	−67746.9	−4.44198
$616$	0	0
$617$	−26651.4	−1.73897	−0.869486	−	0.493958i	$-0.835550\pi$
$617$	−26651.4	−1.73897	−0.869486	+	0.493958i	$0.835550\pi$
$618$	0	0
$619$	254.708	0.0165389	0.00826944	−	0.999966i	$-0.497368\pi$
$619$	254.708	0.0165389	0.00826944	+	0.999966i	$0.497368\pi$
$620$	0	0
$621$	2862.53	0.184975
$622$	0	0
$623$	−24092.1	−1.54932
$624$	0	0
$625$	16100.7	1.03044
$626$	0	0
$627$	11446.7	0.729090
$628$	0	0
$629$	7150.26	0.453259
$630$	0	0
$631$	4973.61	0.313782	0.156891	−	0.987616i	$-0.449853\pi$
$631$	4973.61	0.313782	0.156891	+	0.987616i	$0.449853\pi$
$632$	0	0
$633$	−8903.19	−0.559036
$634$	0	0
$635$	−8653.72	−0.540807
$636$	0	0
$637$	10288.7	0.639955
$638$	0	0
$639$	35093.0	2.17255
$640$	0	0
$641$	−22952.9	−1.41433	−0.707165	−	0.707049i	$-0.750026\pi$
$641$	−22952.9	−1.41433	−0.707165	+	0.707049i	$0.750026\pi$
$642$	0	0
$643$	−24788.7	−1.52033	−0.760164	−	0.649732i	$-0.774881\pi$
$643$	−24788.7	−1.52033	−0.760164	+	0.649732i	$0.774881\pi$
$644$	0	0
$645$	−15758.7	−0.962010
$646$	0	0
$647$	−3325.08	−0.202044	−0.101022	−	0.994884i	$-0.532211\pi$
$647$	−3325.08	−0.202044	−0.101022	+	0.994884i	$0.532211\pi$
$648$	0	0
$649$	33837.5	2.04659
$650$	0	0
$651$	−55249.6	−3.32627
$652$	0	0
$653$	17899.1	1.07266	0.536329	−	0.844009i	$-0.319810\pi$
$653$	17899.1	1.07266	0.536329	+	0.844009i	$0.319810\pi$
$654$	0	0
$655$	−27258.2	−1.62606
$656$	0	0
$657$	23009.9	1.36636
$658$	0	0
$659$	3774.81	0.223134	0.111567	−	0.993757i	$-0.464413\pi$
$659$	3774.81	0.223134	0.111567	+	0.993757i	$0.464413\pi$
$660$	0	0
$661$	−10385.3	−0.611109	−0.305555	−	0.952175i	$-0.598842\pi$
$661$	−10385.3	−0.611109	−0.305555	+	0.952175i	$0.598842\pi$
$662$	0	0
$663$	37351.5	2.18795
$664$	0	0
$665$	−14106.5	−0.822595
$666$	0	0
$667$	−1890.56	−0.109750
$668$	0	0
$669$	14691.5	0.849037
$670$	0	0
$671$	8285.80	0.476706
$672$	0	0
$673$	7805.48	0.447071	0.223536	−	0.974696i	$-0.428240\pi$
$673$	7805.48	0.447071	0.223536	+	0.974696i	$0.428240\pi$
$674$	0	0
$675$	31271.8	1.78319
$676$	0	0
$677$	12114.4	0.687730	0.343865	−	0.939019i	$-0.388264\pi$
$677$	12114.4	0.687730	0.343865	+	0.939019i	$0.388264\pi$
$678$	0	0
$679$	−33966.3	−1.91975
$680$	0	0
$681$	416.370	0.0234293
$682$	0	0
$683$	33165.2	1.85802	0.929012	−	0.370049i	$-0.120659\pi$
$683$	33165.2	1.85802	0.929012	+	0.370049i	$0.120659\pi$
$684$	0	0
$685$	38803.1	2.16436
$686$	0	0
