LMFDB - Embedded newform 1617.4.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.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:	$95.4060884793$
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} - 28x^{3} - 11x^{2} + 108x - 64 $ x^5 - 28x^3 - 11x^2 + 108*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.767088$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.232912 q^{2} -3.00000 q^{3} -7.94575 q^{4} -7.75746 q^{5} -0.698736 q^{6} -3.71396 q^{8} +9.00000 q^{9} -1.80680 q^{10} +11.0000 q^{11} +23.8373 q^{12} -59.4737 q^{13} +23.2724 q^{15} +62.7010 q^{16} +45.4400 q^{17} +2.09621 q^{18} -111.752 q^{19} +61.6388 q^{20} +2.56203 q^{22} +105.326 q^{23} +11.1419 q^{24} -64.8219 q^{25} -13.8521 q^{26} -27.0000 q^{27} +10.0244 q^{29} +5.42041 q^{30} -315.364 q^{31} +44.3154 q^{32} -33.0000 q^{33} +10.5835 q^{34} -71.5118 q^{36} -182.176 q^{37} -26.0283 q^{38} +178.421 q^{39} +28.8108 q^{40} -487.944 q^{41} -358.348 q^{43} -87.4033 q^{44} -69.8171 q^{45} +24.5317 q^{46} -205.857 q^{47} -188.103 q^{48} -15.0978 q^{50} -136.320 q^{51} +472.563 q^{52} +134.518 q^{53} -6.28862 q^{54} -85.3320 q^{55} +335.255 q^{57} +2.33481 q^{58} +891.997 q^{59} -184.916 q^{60} -654.613 q^{61} -73.4520 q^{62} -491.286 q^{64} +461.364 q^{65} -7.68609 q^{66} -102.298 q^{67} -361.055 q^{68} -315.978 q^{69} +119.126 q^{71} -33.4256 q^{72} +346.254 q^{73} -42.4310 q^{74} +194.466 q^{75} +887.951 q^{76} +41.5564 q^{78} -774.834 q^{79} -486.400 q^{80} +81.0000 q^{81} -113.648 q^{82} -1040.89 q^{83} -352.499 q^{85} -83.4636 q^{86} -30.0732 q^{87} -40.8535 q^{88} -502.925 q^{89} -16.2612 q^{90} -836.896 q^{92} +946.091 q^{93} -47.9465 q^{94} +866.908 q^{95} -132.946 q^{96} +939.285 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9} - 55 q^{10} + 55 q^{11} - 63 q^{12} - 111 q^{13} + 21 q^{15} + 201 q^{16} - 136 q^{17} + 45 q^{18} - 111 q^{19} - 219 q^{20}+ \cdots + 495 q^{99}+O(q^{100}) $ 5 * q + 5 * q^2 - 15 * q^3 + 21 * q^4 - 7 * q^5 - 15 * q^6 + 60 * q^8 + 45 * q^9 - 55 * q^10 + 55 * q^11 - 63 * q^12 - 111 * q^13 + 21 * q^15 + 201 * q^16 - 136 * q^17 + 45 * q^18 - 111 * q^19 - 219 * q^20 + 55 * q^22 - 28 * q^23 - 180 * q^24 + 190 * q^25 + q^26 - 135 * q^27 + 61 * q^29 + 165 * q^30 + 280 * q^31 + 535 * q^32 - 165 * q^33 + 572 * q^34 + 189 * q^36 - 41 * q^37 - 267 * q^38 + 333 * q^39 + 336 * q^40 - 426 * q^41 + 424 * q^43 + 231 * q^44 - 63 * q^45 + 140 * q^46 - 75 * q^47 - 603 * q^48 + 490 * q^50 + 408 * q^51 + 269 * q^52 + 1500 * q^53 - 135 * q^54 - 77 * q^55 + 333 * q^57 - 1767 * q^58 - 757 * q^59 + 657 * q^60 - 658 * q^61 + 568 * q^62 - 748 * q^64 + 537 * q^65 - 165 * q^66 - 583 * q^67 + 1650 * q^68 + 84 * q^69 - 764 * q^71 + 540 * q^72 - 875 * q^73 - 825 * q^74 - 570 * q^75 - 213 * q^76 - 3 * q^78 - 244 * q^79 + 2577 * q^80 + 405 * q^81 + 2006 * q^82 - 924 * q^83 - 1402 * q^85 + 1272 * q^86 - 183 * q^87 + 660 * q^88 + 1110 * q^89 - 495 * q^90 - 2046 * q^92 - 840 * q^93 + 3349 * q^94 + 1923 * q^95 - 1605 * q^96 + 852 * q^97 + 495 * 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 $

$2$	0.232912	0.0823468	0.0411734	−	0.999152i	$-0.486890\pi$
$2$	0.232912	0.0823468	0.0411734	+	0.999152i	$0.486890\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.94575	−0.993219
$5$	−7.75746	−0.693848	−0.346924	−	0.937893i	$-0.612774\pi$
$5$	−7.75746	−0.693848	−0.346924	+	0.937893i	$0.612774\pi$
$6$	−0.698736	−0.0475429
$7$	0	0
$7$	0	0
$8$	−3.71396	−0.164135
$9$	9.00000	0.333333
$10$	−1.80680	−0.0571362
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	23.8373	0.573435
$13$	−59.4737	−1.26885	−0.634424	−	0.772985i	$-0.718763\pi$
$13$	−59.4737	−1.26885	−0.634424	+	0.772985i	$0.718763\pi$
$14$	0	0
$15$	23.2724	0.400593
$16$	62.7010	0.979703
$17$	45.4400	0.648284	0.324142	−	0.946008i	$-0.394924\pi$
$17$	45.4400	0.648284	0.324142	+	0.946008i	$0.394924\pi$
$18$	2.09621	0.0274489
$19$	−111.752	−1.34935	−0.674673	−	0.738117i	$-0.735715\pi$
$19$	−111.752	−1.34935	−0.674673	+	0.738117i	$0.735715\pi$
$20$	61.6388	0.689143
$21$	0	0
$22$	2.56203	0.0248285
$23$	105.326	0.954871	0.477435	−	0.878667i	$-0.341567\pi$
$23$	105.326	0.954871	0.477435	+	0.878667i	$0.341567\pi$
$24$	11.1419	0.0947635
$25$	−64.8219	−0.518575
$26$	−13.8521	−0.104486
$27$	−27.0000	−0.192450
$28$	0	0
$29$	10.0244	0.0641892	0.0320946	−	0.999485i	$-0.489782\pi$
$29$	10.0244	0.0641892	0.0320946	+	0.999485i	$0.489782\pi$
$30$	5.42041	0.0329876
$31$	−315.364	−1.82713	−0.913565	−	0.406694i	$-0.866682\pi$
$31$	−315.364	−1.82713	−0.913565	+	0.406694i	$0.866682\pi$
$32$	44.3154	0.244811
$33$	−33.0000	−0.174078
$34$	10.5835	0.0533841
$35$	0	0
$36$	−71.5118	−0.331073
$37$	−182.176	−0.809448	−0.404724	−	0.914439i	$-0.632632\pi$
$37$	−182.176	−0.809448	−0.404724	+	0.914439i	$0.632632\pi$
$38$	−26.0283	−0.111114
$39$	178.421	0.732570
$40$	28.8108	0.113885
$41$	−487.944	−1.85864	−0.929318	−	0.369280i	$-0.879604\pi$
$41$	−487.944	−1.85864	−0.929318	+	0.369280i	$0.879604\pi$
$42$	0	0
$43$	−358.348	−1.27087	−0.635437	−	0.772152i	$-0.719180\pi$
$43$	−358.348	−1.27087	−0.635437	+	0.772152i	$0.719180\pi$
$44$	−87.4033	−0.299467
$45$	−69.8171	−0.231283
$46$	24.5317	0.0786305
$47$	−205.857	−0.638879	−0.319439	−	0.947607i	$-0.603495\pi$
$47$	−205.857	−0.638879	−0.319439	+	0.947607i	$0.603495\pi$
$48$	−188.103	−0.565632
$49$	0	0
$50$	−15.0978	−0.0427030
$51$	−136.320	−0.374287
$52$	472.563	1.26024
$53$	134.518	0.348632	0.174316	−	0.984690i	$-0.444229\pi$
$53$	134.518	0.348632	0.174316	+	0.984690i	$0.444229\pi$
$54$	−6.28862	−0.0158476
$55$	−85.3320	−0.209203
$56$	0	0
