LMFDB - Embedded newform 1617.4.a.i.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	\(q-3.56155 q^{2} -3.00000 q^{3} +4.68466 q^{4} +15.6847 q^{5} +10.6847 q^{6} +11.8078 q^{8} +9.00000 q^{9} -55.8617 q^{10} +11.0000 q^{11} -14.0540 q^{12} -52.9157 q^{13} -47.0540 q^{15} -79.5312 q^{16} +77.1231 q^{17} -32.0540 q^{18} -13.4081 q^{19} +73.4773 q^{20} -39.1771 q^{22} -59.0152 q^{23} -35.4233 q^{24} +121.009 q^{25} +188.462 q^{26} -27.0000 q^{27} -69.3305 q^{29} +167.585 q^{30} -75.9697 q^{31} +188.793 q^{32} -33.0000 q^{33} -274.678 q^{34} +42.1619 q^{36} -335.948 q^{37} +47.7538 q^{38} +158.747 q^{39} +185.201 q^{40} +318.617 q^{41} -57.2614 q^{43} +51.5312 q^{44} +141.162 q^{45} +210.186 q^{46} +577.779 q^{47} +238.594 q^{48} -430.978 q^{50} -231.369 q^{51} -247.892 q^{52} +315.555 q^{53} +96.1619 q^{54} +172.531 q^{55} +40.2244 q^{57} +246.924 q^{58} +598.717 q^{59} -220.432 q^{60} +337.879 q^{61} +270.570 q^{62} -36.1449 q^{64} -829.965 q^{65} +117.531 q^{66} -107.779 q^{67} +361.295 q^{68} +177.045 q^{69} +405.390 q^{71} +106.270 q^{72} +133.300 q^{73} +1196.50 q^{74} -363.026 q^{75} -62.8125 q^{76} -565.386 q^{78} -922.038 q^{79} -1247.42 q^{80} +81.0000 q^{81} -1134.77 q^{82} +1221.17 q^{83} +1209.65 q^{85} +203.939 q^{86} +207.991 q^{87} +129.885 q^{88} -1580.19 q^{89} -502.756 q^{90} -276.466 q^{92} +227.909 q^{93} -2057.79 q^{94} -210.302 q^{95} -566.378 q^{96} +287.097 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 6 q^{3} - 3 q^{4} + 19 q^{5} + 9 q^{6} + 3 q^{8} + 18 q^{9} - 54 q^{10} + 22 q^{11} + 9 q^{12} - 11 q^{13} - 57 q^{15} - 23 q^{16} + 146 q^{17} - 27 q^{18} + 101 q^{19} + 48 q^{20} - 33 q^{22}+ \cdots + 198 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 - 6 * q^3 - 3 * q^4 + 19 * q^5 + 9 * q^6 + 3 * q^8 + 18 * q^9 - 54 * q^10 + 22 * q^11 + 9 * q^12 - 11 * q^13 - 57 * q^15 - 23 * q^16 + 146 * q^17 - 27 * q^18 + 101 * q^19 + 48 * q^20 - 33 * q^22 - 184 * q^23 - 9 * q^24 + 7 * q^25 + 212 * q^26 - 54 * q^27 - 217 * q^29 + 162 * q^30 - 20 * q^31 + 291 * q^32 - 66 * q^33 - 236 * q^34 - 27 * q^36 - 173 * q^37 + 112 * q^38 + 33 * q^39 + 156 * q^40 + 60 * q^41 - 164 * q^43 - 33 * q^44 + 171 * q^45 + 140 * q^46 + 467 * q^47 + 69 * q^48 - 495 * q^50 - 438 * q^51 - 570 * q^52 + 326 * q^53 + 81 * q^54 + 209 * q^55 - 303 * q^57 + 164 * q^58 + 781 * q^59 - 144 * q^60 + 148 * q^61 + 302 * q^62 - 431 * q^64 - 691 * q^65 + 99 * q^66 + 473 * q^67 - 168 * q^68 + 552 * q^69 - 756 * q^71 + 27 * q^72 + 213 * q^73 + 1288 * q^74 - 21 * q^75 - 942 * q^76 - 636 * q^78 + 168 * q^79 - 1060 * q^80 + 162 * q^81 - 1280 * q^82 + 2096 * q^83 + 1438 * q^85 + 144 * q^86 + 651 * q^87 + 33 * q^88 - 736 * q^89 - 486 * q^90 + 684 * q^92 + 60 * q^93 - 2120 * q^94 + 169 * q^95 - 873 * q^96 - 638 * q^97 + 198 * 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 $

$2$	−3.56155	−1.25920	−0.629600	−	0.776920i	$-0.716781\pi$
$2$	−3.56155	−1.25920	−0.629600	+	0.776920i	$0.716781\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.68466	0.585582