$687$	−22523.7	−1.25085
$688$	0	0
$689$	30871.6	1.70699
$690$	0	0
$691$	−25510.7	−1.40445	−0.702223	−	0.711958i	$-0.747809\pi$
$691$	−25510.7	−1.40445	−0.702223	+	0.711958i	$0.747809\pi$
$692$	0	0
$693$	−40595.6	−2.22525
$694$	0	0
$695$	32187.0	1.75672
$696$	0	0
$697$	−30734.2	−1.67022
$698$	0	0
$699$	−8789.11	−0.475586
$700$	0	0
$701$	−18006.0	−0.970152	−0.485076	−	0.874472i	$-0.661208\pi$
$701$	−18006.0	−0.970152	−0.485076	+	0.874472i	$0.661208\pi$
$702$	0	0
$703$	−3149.64	−0.168977
$704$	0	0
$705$	−12648.8	−0.675721
$706$	0	0
$707$	−34398.9	−1.82985
$708$	0	0
$709$	−11732.8	−0.621490	−0.310745	−	0.950493i	$-0.600578\pi$
$709$	−11732.8	−0.621490	−0.310745	+	0.950493i	$0.600578\pi$
$710$	0	0
$711$	−6917.66	−0.364884
$712$	0	0
$713$	−6773.34	−0.355769
$714$	0	0
$715$	−51089.9	−2.67225
$716$	0	0
$717$	26863.2	1.39920
$718$	0	0
$719$	−4592.86	−0.238226	−0.119113	−	0.992881i	$-0.538005\pi$
$719$	−4592.86	−0.238226	−0.119113	+	0.992881i	$0.538005\pi$
$720$	0	0
$721$	−15366.3	−0.793719
$722$	0	0
$723$	−4987.94	−0.256574
$724$	0	0
$725$	−20653.5	−1.05800
$726$	0	0
$727$	12779.3	0.651935	0.325967	−	0.945381i	$-0.394310\pi$
$727$	12779.3	0.651935	0.325967	+	0.945381i	$0.394310\pi$
$728$	0	0
$729$	−32103.4	−1.63102
$730$	0	0
$731$	−7149.10	−0.361723
$732$	0	0
$733$	−12434.2	−0.626559	−0.313279	−	0.949661i	$-0.601428\pi$
$733$	−12434.2	−0.626559	−0.313279	+	0.949661i	$0.601428\pi$
$734$	0	0
$735$	26940.4	1.35199
$736$	0	0
$737$	−24058.5	−1.20245
$738$	0	0
$739$	−31481.1	−1.56705	−0.783526	−	0.621359i	$-0.786581\pi$
$739$	−31481.1	−1.56705	−0.783526	+	0.621359i	$0.786581\pi$
$740$	0	0
$741$	−16453.1	−0.815682
$742$	0	0
$743$	−31833.2	−1.57180	−0.785900	−	0.618353i	$-0.787800\pi$
$743$	−31833.2	−1.57180	−0.785900	+	0.618353i	$0.787800\pi$
$744$	0	0
$745$	−20718.3	−1.01887
$746$	0	0
$747$	34079.4	1.66921
$748$	0	0
$749$	−21163.2	−1.03243
$750$	0	0
$751$	33233.8	1.61480	0.807402	−	0.590001i	$-0.200873\pi$
$751$	33233.8	1.61480	0.807402	+	0.590001i	$0.200873\pi$
$752$	0	0
$753$	−14671.1	−0.710017
$754$	0	0
$755$	−56292.8	−2.71352
$756$	0	0
$757$	2721.56	0.130669	0.0653347	−	0.997863i	$-0.479189\pi$
$757$	2721.56	0.130669	0.0653347	+	0.997863i	$0.479189\pi$
$758$	0	0
$759$	−8177.40	−0.391068
$760$	0	0
$761$	−24190.1	−1.15229	−0.576143	−	0.817349i	$-0.695443\pi$
$761$	−24190.1	−1.15229	−0.576143	+	0.817349i	$0.695443\pi$