$57$	335.255	0.779046
$58$	2.33481	0.00528578
$59$	891.997	1.96827	0.984137	−	0.177410i	$-0.0567719\pi$
$59$	891.997	1.96827	0.984137	+	0.177410i	$0.0567719\pi$
$60$	−184.916	−0.397877
$61$	−654.613	−1.37401	−0.687005	−	0.726653i	$-0.741075\pi$
$61$	−654.613	−1.37401	−0.687005	+	0.726653i	$0.741075\pi$
$62$	−73.4520	−0.150458
$63$	0	0
$64$	−491.286	−0.959544
$65$	461.364	0.880388
$66$	−7.68609	−0.0143347
$67$	−102.298	−0.186532	−0.0932659	−	0.995641i	$-0.529731\pi$
$67$	−102.298	−0.186532	−0.0932659	+	0.995641i	$0.529731\pi$
$68$	−361.055	−0.643888
$69$	−315.978	−0.551295
$70$	0	0
$71$	119.126	0.199122	0.0995608	−	0.995031i	$-0.468256\pi$
$71$	119.126	0.199122	0.0995608	+	0.995031i	$0.468256\pi$
$72$	−33.4256	−0.0547117
$73$	346.254	0.555151	0.277575	−	0.960704i	$-0.410469\pi$
$73$	346.254	0.555151	0.277575	+	0.960704i	$0.410469\pi$
$74$	−42.4310	−0.0666554
$75$	194.466	0.299399
$76$	887.951	1.34020
$77$	0	0
$78$	41.5564	0.0603248
$79$	−774.834	−1.10349	−0.551744	−	0.834013i	$-0.686038\pi$
$79$	−774.834	−1.10349	−0.551744	+	0.834013i	$0.686038\pi$
$80$	−486.400	−0.679765
$81$	81.0000	0.111111
$82$	−113.648	−0.153053
$83$	−1040.89	−1.37653	−0.688265	−	0.725459i	$-0.741627\pi$
$83$	−1040.89	−1.37653	−0.688265	+	0.725459i	$0.741627\pi$
$84$	0	0
$85$	−352.499	−0.449811
$86$	−83.4636	−0.104652
$87$	−30.0732	−0.0370597
$88$	−40.8535	−0.0494886
$89$	−502.925	−0.598988	−0.299494	−	0.954098i	$-0.596818\pi$
$89$	−502.925	−0.598988	−0.299494	+	0.954098i	$0.596818\pi$
$90$	−16.2612	−0.0190454
$91$	0	0
$92$	−836.896	−0.948396
$93$	946.091	1.05489
$94$	−47.9465	−0.0526096
$95$	866.908	0.936241
$96$	−132.946	−0.141341
$97$	939.285	0.983195	0.491598	−	0.870822i	$-0.336413\pi$
$97$	939.285	0.983195	0.491598	+	0.870822i	$0.336413\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	515.058	0.515058
$101$	664.250	0.654409	0.327205	−	0.944953i	$-0.393893\pi$
$101$	664.250	0.654409	0.327205	+	0.944953i	$0.393893\pi$
$102$	−31.7506	−0.0308213
$103$	1500.66	1.43558	0.717789	−	0.696261i	$-0.245154\pi$
$103$	1500.66	1.43558	0.717789	+	0.696261i	$0.245154\pi$
$104$	220.882	0.208263
$105$	0	0
$106$	31.3309	0.0287087
$107$	−1090.60	−0.985353	−0.492676	−	0.870213i	$-0.663981\pi$
$107$	−1090.60	−0.985353	−0.492676	+	0.870213i	$0.663981\pi$
$108$	214.535	0.191145
$109$	525.673	0.461930	0.230965	−	0.972962i	$-0.425812\pi$
$109$	525.673	0.461930	0.230965	+	0.972962i	$0.425812\pi$
$110$	−19.8748	−0.0172272
$111$	546.528	0.467335
$112$	0	0
$113$	−1978.09	−1.64675	−0.823377	−	0.567494i	$-0.807913\pi$
$113$	−1978.09	−1.64675	−0.823377	+	0.567494i	$0.807913\pi$
$114$	78.0848	0.0641519
$115$	−817.063	−0.662535
$116$	−79.6515	−0.0637539
$117$	−535.263	−0.422949
$118$	207.757	0.162081
$119$	0	0
$120$	−86.4325	−0.0657515
$121$	121.000	0.0909091
$122$	−152.467	−0.113145
$123$	1463.83	1.07308
$124$	2505.80	1.81474
$125$	1472.53	1.05366
$126$	0	0
$127$	−2464.93	−1.72226	−0.861131	−	0.508383i	$-0.830243\pi$
$127$	−2464.93	−1.72226	−0.861131	+	0.508383i	$0.830243\pi$
$128$	−468.950	−0.323826
$129$	1075.05	0.733740
$130$	107.457	0.0724971
$131$	552.047	0.368188	0.184094	−	0.982909i	$-0.441065\pi$
$131$	552.047	0.368188	0.184094	+	0.982909i	$0.441065\pi$
$132$	262.210	0.172897
$133$	0	0
$134$	−23.8263	−0.0153603
$135$	209.451	0.133531
$136$	−168.762	−0.106406
$137$	794.265	0.495318	0.247659	−	0.968847i	$-0.420339\pi$
$137$	794.265	0.495318	0.247659	+	0.968847i	$0.420339\pi$
$138$	−73.5951	−0.0453974
$139$	−430.970	−0.262982	−0.131491	−	0.991317i	$-0.541976\pi$
$139$	−430.970	−0.262982	−0.131491	+	0.991317i	$0.541976\pi$
$140$	0	0
$141$	617.571	0.368857
$142$	27.7458	0.0163970
$143$	−654.210	−0.382572
$144$	564.309	0.326568
$145$	−77.7640	−0.0445376
$146$	80.6468	0.0457149
$147$	0	0
$148$	1447.53	0.803959
$149$	2645.94	1.45479	0.727397	−	0.686217i	$-0.240730\pi$
$149$	2645.94	1.45479	0.727397	+	0.686217i	$0.240730\pi$
$150$	45.2933	0.0246546
$151$	−538.071	−0.289984	−0.144992	−	0.989433i	$-0.546316\pi$
$151$	−538.071	−0.289984	−0.144992	+	0.989433i	$0.546316\pi$
$152$	415.041	0.221475
$153$	408.960	0.216095
$154$	0	0
$155$	2446.42	1.26775
$156$	−1417.69	−0.727602
$157$	−2052.43	−1.04332	−0.521661	−	0.853153i	$-0.674687\pi$
$157$	−2052.43	−1.04332	−0.521661	+	0.853153i	$0.674687\pi$
$158$	−180.468	−0.0908688
$159$	−403.554	−0.201283
$160$	−343.775	−0.169861
$161$	0	0
$162$	18.8659	0.00914964
$163$	2191.90	1.05327	0.526634	−	0.850092i	$-0.323454\pi$
$163$	2191.90	1.05327	0.526634	+	0.850092i	$0.323454\pi$
$164$	3877.08	1.84603
$165$	255.996	0.120783
$166$	−242.435	−0.113353
$167$	−40.0010	−0.0185351	−0.00926757	−	0.999957i	$-0.502950\pi$
$167$	−40.0010	−0.0185351	−0.00926757	+	0.999957i	$0.502950\pi$
$168$	0	0
$169$	1340.12	0.609975
$170$	−82.1012	−0.0370405
$171$	−1005.76	−0.449782
$172$	2847.35	1.26226
$173$	−4251.43	−1.86838	−0.934192	−	0.356770i	$-0.883878\pi$
$173$	−4251.43	−1.86838	−0.934192	+	0.356770i	$0.883878\pi$
$174$	−7.00442	−0.00305174
$175$	0	0
$176$	689.711	0.295392
$177$	−2675.99	−1.13638
$178$	−117.137	−0.0493247
$179$	4768.34	1.99108	0.995538	−	0.0943649i	$-0.0300820\pi$
$179$	4768.34	1.99108	0.995538	+	0.0943649i	$0.0300820\pi$
$180$	554.749	0.229714
$181$	2955.03	1.21351	0.606756	−	0.794888i	$-0.292470\pi$
$181$	2955.03	1.21351	0.606756	+	0.794888i	$0.292470\pi$
$182$	0	0
$183$	1963.84	0.793285
$184$	−391.177	−0.156728
$185$	1413.22	0.561634
$186$	220.356	0.0868671
$187$	499.840	0.195465
$188$	1635.69	0.634547
$189$	0	0
$190$	201.913	0.0770965
$191$	715.502	0.271057	0.135529	−	0.990773i	$-0.456727\pi$