$5$	15.6847	1.40288	0.701439	−	0.712729i	$-0.252541\pi$
$5$	15.6847	1.40288	0.701439	+	0.712729i	$0.252541\pi$
$6$	10.6847	0.726999
$7$	0	0
$7$	0	0
$8$	11.8078	0.521834
$9$	9.00000	0.333333
$10$	−55.8617	−1.76650
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−14.0540	−0.338086
$13$	−52.9157	−1.12894	−0.564468	−	0.825455i	$-0.690919\pi$
$13$	−52.9157	−1.12894	−0.564468	+	0.825455i	$0.690919\pi$
$14$	0	0
$15$	−47.0540	−0.809952
$16$	−79.5312	−1.24268
$17$	77.1231	1.10030	0.550150	−	0.835066i	$-0.314571\pi$
$17$	77.1231	1.10030	0.550150	+	0.835066i	$0.314571\pi$
$18$	−32.0540	−0.419733
$19$	−13.4081	−0.161897	−0.0809484	−	0.996718i	$-0.525795\pi$
$19$	−13.4081	−0.161897	−0.0809484	+	0.996718i	$0.525795\pi$
$20$	73.4773	0.821501
$21$	0	0
$22$	−39.1771	−0.379663
$23$	−59.0152	−0.535022	−0.267511	−	0.963555i	$-0.586201\pi$
$23$	−59.0152	−0.535022	−0.267511	+	0.963555i	$0.586201\pi$
$24$	−35.4233	−0.301281
$25$	121.009	0.968068
$26$	188.462	1.42156
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−69.3305	−0.443943	−0.221972	−	0.975053i	$-0.571249\pi$
$29$	−69.3305	−0.443943	−0.221972	+	0.975053i	$0.571249\pi$
$30$	167.585	1.01989
$31$	−75.9697	−0.440147	−0.220074	−	0.975483i	$-0.570630\pi$
$31$	−75.9697	−0.440147	−0.220074	+	0.975483i	$0.570630\pi$
$32$	188.793	1.04294
$33$	−33.0000	−0.174078
$34$	−274.678	−1.38550
$35$	0	0
$36$	42.1619	0.195194
$37$	−335.948	−1.49269	−0.746344	−	0.665560i	$-0.768193\pi$
$37$	−335.948	−1.49269	−0.746344	+	0.665560i	$0.768193\pi$
$38$	47.7538	0.203860
$39$	158.747	0.651792
$40$	185.201	0.732070
$41$	318.617	1.21365	0.606825	−	0.794835i	$-0.292443\pi$
$41$	318.617	1.21365	0.606825	+	0.794835i	$0.292443\pi$
$42$	0	0
$43$	−57.2614	−0.203076	−0.101538	−	0.994832i	$-0.532376\pi$
$43$	−57.2614	−0.203076	−0.101538	+	0.994832i	$0.532376\pi$
$44$	51.5312	0.176560
$45$	141.162	0.467626
$46$	210.186	0.673699
$47$	577.779	1.79314	0.896572	−	0.442898i	$-0.146050\pi$
$47$	577.779	1.79314	0.896572	+	0.442898i	$0.146050\pi$
$48$	238.594	0.717459
$49$	0	0
$50$	−430.978	−1.21899
$51$	−231.369	−0.635259
$52$	−247.892	−0.661085
$53$	315.555	0.817826	0.408913	−	0.912573i	$-0.365908\pi$
$53$	315.555	0.817826	0.408913	+	0.912573i	$0.365908\pi$
$54$	96.1619	0.242333
$55$	172.531	0.422984
$56$	0	0
$57$	40.2244	0.0934711
$58$	246.924	0.559013
$59$	598.717	1.32112	0.660562	−	0.750772i	$-0.270318\pi$
$59$	598.717	1.32112	0.660562	+	0.750772i	$0.270318\pi$
$60$	−220.432	−0.474294
$61$	337.879	0.709196	0.354598	−	0.935019i	$-0.384618\pi$
$61$	337.879	0.709196	0.354598	+	0.935019i	$0.384618\pi$
$62$	270.570	0.554233
$63$	0	0
$64$	−36.1449	−0.0705955
$65$	−829.965	−1.58376
$66$	117.531	0.219198
$67$	−107.779	−0.196527	−0.0982637	−	0.995160i	$-0.531329\pi$
$67$	−107.779	−0.196527	−0.0982637	+	0.995160i	$0.531329\pi$
$68$	361.295	0.644316
$69$	177.045	0.308895
$70$	0	0
$71$	405.390	0.677619	0.338810	−	0.940855i	$-0.389976\pi$