$762$	0	0
$763$	9676.17	0.459110
$764$	0	0
$765$	59523.9	2.81319
$766$	0	0
$767$	−48636.7	−2.28966
$768$	0	0
$769$	−10310.2	−0.483478	−0.241739	−	0.970341i	$-0.577718\pi$
$769$	−10310.2	−0.483478	−0.241739	+	0.970341i	$0.577718\pi$
$770$	0	0
$771$	−21069.5	−0.984175
$772$	0	0
$773$	−1202.02	−0.0559298	−0.0279649	−	0.999609i	$-0.508903\pi$
$773$	−1202.02	−0.0559298	−0.0279649	+	0.999609i	$0.508903\pi$
$774$	0	0
$775$	−73995.5	−3.42968
$776$	0	0
$777$	18353.6	0.847402
$778$	0	0
$779$	13538.2	0.622666
$780$	0	0
$781$	−35780.1	−1.63933
$782$	0	0
$783$	−10230.2	−0.466920
$784$	0	0
$785$	62938.9	2.86164
$786$	0	0
$787$	12086.9	0.547460	0.273730	−	0.961807i	$-0.411743\pi$
$787$	12086.9	0.547460	0.273730	+	0.961807i	$0.411743\pi$
$788$	0	0
$789$	−18517.6	−0.835544
$790$	0	0
$791$	−19391.0	−0.871636
$792$	0	0
$793$	−11909.7	−0.533323
$794$	0	0
$795$	80836.0	3.60624
$796$	0	0
$797$	−25106.8	−1.11584	−0.557922	−	0.829893i	$-0.688401\pi$
$797$	−25106.8	−1.11584	−0.557922	+	0.829893i	$0.688401\pi$
$798$	0	0
$799$	−5738.30	−0.254076
$800$	0	0
$801$	−44780.3	−1.97532
$802$	0	0
$803$	−23460.4	−1.03101
$804$	0	0
$805$	10077.5	0.441223
$806$	0	0
$807$	59458.2	2.59359
$808$	0	0
$809$	−19944.8	−0.866774	−0.433387	−	0.901208i	$-0.642682\pi$
$809$	−19944.8	−0.866774	−0.433387	+	0.901208i	$0.642682\pi$
$810$	0	0
$811$	4996.88	0.216355	0.108178	−	0.994132i	$-0.465498\pi$
$811$	4996.88	0.216355	0.108178	+	0.994132i	$0.465498\pi$
$812$	0	0
$813$	−9515.94	−0.410503
$814$	0	0
$815$	32798.7	1.40968
$816$	0	0
$817$	3149.13	0.134852
$818$	0	0
$819$	58350.6	2.48954
$820$	0	0
$821$	−933.972	−0.0397026	−0.0198513	−	0.999803i	$-0.506319\pi$
$821$	−933.972	−0.0397026	−0.0198513	+	0.999803i	$0.506319\pi$
$822$	0	0
$823$	−40805.7	−1.72831	−0.864153	−	0.503229i	$-0.832145\pi$
$823$	−40805.7	−1.72831	−0.864153	+	0.503229i	$0.832145\pi$
$824$	0	0
$825$	−89334.3	−3.76996
$826$	0	0
$827$	38919.3	1.63647	0.818234	−	0.574886i	$-0.194954\pi$
$827$	38919.3	1.63647	0.818234	+	0.574886i	$0.194954\pi$
$828$	0	0
$829$	41460.2	1.73700	0.868499	−	0.495691i	$-0.165085\pi$
$829$	41460.2	1.73700	0.868499	+	0.495691i	$0.165085\pi$
$830$	0	0
$831$	59685.8	2.49155
$832$	0	0
$833$	12221.8	0.508357
$834$	0	0
$835$	−16458.3	−0.682112
$836$	0	0
$837$	−36651.9	−1.51359
$838$	0	0
$839$	−2301.38	−0.0946990	−0.0473495	−	0.998878i	$-0.515077\pi$