$191$	715.502	0.271057	0.135529	+	0.990773i	$0.456727\pi$
$192$	1473.86	0.553993
$193$	1799.38	0.671098	0.335549	−	0.942023i	$-0.391078\pi$
$193$	1799.38	0.671098	0.335549	+	0.942023i	$0.391078\pi$
$194$	218.771	0.0809630
$195$	−1384.09	−0.508292
$196$	0	0
$197$	2978.55	1.07722	0.538612	−	0.842554i	$-0.318949\pi$
$197$	2978.55	1.07722	0.538612	+	0.842554i	$0.318949\pi$
$198$	23.0583	0.00827616
$199$	2303.16	0.820434	0.410217	−	0.911988i	$-0.365453\pi$
$199$	2303.16	0.820434	0.410217	+	0.911988i	$0.365453\pi$
$200$	240.745	0.0851164
$201$	306.893	0.107694
$202$	154.712	0.0538885
$203$	0	0
$204$	1083.17	0.371749
$205$	3785.21	1.28961
$206$	349.522	0.118215
$207$	947.935	0.318290
$208$	−3729.06	−1.24309
$209$	−1229.27	−0.406843
$210$	0	0
$211$	−4258.59	−1.38945	−0.694724	−	0.719277i	$-0.744473\pi$
$211$	−4258.59	−1.38945	−0.694724	+	0.719277i	$0.744473\pi$
$212$	−1068.85	−0.346268
$213$	−357.377	−0.114963
$214$	−254.015	−0.0811406
$215$	2779.87	0.881794
$216$	100.277	0.0315878
$217$	0	0
$218$	122.435	0.0380384
$219$	−1038.76	−0.320517
$220$	678.027	0.207784
$221$	−2702.49	−0.822574
$222$	127.293	0.0384835
$223$	2739.54	0.822660	0.411330	−	0.911486i	$-0.365064\pi$
$223$	2739.54	0.822660	0.411330	+	0.911486i	$0.365064\pi$
$224$	0	0
$225$	−583.397	−0.172858
$226$	−460.721	−0.135605
$227$	−3373.57	−0.986395	−0.493198	−	0.869917i	$-0.664172\pi$
$227$	−3373.57	−0.986395	−0.493198	+	0.869917i	$0.664172\pi$
$228$	−2663.85	−0.773763
$229$	−349.100	−0.100739	−0.0503693	−	0.998731i	$-0.516040\pi$
$229$	−349.100	−0.100739	−0.0503693	+	0.998731i	$0.516040\pi$
$230$	−190.304	−0.0545576
$231$	0	0
$232$	−37.2302	−0.0105357
$233$	2210.59	0.621548	0.310774	−	0.950484i	$-0.399412\pi$
$233$	2210.59	0.621548	0.310774	+	0.950484i	$0.399412\pi$
$234$	−124.669	−0.0348285
$235$	1596.93	0.443285
$236$	−7087.59	−1.95493
$237$	2324.50	0.637100
$238$	0	0
$239$	−4073.04	−1.10236	−0.551178	−	0.834388i	$-0.685821\pi$
$239$	−4073.04	−1.10236	−0.551178	+	0.834388i	$0.685821\pi$
$240$	1459.20	0.392463
$241$	−1967.60	−0.525910	−0.262955	−	0.964808i	$-0.584697\pi$
$241$	−1967.60	−0.525910	−0.262955	+	0.964808i	$0.584697\pi$
$242$	28.1823	0.00748607
$243$	−243.000	−0.0641500
$244$	5201.39	1.36469
$245$	0	0
$246$	340.944	0.0883650
$247$	6646.28	1.71212
$248$	1171.25	0.299896
$249$	3122.66	0.794740
$250$	342.971	0.0867655
$251$	7019.95	1.76532	0.882660	−	0.470011i	$-0.155750\pi$
$251$	7019.95	1.76532	0.882660	+	0.470011i	$0.155750\pi$
$252$	0	0
$253$	1158.59	0.287904
$254$	−574.112	−0.141823
$255$	1057.50	0.259698
$256$	3821.07	0.932878
$257$	2810.13	0.682066	0.341033	−	0.940051i	$-0.389223\pi$
$257$	2810.13	0.682066	0.341033	+	0.940051i	$0.389223\pi$
$258$	250.391	0.0604211
$259$	0	0
$260$	−3665.89	−0.874418
$261$	90.2197	0.0213964
$262$	128.578	0.0303191
$263$	5388.34	1.26334	0.631672	−	0.775236i	$-0.282369\pi$
$263$	5388.34	1.26334	0.631672	+	0.775236i	$0.282369\pi$
$264$	122.561	0.0285723
$265$	−1043.52	−0.241898
$266$	0	0
$267$	1508.77	0.345826
$268$	812.831	0.185267
$269$	839.014	0.190169	0.0950847	−	0.995469i	$-0.469688\pi$
$269$	839.014	0.190169	0.0950847	+	0.995469i	$0.469688\pi$
$270$	48.7837	0.0109959
$271$	−3831.43	−0.858830	−0.429415	−	0.903107i	$-0.641280\pi$
$271$	−3831.43	−0.858830	−0.429415	+	0.903107i	$0.641280\pi$
$272$	2849.14	0.635126
$273$	0	0
$274$	184.994	0.0407879
$275$	−713.040	−0.156356
$276$	2510.69	0.547556
$277$	−800.595	−0.173657	−0.0868287	−	0.996223i	$-0.527673\pi$
$277$	−800.595	−0.173657	−0.0868287	+	0.996223i	$0.527673\pi$
$278$	−100.378	−0.0216557
$279$	−2838.27	−0.609043
$280$	0	0
$281$	3988.10	0.846655	0.423327	−	0.905977i	$-0.360862\pi$
$281$	3988.10	0.846655	0.423327	+	0.905977i	$0.360862\pi$
$282$	143.840	0.0303742
$283$	−218.036	−0.0457982	−0.0228991	−	0.999738i	$-0.507290\pi$
$283$	−218.036	−0.0457982	−0.0228991	+	0.999738i	$0.507290\pi$
$284$	−946.544	−0.197771
$285$	−2600.73	−0.540539
$286$	−152.373	−0.0315036
$287$	0	0
$288$	398.839	0.0816035
$289$	−2848.20	−0.579728
$290$	−18.1122	−0.00366752
$291$	−2817.86	−0.567648
$292$	−2751.25	−0.551386
$293$	−2886.47	−0.575527	−0.287763	−	0.957702i	$-0.592912\pi$
$293$	−2886.47	−0.575527	−0.287763	+	0.957702i	$0.592912\pi$
$294$	0	0
$295$	−6919.63	−1.36568
$296$	676.594	0.132859
$297$	−297.000	−0.0580259
$298$	616.272	0.119798
$299$	−6264.13	−1.21159
$300$	−1545.18	−0.297369
$301$	0	0
$302$	−125.323	−0.0238792
$303$	−1992.75	−0.377823
$304$	−7006.94	−1.32196
$305$	5078.13	0.953354
$306$	95.2517	0.0177947
$307$	−1453.05	−0.270130	−0.135065	−	0.990837i	$-0.543124\pi$
$307$	−1453.05	−0.270130	−0.135065	+	0.990837i	$0.543124\pi$
$308$	0	0
$309$	−4501.98	−0.828831
$310$	569.800	0.104395
$311$	10772.3	1.96412	0.982062	−	0.188556i	$-0.0603806\pi$
$311$	10772.3	1.96412	0.982062	+	0.188556i	$0.0603806\pi$
$312$	−662.647	−0.120240
$313$	2382.11	0.430175	0.215087	−	0.976595i	$-0.430996\pi$
$313$	2382.11	0.430175	0.215087	+	0.976595i	$0.430996\pi$
$314$	−478.035	−0.0859142
$315$	0	0
$316$	6156.64	1.09601
$317$	−7870.28	−1.39444	−0.697222	−	0.716855i	$-0.745581\pi$
$317$	−7870.28	−1.39444	−0.697222	+	0.716855i	$0.745581\pi$
$318$	−93.9926	−0.0165750
$319$	110.269	0.0193538
$320$	3811.13	0.665777
$321$	3271.81	0.568894
$322$	0	0
$323$	−5078.00	−0.874760
$324$	−643.606	−0.110358
$325$	3855.19	0.657993
$326$	510.519	0.0867333
$327$	−1577.02	−0.266695
$328$	1812.20	0.305068
$329$	0	0
$330$	59.6245	0.00994613
$331$	−6237.15	−1.03572	−0.517862	−	0.855464i	$-0.673272\pi$
$331$	−6237.15	−1.03572	−0.517862	+	0.855464i	$0.673272\pi$
$332$	8270.62	1.36720
$333$	−1639.59	−0.269816
$334$	−9.31670	−0.00152631