$71$	405.390	0.677619	0.338810	+	0.940855i	$0.389976\pi$
$72$	106.270	0.173945
$73$	133.300	0.213721	0.106860	−	0.994274i	$-0.465920\pi$
$73$	133.300	0.213721	0.106860	+	0.994274i	$0.465920\pi$
$74$	1196.50	1.87959
$75$	−363.026	−0.558914
$76$	−62.8125	−0.0948039
$77$	0	0
$78$	−565.386	−0.820736
$79$	−922.038	−1.31313	−0.656566	−	0.754269i	$-0.727991\pi$
$79$	−922.038	−1.31313	−0.656566	+	0.754269i	$0.727991\pi$
$80$	−1247.42	−1.74332
$81$	81.0000	0.111111
$82$	−1134.77	−1.52823
$83$	1221.17	1.61495	0.807475	−	0.589902i	$-0.200833\pi$
$83$	1221.17	1.61495	0.807475	+	0.589902i	$0.200833\pi$
$84$	0	0
$85$	1209.65	1.54359
$86$	203.939	0.255713
$87$	207.991	0.256311
$88$	129.885	0.157339
$89$	−1580.19	−1.88202	−0.941012	−	0.338373i	$-0.890123\pi$
$89$	−1580.19	−1.88202	−0.941012	+	0.338373i	$0.890123\pi$
$90$	−502.756	−0.588834
$91$	0	0
$92$	−276.466	−0.313300
$93$	227.909	0.254119
$94$	−2057.79	−2.25793
$95$	−210.302	−0.227121
$96$	−566.378	−0.602143
$97$	287.097	0.300518	0.150259	−	0.988647i	$-0.451989\pi$
$97$	287.097	0.300518	0.150259	+	0.988647i	$0.451989\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	566.884	0.566884
$101$	−780.867	−0.769299	−0.384650	−	0.923063i	$-0.625678\pi$
$101$	−780.867	−0.769299	−0.384650	+	0.923063i	$0.625678\pi$
$102$	824.034	0.799917
$103$	−1218.22	−1.16538	−0.582691	−	0.812694i	$-0.698000\pi$
$103$	−1218.22	−1.16538	−0.582691	+	0.812694i	$0.698000\pi$
$104$	−624.816	−0.589118
$105$	0	0
$106$	−1123.87	−1.02981
$107$	−873.196	−0.788926	−0.394463	−	0.918912i	$-0.629069\pi$
$107$	−873.196	−0.788926	−0.394463	+	0.918912i	$0.629069\pi$
$108$	−126.486	−0.112695
$109$	126.424	0.111094	0.0555470	−	0.998456i	$-0.482310\pi$
$109$	126.424	0.111094	0.0555470	+	0.998456i	$0.482310\pi$
$110$	−614.479	−0.532621
$111$	1007.84	0.861804
$112$	0	0
$113$	257.318	0.214217	0.107108	−	0.994247i	$-0.465841\pi$
$113$	257.318	0.214217	0.107108	+	0.994247i	$0.465841\pi$
$114$	−143.261	−0.117699
$115$	−925.633	−0.750571
$116$	−324.790	−0.259965
$117$	−476.241	−0.376312
$118$	−2132.36	−1.66356
$119$	0	0
$120$	−555.602	−0.422661
$121$	121.000	0.0909091
$122$	−1203.37	−0.893019
$123$	−955.852	−0.700702
$124$	−355.892	−0.257742
$125$	−62.6052	−0.0447966
$126$	0	0
$127$	74.9450	0.0523645	0.0261822	−	0.999657i	$-0.491665\pi$
$127$	74.9450	0.0523645	0.0261822	+	0.999657i	$0.491665\pi$
$128$	−1381.61	−0.954048
$129$	171.784	0.117246
$130$	2955.96	1.99427
$131$	2636.30	1.75828	0.879141	−	0.476561i	$-0.158117\pi$
$131$	2636.30	1.75828	0.879141	+	0.476561i	$0.158117\pi$
$132$	−154.594	−0.101937
$133$	0	0
$134$	383.862	0.247467
$135$	−423.486	−0.269984
$136$	910.651	0.574174
$137$	1146.87	0.715210	0.357605	−	0.933873i	$-0.383593\pi$
$137$	1146.87	0.715210	0.357605	+	0.933873i	$0.383593\pi$
$138$	−630.557	−0.388961
$139$	1013.00	0.618139	0.309070	−	0.951039i	$-0.399982\pi$
$139$	1013.00	0.618139	0.309070	+	0.951039i	$0.399982\pi$
$140$	0	0
$141$	−1733.34	−1.03527
$142$	−1443.82	−0.853257