$839$	−2301.38	−0.0946990	−0.0473495	+	0.998878i	$0.515077\pi$
$840$	0	0
$841$	−17632.4	−0.722966
$842$	0	0
$843$	−4737.09	−0.193540
$844$	0	0
$845$	30818.3	1.25465
$846$	0	0
$847$	11325.9	0.459459
$848$	0	0
$849$	34853.9	1.40893
$850$	0	0
$851$	2250.06	0.0906360
$852$	0	0
$853$	15728.1	0.631326	0.315663	−	0.948871i	$-0.397773\pi$
$853$	15728.1	0.631326	0.315663	+	0.948871i	$0.397773\pi$
$854$	0	0
$855$	−26219.9	−1.04877
$856$	0	0
$857$	16543.0	0.659389	0.329695	−	0.944088i	$-0.393054\pi$
$857$	16543.0	0.659389	0.329695	+	0.944088i	$0.393054\pi$
$858$	0	0
$859$	6134.46	0.243661	0.121831	−	0.992551i	$-0.461124\pi$
$859$	6134.46	0.243661	0.121831	+	0.992551i	$0.461124\pi$
$860$	0	0
$861$	−78889.7	−3.12260
$862$	0	0
$863$	−7175.54	−0.283034	−0.141517	−	0.989936i	$-0.545198\pi$
$863$	−7175.54	−0.283034	−0.141517	+	0.989936i	$0.545198\pi$
$864$	0	0
$865$	50175.2	1.97227
$866$	0	0
$867$	3563.75	0.139598
$868$	0	0
$869$	7053.11	0.275328
$870$	0	0
$871$	34580.8	1.34526
$872$	0	0
$873$	−63133.7	−2.44760
$874$	0	0
$875$	55322.8	2.13743
$876$	0	0
$877$	−8685.50	−0.334423	−0.167211	−	0.985921i	$-0.553476\pi$
$877$	−8685.50	−0.334423	−0.167211	+	0.985921i	$0.553476\pi$
$878$	0	0
$879$	7529.95	0.288941
$880$	0	0
$881$	−45118.5	−1.72540	−0.862702	−	0.505713i	$-0.831230\pi$
$881$	−45118.5	−1.72540	−0.862702	+	0.505713i	$0.831230\pi$
$882$	0	0
$883$	27466.4	1.04679	0.523396	−	0.852090i	$-0.324665\pi$
$883$	27466.4	1.04679	0.523396	+	0.852090i	$0.324665\pi$
$884$	0	0
$885$	−127353.	−4.83720
$886$	0	0
$887$	17664.0	0.668656	0.334328	−	0.942457i	$-0.391491\pi$
$887$	17664.0	0.668656	0.334328	+	0.942457i	$0.391491\pi$
$888$	0	0
$889$	−10077.1	−0.380173
$890$	0	0
$891$	4275.38	0.160753
$892$	0	0
$893$	2527.69	0.0947209
$894$	0	0
$895$	−11195.5	−0.418129
$896$	0	0
$897$	11753.9	0.437515
$898$	0	0
$899$	24206.9	0.898047
$900$	0	0
$901$	36672.2	1.35597
$902$	0	0
$903$	−18350.6	−0.676268
$904$	0	0
$905$	36647.6	1.34609
$906$	0	0
$907$	−5918.67	−0.216677	−0.108339	−	0.994114i	$-0.534553\pi$
$907$	−5918.67	−0.216677	−0.108339	+	0.994114i	$0.534553\pi$
$908$	0	0
$909$	−63937.8	−2.33298
$910$	0	0
$911$	−21288.3	−0.774217	−0.387108	−	0.922034i	$-0.626526\pi$
$911$	−21288.3	−0.774217	−0.387108	+	0.922034i	$0.626526\pi$
$912$	0	0
$913$	−34746.7	−1.25953
$914$	0	0
$915$	−31185.0	−1.12671
$916$	0	0
$917$	−31741.6	−1.14308
$918$	0	0