$335$	793.569	0.129425
$336$	0	0
$337$	−5496.22	−0.888422	−0.444211	−	0.895922i	$-0.646516\pi$
$337$	−5496.22	−0.888422	−0.444211	+	0.895922i	$0.646516\pi$
$338$	312.129	0.0502295
$339$	5934.28	0.950754
$340$	2800.87	0.446760
$341$	−3469.00	−0.550900
$342$	−234.255	−0.0370381
$343$	0	0
$344$	1330.89	0.208595
$345$	2451.19	0.382515
$346$	−990.210	−0.153855
$347$	−10566.3	−1.63466	−0.817332	−	0.576167i	$-0.804548\pi$
$347$	−10566.3	−1.63466	−0.817332	+	0.576167i	$0.804548\pi$
$348$	238.955	0.0368084
$349$	2553.15	0.391596	0.195798	−	0.980644i	$-0.437270\pi$
$349$	2553.15	0.391596	0.195798	+	0.980644i	$0.437270\pi$
$350$	0	0
$351$	1605.79	0.244190
$352$	487.470	0.0738132
$353$	−1724.64	−0.260037	−0.130019	−	0.991512i	$-0.541504\pi$
$353$	−1724.64	−0.260037	−0.130019	+	0.991512i	$0.541504\pi$
$354$	−623.270	−0.0935775
$355$	−924.113	−0.138160
$356$	3996.11	0.594926
$357$	0	0
$358$	1110.60	0.163959
$359$	2686.78	0.394994	0.197497	−	0.980303i	$-0.436719\pi$
$359$	2686.78	0.394994	0.197497	+	0.980303i	$0.436719\pi$
$360$	259.298	0.0379616
$361$	5629.43	0.820736
$362$	688.262	0.0999289
$363$	−363.000	−0.0524864
$364$	0	0
$365$	−2686.05	−0.385190
$366$	457.401	0.0653245
$367$	2379.21	0.338403	0.169201	−	0.985582i	$-0.445881\pi$
$367$	2379.21	0.338403	0.169201	+	0.985582i	$0.445881\pi$
$368$	6604.05	0.935490
$369$	−4391.50	−0.619545
$370$	329.157	0.0462487
$371$	0	0
$372$	−7517.41	−1.04774
$373$	9982.30	1.38569	0.692847	−	0.721085i	$-0.256356\pi$
$373$	9982.30	1.38569	0.692847	+	0.721085i	$0.256356\pi$
$374$	116.419	0.0160959
$375$	−4417.60	−0.608331
$376$	764.543	0.104863
$377$	−596.189	−0.0814464
$378$	0	0
$379$	−1197.67	−0.162323	−0.0811613	−	0.996701i	$-0.525863\pi$
$379$	−1197.67	−0.162323	−0.0811613	+	0.996701i	$0.525863\pi$
$380$	−6888.24	−0.929893
$381$	7394.79	0.994349
$382$	166.649	0.0223207
$383$	3954.76	0.527621	0.263811	−	0.964575i	$-0.415021\pi$
$383$	3954.76	0.527621	0.263811	+	0.964575i	$0.415021\pi$
$384$	1406.85	0.186961
$385$	0	0
$386$	419.096	0.0552627
$387$	−3225.14	−0.423625
$388$	−7463.33	−0.976528
$389$	−14191.8	−1.84976	−0.924878	−	0.380263i	$-0.875833\pi$
$389$	−14191.8	−1.84976	−0.924878	+	0.380263i	$0.875833\pi$
$390$	−322.372	−0.0418562
$391$	4786.03	0.619027
$392$	0	0
$393$	−1656.14	−0.212573
$394$	693.740	0.0887059
$395$	6010.74	0.765654
$396$	−786.629	−0.0998223
$397$	8724.90	1.10300	0.551499	−	0.834176i	$-0.314056\pi$
$397$	8724.90	1.10300	0.551499	+	0.834176i	$0.314056\pi$
$398$	536.432	0.0675601
$399$	0	0
$400$	−4064.40	−0.508049
$401$	9736.18	1.21247	0.606236	−	0.795285i	$-0.292679\pi$
$401$	9736.18	1.21247	0.606236	+	0.795285i	$0.292679\pi$
$402$	71.4789	0.00886827
$403$	18755.8	2.31835
$404$	−5277.97	−0.649972
$405$	−628.354	−0.0770942
$406$	0	0
$407$	−2003.94	−0.244058
$408$	506.287	0.0614337
$409$	10284.9	1.24342	0.621708	−	0.783249i	$-0.286439\pi$
$409$	10284.9	1.24342	0.621708	+	0.783249i	$0.286439\pi$
$410$	881.620	0.106195
$411$	−2382.79	−0.285972
$412$	−11923.9	−1.42584
$413$	0	0
$414$	220.785	0.0262102
$415$	8074.62	0.955103
$416$	−2635.60	−0.310627
$417$	1292.91	0.151832
$418$	−286.311	−0.0335022
$419$	−12837.5	−1.49679	−0.748393	−	0.663255i	$-0.769174\pi$
$419$	−12837.5	−1.49679	−0.748393	+	0.663255i	$0.769174\pi$
$420$	0	0
$421$	3108.45	0.359849	0.179925	−	0.983680i	$-0.442415\pi$
$421$	3108.45	0.359849	0.179925	+	0.983680i	$0.442415\pi$
$422$	−991.876	−0.114417
$423$	−1852.71	−0.212960
$424$	−499.594	−0.0572227
$425$	−2945.51	−0.336184
$426$	−83.2374	−0.00946682
$427$	0	0
$428$	8665.67	0.978671
$429$	1962.63	0.220878
$430$	647.465	0.0726129
$431$	−414.489	−0.0463231	−0.0231615	−	0.999732i	$-0.507373\pi$
$431$	−414.489	−0.0463231	−0.0231615	+	0.999732i	$0.507373\pi$
$432$	−1692.93	−0.188544
$433$	−7591.87	−0.842591	−0.421296	−	0.906923i	$-0.638425\pi$
$433$	−7591.87	−0.842591	−0.421296	+	0.906923i	$0.638425\pi$
$434$	0	0
$435$	233.292	0.0257138
$436$	−4176.87	−0.458797
$437$	−11770.4	−1.28845
$438$	−241.940	−0.0263935
$439$	1585.71	0.172396	0.0861982	−	0.996278i	$-0.472528\pi$
$439$	1585.71	0.172396	0.0861982	+	0.996278i	$0.472528\pi$
$440$	316.919	0.0343376
$441$	0	0
$442$	−629.441	−0.0677363
$443$	3305.39	0.354501	0.177250	−	0.984166i	$-0.443280\pi$
$443$	3305.39	0.354501	0.177250	+	0.984166i	$0.443280\pi$
$444$	−4342.58	−0.464166
$445$	3901.42	0.415606
$446$	638.072	0.0677434
$447$	−7937.83	−0.839925
$448$	0	0
$449$	−2470.77	−0.259694	−0.129847	−	0.991534i	$-0.541449\pi$
$449$	−2470.77	−0.259694	−0.129847	+	0.991534i	$0.541449\pi$
$450$	−135.880	−0.0142343
$451$	−5367.39	−0.560400
$452$	15717.4	1.63559
$453$	1614.21	0.167422
$454$	−785.745	−0.0812265
$455$	0	0
$456$	−1245.12	−0.127869
$457$	2916.85	0.298566	0.149283	−	0.988795i	$-0.452303\pi$
$457$	2916.85	0.298566	0.149283	+	0.988795i	$0.452303\pi$
$458$	−81.3095	−0.00829551
$459$	−1226.88	−0.124762
$460$	6492.18	0.658042
$461$	6185.85	0.624954	0.312477	−	0.949925i	$-0.398841\pi$
$461$	6185.85	0.624954	0.312477	+	0.949925i	$0.398841\pi$
$462$	0	0
$463$	−19037.0	−1.91086	−0.955428	−	0.295224i	$-0.904606\pi$
$463$	−19037.0	−1.91086	−0.955428	+	0.295224i	$0.904606\pi$
$464$	628.541	0.0628864
$465$	−7339.26	−0.731936
$466$	514.873	0.0511825
$467$	3241.03	0.321150	0.160575	−	0.987024i	$-0.448665\pi$
$467$	3241.03	0.321150	0.160575	+	0.987024i	$0.448665\pi$
$468$	4253.07	0.420081
$469$	0	0
$470$	371.943	0.0365031
$471$	6157.28	0.602362
$472$	−3312.84	−0.323063
$473$	−3941.83	−0.383183
$474$	541.404	0.0524631
$475$	7243.95	0.699737
$476$	0	0
$477$	1210.66	0.116211
$478$	−948.659	−0.0907754
$479$	−17662.0	−1.68475	−0.842377	−	0.538889i	$-0.818844\pi$
$479$	−17662.0	−1.68475	−0.842377	+	0.538889i	$0.818844\pi$
$480$	1031.33	0.0980695
$481$	10834.7	1.02707
$482$	−458.278	−0.0433070
$483$	0	0
$484$	−961.436	−0.0902926
$485$	−7286.46	−0.682188
$486$	−56.5976	−0.00528255
$487$	−11143.6	−1.03689	−0.518445	−	0.855111i	$-0.673489\pi$
$487$	−11143.6	−1.03689	−0.518445	+	0.855111i	$0.673489\pi$
$488$	2431.20	0.225523
$489$	−6575.70	−0.608105
$490$	0	0
$491$	7223.60	0.663944	0.331972	−	0.943289i	$-0.392286\pi$
$491$	7223.60	0.663944	0.331972	+	0.943289i	$0.392286\pi$
$492$	−11631.3	−1.06581
$493$	455.510	0.0416128
$494$	1548.00	0.140987
$495$	−767.988	−0.0697344
$496$	−19773.6	−1.79004
$497$	0	0
$498$	727.304	0.0654443
$499$	−13771.3	−1.23545	−0.617725	−	0.786394i	$-0.711946\pi$
$499$	−13771.3	−1.23545	−0.617725	+	0.786394i	$0.711946\pi$
$500$	−11700.4	−1.04652
$501$	120.003	0.0107013
$502$	1635.03	0.145368
$503$	16826.6	1.49157	0.745787	−	0.666185i	$-0.232074\pi$
$503$	16826.6	1.49157	0.745787	+	0.666185i	$0.232074\pi$
$504$	0	0
$505$	−5152.89	−0.454061
$506$	269.849	0.0237080
$507$	−4020.35	−0.352169
$508$	19585.7	1.71058
$509$	−17447.3	−1.51932	−0.759662	−	0.650318i	$-0.774636\pi$
$509$	−17447.3	−1.51932	−0.759662	+	0.650318i	$0.774636\pi$
$510$	246.304	0.0213853
$511$	0	0
$512$	4641.57	0.400645
$513$	3017.29	0.259682
$514$	654.513	0.0561660
$515$	−11641.3	−0.996073
$516$	−8542.04	−0.728764
$517$	−2264.43	−0.192629
$518$	0	0
$519$	12754.3	1.07871
$520$	−1713.49	−0.144503
$521$	1688.35	0.141973	0.0709866	−	0.997477i	$-0.477385\pi$
$521$	1688.35	0.141973	0.0709866	+	0.997477i	$0.477385\pi$
$522$	21.0132	0.00176193
$523$	21089.3	1.76323	0.881616	−	0.471967i	$-0.156456\pi$
$523$	21089.3	1.76323	0.881616	+	0.471967i	$0.156456\pi$
$524$	−4386.43	−0.365691
$525$	0	0
$526$	1255.01	0.104032
$527$	−14330.1	−1.18450
$528$	−2069.13	−0.170544
$529$	−1073.40	−0.0882223
$530$	−243.048	−0.0199195
$531$	8027.98	0.656091
$532$	0	0
$533$	29019.8	2.35833
$534$	351.411	0.0284776
$535$	8460.32	0.683685
$536$	379.928	0.0306164
$537$	−14305.0	−1.14955
$538$	195.416	0.0156598
$539$	0	0
$540$	−1664.25	−0.132626
$541$	15205.5	1.20838	0.604191	−	0.796840i	$-0.293496\pi$
$541$	15205.5	1.20838	0.604191	+	0.796840i	$0.293496\pi$
$542$	−892.386	−0.0707219
$543$	−8865.10	−0.700622
$544$	2013.70	0.158707
$545$	−4077.88	−0.320509
$546$	0	0
$547$	−6915.32	−0.540544	−0.270272	−	0.962784i	$-0.587114\pi$
$547$	−6915.32	−0.540544	−0.270272	+	0.962784i	$0.587114\pi$
$548$	−6311.03	−0.491960
$549$	−5891.52	−0.458003
$550$	−166.076	−0.0128754
$551$	−1120.24	−0.0866135
$552$	1173.53	0.0904869
$553$	0	0
$554$	−186.468	−0.0143001
$555$	−4239.67	−0.324259
$556$	3424.38	0.261198
$557$	−2243.06	−0.170631	−0.0853156	−	0.996354i	$-0.527190\pi$
$557$	−2243.06	−0.170631	−0.0853156	+	0.996354i	$0.527190\pi$
$558$	−661.068	−0.0501527
$559$	21312.3	1.61255
$560$	0	0
$561$	−1499.52	−0.112852
$562$	928.875	0.0697193
$563$	1533.10	0.114765	0.0573824	−	0.998352i	$-0.481725\pi$
$563$	1533.10	0.114765	0.0573824	+	0.998352i	$0.481725\pi$
$564$	−4907.06	−0.366356
$565$	15345.0	1.14260
$566$	−50.7832	−0.00377134
$567$	0	0
$568$	−442.428	−0.0326828
$569$	8226.60	0.606110	0.303055	−	0.952973i	$-0.401993\pi$
$569$	8226.60	0.606110	0.303055	+	0.952973i	$0.401993\pi$
$570$	−605.740	−0.0445117
$571$	565.628	0.0414550	0.0207275	−	0.999785i	$-0.493402\pi$
$571$	565.628	0.0414550	0.0207275	+	0.999785i	$0.493402\pi$
$572$	5198.19	0.379978
$573$	−2146.51	−0.156495
$574$	0	0
$575$	−6827.44	−0.495172
$576$	−4421.58	−0.319848
$577$	7258.21	0.523680	0.261840	−	0.965111i	$-0.415671\pi$
$577$	7258.21	0.523680	0.261840	+	0.965111i	$0.415671\pi$
$578$	−663.380	−0.0477387
$579$	−5398.13	−0.387458
$580$	617.893	0.0442356
$581$	0	0
$582$	−656.312	−0.0467440
$583$	1479.70	0.105116
$584$	−1285.97	−0.0911198
$585$	4152.28	0.293463
$586$	−672.293	−0.0473928
$587$	−17458.6	−1.22758	−0.613792	−	0.789468i	$-0.710357\pi$
$587$	−17458.6	−1.22758	−0.613792	+	0.789468i	$0.710357\pi$
$588$	0	0
$589$	35242.4	2.46543
$590$	−1611.66	−0.112460
$591$	−8935.65	−0.621935
$592$	−11422.6	−0.793019
$593$	13482.3	0.933648	0.466824	−	0.884350i	$-0.345398\pi$
$593$	13482.3	0.933648	0.466824	+	0.884350i	$0.345398\pi$
$594$	−69.1748	−0.00477825
$595$	0	0
$596$	−21024.0	−1.44493
$597$	−6909.47	−0.473678
$598$	−1458.99	−0.0997702
$599$	−5040.45	−0.343819	−0.171909	−	0.985113i	$-0.554994\pi$
$599$	−5040.45	−0.343819	−0.171909	+	0.985113i	$0.554994\pi$
$600$	−722.236	−0.0491420
$601$	−13587.9	−0.922234	−0.461117	−	0.887339i	$-0.652551\pi$
$601$	−13587.9	−0.922234	−0.461117	+	0.887339i	$0.652551\pi$
$602$	0	0
$603$	−920.678	−0.0621773
$604$	4275.38	0.288018
$605$	−938.652	−0.0630771
$606$	−464.135	−0.0311125
$607$	24238.0	1.62074	0.810371	−	0.585917i	$-0.199265\pi$
$607$	24238.0	1.62074	0.810371	+	0.585917i	$0.199265\pi$
$608$	−4952.32	−0.330334
$609$	0	0
$610$	1182.76	0.0785056
$611$	12243.1	0.810640
$612$	−3249.50	−0.214629
$613$	−7239.20	−0.476979	−0.238490	−	0.971145i	$-0.576652\pi$
$613$	−7239.20	−0.476979	−0.238490	+	0.971145i	$0.576652\pi$
$614$	−338.433	−0.0222444
$615$	−11355.6	−0.744557
$616$	0	0
$617$	25095.8	1.63747	0.818736	−	0.574170i	$-0.194675\pi$
$617$	25095.8	1.63747	0.818736	+	0.574170i	$0.194675\pi$
$618$	−1048.57	−0.0682516
$619$	−9983.31	−0.648244	−0.324122	−	0.946015i	$-0.605069\pi$
$619$	−9983.31	−0.648244	−0.324122	+	0.946015i	$0.605069\pi$
$620$	−19438.7	−1.25915
$621$	−2843.81	−0.183765
$622$	2509.00	0.161739
$623$	0	0
$624$	11187.2	0.717701
$625$	−3320.39	−0.212505
$626$	554.821	0.0354235