$919$	41806.3	1.50061	0.750307	−	0.661090i	$-0.229906\pi$
$919$	41806.3	1.50061	0.750307	+	0.661090i	$0.229906\pi$
$920$	0	0
$921$	−513.964	−0.0183884
$922$	0	0
$923$	51429.0	1.83403
$924$	0	0
$925$	24580.9	0.873746
$926$	0	0
$927$	−28561.6	−1.01196
$928$	0	0
$929$	13657.9	0.482348	0.241174	−	0.970482i	$-0.422468\pi$
$929$	13657.9	0.482348	0.241174	+	0.970482i	$0.422468\pi$
$930$	0	0
$931$	−5383.64	−0.189518
$932$	0	0
$933$	20225.6	0.709708
$934$	0	0
$935$	−60689.4	−2.12273
$936$	0	0
$937$	9655.47	0.336639	0.168319	−	0.985733i	$-0.446166\pi$
$937$	9655.47	0.336639	0.168319	+	0.985733i	$0.446166\pi$
$938$	0	0
$939$	24776.7	0.861083
$940$	0	0
$941$	17402.7	0.602883	0.301442	−	0.953485i	$-0.402532\pi$
$941$	17402.7	0.602883	0.301442	+	0.953485i	$0.402532\pi$
$942$	0	0
$943$	−9671.52	−0.333985
$944$	0	0
$945$	54531.3	1.87715
$946$	0	0
$947$	−26078.5	−0.894864	−0.447432	−	0.894318i	$-0.647661\pi$
$947$	−26078.5	−0.894864	−0.447432	+	0.894318i	$0.647661\pi$
$948$	0	0
$949$	33721.1	1.15346
$950$	0	0
$951$	−65592.6	−2.23658
$952$	0	0
$953$	22070.2	0.750182	0.375091	−	0.926988i	$-0.377611\pi$
$953$	22070.2	0.750182	0.375091	+	0.926988i	$0.377611\pi$
$954$	0	0
$955$	−91662.4	−3.10589
$956$	0	0
$957$	29224.8	0.987151
$958$	0	0
$959$	45185.3	1.52149
$960$	0	0
$961$	56935.1	1.91115
$962$	0	0
$963$	−39336.4	−1.31630
$964$	0	0
$965$	5412.12	0.180541
$966$	0	0
$967$	46938.7	1.56096	0.780480	−	0.625181i	$-0.214975\pi$
$967$	46938.7	1.56096	0.780480	+	0.625181i	$0.214975\pi$
$968$	0	0
$969$	−19544.5	−0.647948
$970$	0	0
$971$	−44882.8	−1.48337	−0.741687	−	0.670746i	$-0.765974\pi$
$971$	−44882.8	−1.48337	−0.741687	+	0.670746i	$0.765974\pi$
$972$	0	0
$973$	37481.0	1.23493
$974$	0	0
$975$	128406.	4.21772
$976$	0	0
$977$	−42525.3	−1.39253	−0.696266	−	0.717784i	$-0.745157\pi$
$977$	−42525.3	−1.39253	−0.696266	+	0.717784i	$0.745157\pi$
$978$	0	0
$979$	45657.0	1.49051
$980$	0	0
$981$	17985.2	0.585346
$982$	0	0
$983$	−147.145	−0.00477437	−0.00238719	−	0.999997i	$-0.500760\pi$
$983$	−147.145	−0.00477437	−0.00238719	+	0.999997i	$0.500760\pi$
$984$	0	0
$985$	101321.	3.27750
$986$	0	0
$987$	−14729.3	−0.475014
$988$	0	0
$989$	−2249.70	−0.0723319
$990$	0	0
$991$	−21908.2	−0.702258	−0.351129	−	0.936327i	$-0.614202\pi$
$991$	−21908.2	−0.702258	−0.351129	+	0.936327i	$0.614202\pi$
$992$	0	0
$993$	59523.1	1.90222
$994$	0	0
$995$	48739.7	1.55292
$996$	0	0