$627$	3687.80	0.234891
$628$	16308.1	1.03625
$629$	−8278.09	−0.524752
$630$	0	0
$631$	10735.0	0.677264	0.338632	−	0.940919i	$-0.390036\pi$
$631$	10735.0	0.677264	0.338632	+	0.940919i	$0.390036\pi$
$632$	2877.70	0.181121
$633$	12775.8	0.802198
$634$	−1833.08	−0.114828
$635$	19121.6	1.19499
$636$	3206.54	0.199918
$637$	0	0
$638$	25.6829	0.00159372
$639$	1072.13	0.0663738
$640$	3637.86	0.224686
$641$	12831.2	0.790643	0.395322	−	0.918543i	$-0.370633\pi$
$641$	12831.2	0.790643	0.395322	+	0.918543i	$0.370633\pi$
$642$	762.044	0.0468466
$643$	16536.8	1.01423	0.507113	−	0.861880i	$-0.330713\pi$
$643$	16536.8	1.01423	0.507113	+	0.861880i	$0.330713\pi$
$644$	0	0
$645$	−8339.62	−0.509104
$646$	−1182.73	−0.0720337
$647$	6600.54	0.401073	0.200536	−	0.979686i	$-0.435732\pi$
$647$	6600.54	0.401073	0.200536	+	0.979686i	$0.435732\pi$
$648$	−300.830	−0.0182372
$649$	9811.97	0.593457
$650$	897.920	0.0541836
$651$	0	0
$652$	−17416.3	−1.04613
$653$	−22720.9	−1.36162	−0.680810	−	0.732460i	$-0.738372\pi$
$653$	−22720.9	−1.36162	−0.680810	+	0.732460i	$0.738372\pi$
$654$	−367.306	−0.0219615
$655$	−4282.48	−0.255466
$656$	−30594.6	−1.82091
$657$	3116.29	0.185050
$658$	0	0
$659$	916.372	0.0541681	0.0270841	−	0.999633i	$-0.491378\pi$
$659$	916.372	0.0541681	0.0270841	+	0.999633i	$0.491378\pi$
$660$	−2034.08	−0.119964
$661$	10592.7	0.623312	0.311656	−	0.950195i	$-0.399116\pi$
$661$	10592.7	0.623312	0.311656	+	0.950195i	$0.399116\pi$
$662$	−1452.71	−0.0852886
$663$	8107.46	0.474913
$664$	3865.80	0.225937
$665$	0	0
$666$	−381.879	−0.0222185
$667$	1055.83	0.0612924
$668$	317.838	0.0184095
$669$	−8218.62	−0.474963
$670$	184.832	0.0106577
$671$	−7200.74	−0.414279
$672$	0	0
$673$	10778.2	0.617336	0.308668	−	0.951170i	$-0.400117\pi$
$673$	10778.2	0.617336	0.308668	+	0.951170i	$0.400117\pi$
$674$	−1280.14	−0.0731587
$675$	1750.19	0.0997998
$676$	−10648.2	−0.605839
$677$	−24654.1	−1.39960	−0.699802	−	0.714336i	$-0.746729\pi$
$677$	−24654.1	−1.39960	−0.699802	+	0.714336i	$0.746729\pi$
$678$	1382.16	0.0782915
$679$	0	0
$680$	1309.17	0.0738298
$681$	10120.7	0.569495
$682$	−807.972	−0.0453649
$683$	−21152.8	−1.18505	−0.592524	−	0.805553i	$-0.701868\pi$
$683$	−21152.8	−1.18505	−0.592524	+	0.805553i	$0.701868\pi$
$684$	7991.56	0.446732
$685$	−6161.48	−0.343676
$686$	0	0
$687$	1047.30	0.0581615
$688$	−22468.8	−1.24508
$689$	−8000.29	−0.442361
$690$	570.911	0.0314989
$691$	−3359.89	−0.184973	−0.0924865	−	0.995714i	$-0.529481\pi$
$691$	−3359.89	−0.184973	−0.0924865	+	0.995714i	$0.529481\pi$
$692$	33780.8	1.85572
$693$	0	0
$694$	−2461.02	−0.134609
$695$	3343.23	0.182469
$696$	111.691	0.00608279
$697$	−22172.2	−1.20492
$698$	594.659	0.0322467
$699$	−6631.77	−0.358851
$700$	0	0
$701$	−20003.4	−1.07777	−0.538887	−	0.842378i	$-0.681155\pi$
$701$	−20003.4	−1.07777	−0.538887	+	0.842378i	$0.681155\pi$
$702$	374.007	0.0201083
$703$	20358.5	1.09223
$704$	−5404.15	−0.289313
$705$	−4790.78	−0.255931
$706$	−401.688	−0.0214132
$707$	0	0
$708$	21262.8	1.12868
$709$	6330.56	0.335331	0.167665	−	0.985844i	$-0.446377\pi$
$709$	6330.56	0.335331	0.167665	+	0.985844i	$0.446377\pi$
$710$	−215.237	−0.0113770
$711$	−6973.50	−0.367830
$712$	1867.84	0.0983149
$713$	−33216.1	−1.74467
$714$	0	0
$715$	5075.01	0.265447
$716$	−37888.0	−1.97757
$717$	12219.1	0.636445
$718$	625.784	0.0325265
$719$	111.791	0.00579845	0.00289923	−	0.999996i	$-0.499077\pi$
$719$	111.791	0.00579845	0.00289923	+	0.999996i	$0.499077\pi$
$720$	−4377.60	−0.226588
$721$	0	0
$722$	1311.16	0.0675849
$723$	5902.80	0.303634
$724$	−23480.0	−1.20528
$725$	−649.801	−0.0332869
$726$	−84.5470	−0.00432209
$727$	17266.8	0.880867	0.440433	−	0.897785i	$-0.354825\pi$
$727$	17266.8	0.880867	0.440433	+	0.897785i	$0.354825\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−625.614	−0.0317192
$731$	−16283.4	−0.823888
$732$	−15604.2	−0.787906
$733$	858.639	0.0432668	0.0216334	−	0.999766i	$-0.493113\pi$
$733$	858.639	0.0432668	0.0216334	+	0.999766i	$0.493113\pi$
$734$	554.146	0.0278664
$735$	0	0
$736$	4667.58	0.233762
$737$	−1125.27	−0.0562415
$738$	−1022.83	−0.0510176
$739$	16141.2	0.803471	0.401736	−	0.915756i	$-0.368407\pi$
$739$	16141.2	0.803471	0.401736	+	0.915756i	$0.368407\pi$
$740$	−11229.1	−0.557825
$741$	−19938.8	−0.988490
$742$	0	0
$743$	1249.50	0.0616953	0.0308476	−	0.999524i	$-0.490179\pi$
$743$	1249.50	0.0616953	0.0308476	+	0.999524i	$0.490179\pi$
$744$	−3513.74	−0.173145
$745$	−20525.8	−1.00941
$746$	2325.00	0.114107
$747$	−9367.97	−0.458843
$748$	−3971.61	−0.194140
$749$	0	0
$750$	−1028.91	−0.0500941
$751$	82.9863	0.00403224	0.00201612	−	0.999998i	$-0.499358\pi$
$751$	82.9863	0.00403224	0.00201612	+	0.999998i	$0.499358\pi$
$752$	−12907.4	−0.625912
$753$	−21059.9	−1.01921
$754$	−138.859	−0.00670685
$755$	4174.06	0.201205
$756$	0	0
$757$	−3439.65	−0.165147	−0.0825735	−	0.996585i	$-0.526314\pi$
$757$	−3439.65	−0.165147	−0.0825735	+	0.996585i	$0.526314\pi$
$758$	−278.952	−0.0133667
$759$	−3475.76	−0.166222
$760$	−3219.66	−0.153670
$761$	−5651.24	−0.269195	−0.134597	−	0.990900i	$-0.542974\pi$
$761$	−5651.24	−0.269195	−0.134597	+	0.990900i	$0.542974\pi$
$762$	1722.34	0.0818814
$763$	0	0
$764$	−5685.20	−0.269219
$765$	−3172.49	−0.149937
$766$	921.111	0.0434479
$767$	−53050.3	−2.49744
$768$	−11463.2	−0.538597
$769$	−14381.1	−0.674378	−0.337189	−	0.941437i	$-0.609476\pi$
$769$	−14381.1	−0.674378	−0.337189	+	0.941437i	$0.609476\pi$
$770$	0	0
$771$	−8430.39	−0.393791
$772$	−14297.4	−0.666547
$773$	31316.0	1.45713	0.728563	−	0.684979i	$-0.240189\pi$
$773$	31316.0	1.45713	0.728563	+	0.684979i	$0.240189\pi$
$774$	−751.172	−0.0348841
$775$	20442.5	0.947503
$776$	−3488.46	−0.161377
$777$	0	0
$778$	−3305.45	−0.152321
$779$	54528.6	2.50794
$780$	10997.7	0.504845
$781$	1310.38	0.0600374
$782$	1114.72	0.0509749
$783$	−270.659	−0.0123532
$784$	0	0
$785$	15921.6	0.723907
$786$	−385.735	−0.0175047
$787$	−29847.6	−1.35191	−0.675954	−	0.736944i	$-0.736268\pi$
$787$	−29847.6	−1.35191	−0.675954	+	0.736944i	$0.736268\pi$
$788$	−23666.8	−1.06992
$789$	−16165.0	−0.729391
$790$	1399.97	0.0630491
$791$	0	0
$792$	−367.682	−0.0164962
$793$	38932.2	1.74341
$794$	2032.13	0.0908283
$795$	3130.56	0.139660
$796$	−18300.3	−0.814871
$797$	−7581.40	−0.336947	−0.168474	−	0.985706i	$-0.553884\pi$
$797$	−7581.40	−0.336947	−0.168474	+	0.985706i	$0.553884\pi$
$798$	0	0
$799$	−9354.15	−0.414175
$800$	−2872.61	−0.126953
$801$	−4526.32	−0.199663
$802$	2267.67	0.0998432
$803$	3808.80	0.167384
$804$	−2438.49	−0.106964
$805$	0	0
$806$	4368.46	0.190909
$807$	−2517.04	−0.109794
$808$	−2467.00	−0.107412
$809$	−25260.6	−1.09779	−0.548896	−	0.835891i	$-0.684952\pi$
$809$	−25260.6	−1.09779	−0.548896	+	0.835891i	$0.684952\pi$
$810$	−146.351	−0.00634846
$811$	25594.8	1.10821	0.554103	−	0.832448i	$-0.313061\pi$
$811$	25594.8	1.10821	0.554103	+	0.832448i	$0.313061\pi$
$812$	0	0
$813$	11494.3	0.495846
$814$	−466.741	−0.0200974
$815$	−17003.6	−0.730808
$816$	−8547.41	−0.366690
$817$	40046.0	1.71485
$818$	2395.48	0.102391
$819$	0	0
$820$	−30076.3	−1.28087
$821$	−19077.2	−0.810959	−0.405480	−	0.914104i	$-0.632895\pi$
$821$	−19077.2	−0.810959	−0.405480	+	0.914104i	$0.632895\pi$
$822$	−554.981	−0.0235489
$823$	13889.2	0.588270	0.294135	−	0.955764i	$-0.404969\pi$
$823$	13889.2	0.588270	0.294135	+	0.955764i	$0.404969\pi$
$824$	−5573.39	−0.235629
$825$	2139.12	0.0902723
$826$	0	0
$827$	1354.92	0.0569714	0.0284857	−	0.999594i	$-0.490931\pi$
$827$	1354.92	0.0569714	0.0284857	+	0.999594i	$0.490931\pi$
$828$	−7532.06	−0.316132
$829$	−29760.9	−1.24685	−0.623426	−	0.781882i	$-0.714260\pi$
$829$	−29760.9	−1.24685	−0.623426	+	0.781882i	$0.714260\pi$
$830$	1880.68	0.0786496
$831$	2401.79	0.100261
$832$	29218.6	1.21752
$833$	0	0
$834$	301.134	0.0125029
$835$	310.306	0.0128606
$836$	9767.46	0.404084
$837$	8514.82	0.351631
$838$	−2990.01	−0.123256
$839$	−33789.1	−1.39038	−0.695191	−	0.718825i	$-0.744680\pi$
$839$	−33789.1	−1.39038	−0.695191	+	0.718825i	$0.744680\pi$
$840$	0	0
$841$	−24288.5	−0.995880
$842$	723.995	0.0296324
$843$	−11964.3	−0.488816
$844$	33837.7	1.38003
$845$	−10395.9	−0.423230
$846$	−431.519	−0.0175365
$847$	0	0
$848$	8434.42	0.341556
$849$	654.108	0.0264416
$850$	−686.044	−0.0276837
$851$	−19187.9	−0.772918
$852$	2839.63	0.114183
$853$	−22293.3	−0.894851	−0.447426	−	0.894321i	$-0.647659\pi$
$853$	−22293.3	−0.894851	−0.447426	+	0.894321i	$0.647659\pi$
$854$	0	0
$855$	7802.18	0.312080
$856$	4050.46	0.161731
$857$	43972.3	1.75270	0.876352	−	0.481672i	$-0.159970\pi$
$857$	43972.3	1.75270	0.876352	+	0.481672i	$0.159970\pi$
$858$	457.120	0.0181886
$859$	34582.5	1.37362	0.686810	−	0.726837i	$-0.259010\pi$
$859$	34582.5	1.37362	0.686810	+	0.726837i	$0.259010\pi$
$860$	−22088.2	−0.875814
$861$	0	0
$862$	−96.5395	−0.00381456
$863$	48984.4	1.93215	0.966077	−	0.258254i	$-0.0831472\pi$
$863$	48984.4	1.93215	0.966077	+	0.258254i	$0.0831472\pi$
$864$	−1196.52	−0.0471138
$865$	32980.3	1.29638
$866$	−1768.24	−0.0693847
$867$	8544.61	0.334706
$868$	0	0
$869$	−8523.17	−0.332714
$870$	54.3365	0.00211745
$871$	6084.01	0.236681
$872$	−1952.33	−0.0758189
$873$	8453.57	0.327732
$874$	−2741.46	−0.106100
$875$	0	0
$876$	8253.76	0.318343
$877$	3914.98	0.150741	0.0753703	−	0.997156i	$-0.475986\pi$
$877$	3914.98	0.150741	0.0753703	+	0.997156i	$0.475986\pi$
$878$	369.332	0.0141963
$879$	8659.41	0.332281
$880$	−5350.40	−0.204957
$881$	−4908.77	−0.187719	−0.0938597	−	0.995585i	$-0.529921\pi$
$881$	−4908.77	−0.187719	−0.0938597	+	0.995585i	$0.529921\pi$
$882$	0	0
$883$	8791.82	0.335072	0.167536	−	0.985866i	$-0.446419\pi$
$883$	8791.82	0.335072	0.167536	+	0.985866i	$0.446419\pi$
$884$	21473.3	0.816996
$885$	20758.9	0.788477
$886$	769.865	0.0291920
$887$	10011.0	0.378958	0.189479	−	0.981885i	$-0.439320\pi$
$887$	10011.0	0.378958	0.189479	+	0.981885i	$0.439320\pi$
$888$	−2029.78	−0.0767061
$889$	0	0
$890$	908.686	0.0342239
$891$	891.000	0.0335013
$892$	−21767.7	−0.817082
$893$	23004.8	0.862069
$894$	−1848.82	−0.0691651
$895$	−36990.2	−1.38150
$896$	0	0
$897$	18792.4	0.699509
$898$	−575.471	−0.0213850
$899$	−3161.34	−0.117282
$900$	4635.53	0.171686
$901$	6112.51	0.226012
$902$	−1250.13	−0.0461471
$903$	0	0
$904$	7346.55	0.270290
$905$	−22923.5	−0.841993
$906$	375.969	0.0137867
$907$	−44587.3	−1.63230	−0.816150	−	0.577840i	$-0.803896\pi$
$907$	−44587.3	−1.63230	−0.816150	+	0.577840i	$0.803896\pi$
$908$	26805.5	0.979706
$909$	5978.25	0.218136
$910$	0	0
$911$	22201.6	0.807434	0.403717	−	0.914884i	$-0.367718\pi$
$911$	22201.6	0.807434	0.403717	+	0.914884i	$0.367718\pi$
$912$	21020.8	0.763233
$913$	−11449.7	−0.415039
$914$	679.370	0.0245860
$915$	−15234.4	−0.550419
$916$	2773.86	0.100056
$917$	0	0
$918$	−285.755	−0.0102738
$919$	−4755.83	−0.170708	−0.0853538	−	0.996351i	$-0.527202\pi$
$919$	−4755.83	−0.170708	−0.0853538	+	0.996351i	$0.527202\pi$
$920$	3034.54	0.108745
$921$	4359.15	0.155960
$922$	1440.76	0.0514630
$923$	−7084.84	−0.252655
$924$	0	0
$925$	11809.0	0.419759
$926$	−4433.95	−0.157353
$927$	13505.9	0.478526
$928$	444.236	0.0157142
$929$	22020.9	0.777699	0.388849	−	0.921301i	$-0.372873\pi$
$929$	22020.9	0.777699	0.388849	+	0.921301i	$0.372873\pi$
$930$	−1709.40	−0.0602726
$931$	0	0