$997$	41766.1	1.32673	0.663363	−	0.748298i	$-0.269129\pi$
$997$	41766.1	1.32673	0.663363	+	0.748298i	$0.269129\pi$
$998$	0	0
$999$	12175.6	0.385603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bd.1.4		4
4.3	odd	2		1472.4.a.ba.1.1		4
8.3	odd	2		184.4.a.e.1.4	✓	4
8.5	even	2		368.4.a.n.1.1		4
24.11	even	2		1656.4.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.4.a.e.1.4	✓	4	8.3	odd	2
368.4.a.n.1.1		4	8.5	even	2
1472.4.a.ba.1.1		4	4.3	odd	2
1472.4.a.bd.1.4		4	1.1	even	1	trivial
1656.4.a.n.1.4		4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+8.30570 q^{3} +19.3975 q^{5} +22.5880 q^{7} +41.9846 q^{9} -42.8067 q^{11} +61.5287 q^{13} +161.110 q^{15} +73.0895 q^{17} -32.1955 q^{19} +187.609 q^{21} +23.0000 q^{23} +251.264 q^{25} +124.458 q^{27} -82.1985 q^{29} -294.493 q^{31} -355.539 q^{33} +438.151 q^{35} +97.8289 q^{37} +511.038 q^{39} -420.501 q^{41} -97.8130 q^{43} +814.398 q^{45} -78.5106 q^{47} +167.217 q^{49} +607.059 q^{51} +501.744 q^{53} -830.344 q^{55} -267.406 q^{57} -790.472 q^{59} -193.563 q^{61} +948.348 q^{63} +1193.50 q^{65} +562.027 q^{67} +191.031 q^{69} +835.854 q^{71} +548.055 q^{73} +2086.92 q^{75} -966.917 q^{77} -164.766 q^{79} -99.8765 q^{81} +811.712 q^{83} +1417.76 q^{85} -682.716 q^{87} -1066.59 q^{89} +1389.81 q^{91} -2445.97 q^{93} -624.512 q^{95} -1503.73 q^{97} -1797.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 20 q^{5} + 10 q^{7} + 23 q^{9} - 30 q^{11} + 153 q^{13} + 136 q^{15} - 68 q^{17} - 120 q^{19} + 426 q^{21} + 92 q^{23} - 76 q^{25} + 43 q^{27} + 315 q^{29} - 249 q^{31} - 504 q^{33} + 224 q^{35}+ \cdots - 1470 q^{99}+O(q^{100}) \) 4 * q + q^3 + 20 * q^5 + 10 * q^7 + 23 * q^9 - 30 * q^11 + 153 * q^13 + 136 * q^15 - 68 * q^17 - 120 * q^19 + 426 * q^21 + 92 * q^23 - 76 * q^25 + 43 * q^27 + 315 * q^29 - 249 * q^31 - 504 * q^33 + 224 * q^35 + 348 * q^37 + 61 * q^39 - 929 * q^41 + 50 * q^43 + 1028 * q^45 - 205 * q^47 + 456 * q^49 + 890 * q^51 + 578 * q^53 - 1128 * q^55 + 366 * q^57 - 664 * q^59 + 182 * q^61 - 624 * q^63 + 848 * q^65 + 1078 * q^67 + 23 * q^69 + 739 * q^71 + 921 * q^73 + 2883 * q^75 - 528 * q^77 - 1754 * q^79 + 788 * q^81 + 1020 * q^83 + 1072 * q^85 - 2121 * q^87 - 430 * q^89 + 1334 * q^91 - 1035 * q^93 - 744 * q^95 + 48 * q^97 - 1470 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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