$932$	−17564.8	−0.617333
$933$	−32317.0	−1.13399
$934$	754.874	0.0264457
$935$	−3877.49	−0.135623
$936$	1987.94	0.0694209
$937$	−29862.2	−1.04115	−0.520573	−	0.853817i	$-0.674282\pi$
$937$	−29862.2	−1.04115	−0.520573	+	0.853817i	$0.674282\pi$
$938$	0	0
$939$	−7146.32	−0.248362
$940$	−12688.8	−0.440279
$941$	8090.56	0.280281	0.140141	−	0.990132i	$-0.455245\pi$
$941$	8090.56	0.280281	0.140141	+	0.990132i	$0.455245\pi$
$942$	1434.10	0.0496026
$943$	−51393.3	−1.77476
$944$	55929.1	1.92832
$945$	0	0
$946$	−918.099	−0.0315539
$947$	4398.75	0.150940	0.0754700	−	0.997148i	$-0.475954\pi$
$947$	4398.75	0.150940	0.0754700	+	0.997148i	$0.475954\pi$
$948$	−18469.9	−0.632779
$949$	−20593.0	−0.704402
$950$	1687.20	0.0576211
$951$	23610.8	0.805083
$952$	0	0
$953$	−4288.91	−0.145783	−0.0728917	−	0.997340i	$-0.523223\pi$
$953$	−4288.91	−0.145783	−0.0728917	+	0.997340i	$0.523223\pi$
$954$	281.978	0.00956957
$955$	−5550.48	−0.188073
$956$	32363.3	1.09488
$957$	−330.806	−0.0111739
$958$	−4113.69	−0.138734
$959$	0	0
$960$	−11433.4	−0.384387
$961$	69663.3	2.33840
$962$	2523.53	0.0845756
$963$	−9815.44	−0.328451
$964$	15634.1	0.522344
$965$	−13958.6	−0.465640
$966$	0	0
$967$	34271.8	1.13972	0.569858	−	0.821743i	$-0.306998\pi$
$967$	34271.8	1.13972	0.569858	+	0.821743i	$0.306998\pi$
$968$	−449.389	−0.0149214
$969$	15234.0	0.505043
$970$	−1697.10	−0.0561760
$971$	−36628.3	−1.21056	−0.605282	−	0.796011i	$-0.706940\pi$
$971$	−36628.3	−1.21056	−0.605282	+	0.796011i	$0.706940\pi$
$972$	1930.82	0.0637150
$973$	0	0
$974$	−2595.48	−0.0853845
$975$	−11565.6	−0.379892
$976$	−41044.9	−1.34612
$977$	29087.7	0.952504	0.476252	−	0.879309i	$-0.341995\pi$
$977$	29087.7	0.952504	0.476252	+	0.879309i	$0.341995\pi$
$978$	−1531.56	−0.0500755
$979$	−5532.17	−0.180602
$980$	0	0
$981$	4731.06	0.153977
$982$	1682.46	0.0546737
$983$	32900.7	1.06752	0.533758	−	0.845637i	$-0.320779\pi$
$983$	32900.7	1.06752	0.533758	+	0.845637i	$0.320779\pi$
$984$	−5436.61	−0.176131
$985$	−23106.0	−0.747429
$986$	106.094	0.00342668
$987$	0	0
$988$	−52809.7	−1.70051
$989$	−37743.5	−1.21352
$990$	−178.874	−0.00574240
$991$	7288.46	0.233628	0.116814	−	0.993154i	$-0.462732\pi$
$991$	7288.46	0.233628	0.116814	+	0.993154i	$0.462732\pi$
$992$	−13975.5	−0.447301
$993$	18711.4	0.597976
$994$	0	0
$995$	−17866.6	−0.569257
$996$	−24811.8	−0.789351
$997$	−2983.80	−0.0947822	−0.0473911	−	0.998876i	$-0.515091\pi$
$997$	−2983.80	−0.0947822	−0.0473911	+	0.998876i	$0.515091\pi$
$998$	−3207.51	−0.101735
$999$	4918.76	0.155778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.4.a.p.1.3		5
7.6	odd	2		231.4.a.l.1.3	✓	5
21.20	even	2		693.4.a.n.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.l.1.3	✓	5	7.6	odd	2
693.4.a.n.1.3		5	21.20	even	2
1617.4.a.p.1.3		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+0.232912 q^{2} -3.00000 q^{3} -7.94575 q^{4} -7.75746 q^{5} -0.698736 q^{6} -3.71396 q^{8} +9.00000 q^{9} -1.80680 q^{10} +11.0000 q^{11} +23.8373 q^{12} -59.4737 q^{13} +23.2724 q^{15} +62.7010 q^{16} +45.4400 q^{17} +2.09621 q^{18} -111.752 q^{19} +61.6388 q^{20} +2.56203 q^{22} +105.326 q^{23} +11.1419 q^{24} -64.8219 q^{25} -13.8521 q^{26} -27.0000 q^{27} +10.0244 q^{29} +5.42041 q^{30} -315.364 q^{31} +44.3154 q^{32} -33.0000 q^{33} +10.5835 q^{34} -71.5118 q^{36} -182.176 q^{37} -26.0283 q^{38} +178.421 q^{39} +28.8108 q^{40} -487.944 q^{41} -358.348 q^{43} -87.4033 q^{44} -69.8171 q^{45} +24.5317 q^{46} -205.857 q^{47} -188.103 q^{48} -15.0978 q^{50} -136.320 q^{51} +472.563 q^{52} +134.518 q^{53} -6.28862 q^{54} -85.3320 q^{55} +335.255 q^{57} +2.33481 q^{58} +891.997 q^{59} -184.916 q^{60} -654.613 q^{61} -73.4520 q^{62} -491.286 q^{64} +461.364 q^{65} -7.68609 q^{66} -102.298 q^{67} -361.055 q^{68} -315.978 q^{69} +119.126 q^{71} -33.4256 q^{72} +346.254 q^{73} -42.4310 q^{74} +194.466 q^{75} +887.951 q^{76} +41.5564 q^{78} -774.834 q^{79} -486.400 q^{80} +81.0000 q^{81} -113.648 q^{82} -1040.89 q^{83} -352.499 q^{85} -83.4636 q^{86} -30.0732 q^{87} -40.8535 q^{88} -502.925 q^{89} -16.2612 q^{90} -836.896 q^{92} +946.091 q^{93} -47.9465 q^{94} +866.908 q^{95} -132.946 q^{96} +939.285 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9} - 55 q^{10} + 55 q^{11} - 63 q^{12} - 111 q^{13} + 21 q^{15} + 201 q^{16} - 136 q^{17} + 45 q^{18} - 111 q^{19} - 219 q^{20}+ \cdots + 495 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 - 15 * q^3 + 21 * q^4 - 7 * q^5 - 15 * q^6 + 60 * q^8 + 45 * q^9 - 55 * q^10 + 55 * q^11 - 63 * q^12 - 111 * q^13 + 21 * q^15 + 201 * q^16 - 136 * q^17 + 45 * q^18 - 111 * q^19 - 219 * q^20 + 55 * q^22 - 28 * q^23 - 180 * q^24 + 190 * q^25 + q^26 - 135 * q^27 + 61 * q^29 + 165 * q^30 + 280 * q^31 + 535 * q^32 - 165 * q^33 + 572 * q^34 + 189 * q^36 - 41 * q^37 - 267 * q^38 + 333 * q^39 + 336 * q^40 - 426 * q^41 + 424 * q^43 + 231 * q^44 - 63 * q^45 + 140 * q^46 - 75 * q^47 - 603 * q^48 + 490 * q^50 + 408 * q^51 + 269 * q^52 + 1500 * q^53 - 135 * q^54 - 77 * q^55 + 333 * q^57 - 1767 * q^58 - 757 * q^59 + 657 * q^60 - 658 * q^61 + 568 * q^62 - 748 * q^64 + 537 * q^65 - 165 * q^66 - 583 * q^67 + 1650 * q^68 + 84 * q^69 - 764 * q^71 + 540 * q^72 - 875 * q^73 - 825 * q^74 - 570 * q^75 - 213 * q^76 - 3 * q^78 - 244 * q^79 + 2577 * q^80 + 405 * q^81 + 2006 * q^82 - 924 * q^83 - 1402 * q^85 + 1272 * q^86 - 183 * q^87 + 660 * q^88 + 1110 * q^89 - 495 * q^90 - 2046 * q^92 - 840 * q^93 + 3349 * q^94 + 1923 * q^95 - 1605 * q^96 + 852 * q^97 + 495 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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