$143$	−582.073	−0.340387
$144$	−715.781	−0.414225
$145$	−1087.43	−0.622798
$146$	−474.756	−0.269117
$147$	0	0
$148$	−1573.80	−0.874092
$149$	−2236.91	−1.22990	−0.614948	−	0.788568i	$-0.710823\pi$
$149$	−2236.91	−1.22990	−0.614948	+	0.788568i	$0.710823\pi$
$150$	1292.93	0.703784
$151$	636.657	0.343115	0.171558	−	0.985174i	$-0.445120\pi$
$151$	636.657	0.343115	0.171558	+	0.985174i	$0.445120\pi$
$152$	−158.320	−0.0844833
$153$	694.108	0.366767
$154$	0	0
$155$	−1191.56	−0.617473
$156$	743.676	0.381678
$157$	3561.59	1.81048	0.905241	−	0.424899i	$-0.139690\pi$
$157$	3561.59	1.81048	0.905241	+	0.424899i	$0.139690\pi$
$158$	3283.89	1.65349
$159$	−946.665	−0.472172
$160$	2961.15	1.46312
$161$	0	0
$162$	−288.486	−0.139911
$163$	2540.60	1.22083	0.610414	−	0.792082i	$-0.291003\pi$
$163$	2540.60	1.22083	0.610414	+	0.792082i	$0.291003\pi$
$164$	1492.61	0.710692
$165$	−517.594	−0.244210
$166$	−4349.26	−2.03354
$167$	−1737.09	−0.804913	−0.402456	−	0.915439i	$-0.631843\pi$
$167$	−1737.09	−0.804913	−0.402456	+	0.915439i	$0.631843\pi$
$168$	0	0
$169$	603.073	0.274498
$170$	−4308.23	−1.94368
$171$	−120.673	−0.0539656
$172$	−268.250	−0.118918
$173$	−4194.96	−1.84357	−0.921784	−	0.387704i	$-0.873268\pi$
$173$	−4194.96	−1.84357	−0.921784	+	0.387704i	$0.873268\pi$
$174$	−740.773	−0.322746
$175$	0	0
$176$	−874.844	−0.374681
$177$	−1796.15	−0.762751
$178$	5627.94	2.36984
$179$	1349.85	0.563645	0.281822	−	0.959467i	$-0.409061\pi$
$179$	1349.85	0.563645	0.281822	+	0.959467i	$0.409061\pi$
$180$	661.295	0.273834
$181$	1198.06	0.491994	0.245997	−	0.969271i	$-0.420885\pi$
$181$	1198.06	0.491994	0.245997	+	0.969271i	$0.420885\pi$
$182$	0	0
$183$	−1013.64	−0.409454
$184$	−696.837	−0.279193
$185$	−5269.23	−2.09406
$186$	−811.710	−0.319986
$187$	848.354	0.331753
$188$	2706.70	1.05003
$189$	0	0
$190$	749.002	0.285991
$191$	764.873	0.289761	0.144880	−	0.989449i	$-0.453720\pi$
$191$	764.873	0.289761	0.144880	+	0.989449i	$0.453720\pi$
$192$	108.435	0.0407583
$193$	−2019.52	−0.753204	−0.376602	−	0.926375i	$-0.622907\pi$
$193$	−2019.52	−0.753204	−0.376602	+	0.926375i	$0.622907\pi$
$194$	−1022.51	−0.378412
$195$	2489.89	0.914385
$196$	0	0
$197$	1499.94	0.542470	0.271235	−	0.962513i	$-0.412568\pi$
$197$	1499.94	0.542470	0.271235	+	0.962513i	$0.412568\pi$
$198$	−352.594	−0.126554
$199$	4574.41	1.62950	0.814752	−	0.579809i	$-0.196873\pi$
$199$	4574.41	1.62950	0.814752	+	0.579809i	$0.196873\pi$
$200$	1428.84	0.505171
$201$	323.338	0.113465
$202$	2781.10	0.968701
$203$	0	0
$204$	−1083.89	−0.371996
$205$	4997.40	1.70260
$206$	4338.74	1.46745
$207$	−531.136	−0.178341
$208$	4208.45	1.40290
$209$	−147.490	−0.0488137
$210$	0	0
$211$	−4456.96	−1.45417	−0.727084	−	0.686548i	$-0.759125\pi$
$211$	−4456.96	−1.45417	−0.727084	+	0.686548i	$0.759125\pi$
$212$	1478.27	0.478905
$213$	−1216.17	−0.391224
$214$	3109.93	0.993414
$215$	−898.125	−0.284891
$216$	−318.810	−0.100427
$217$	0	0
$218$	−450.267	−0.139890
$219$	−399.901	−0.123392
$220$	808.250	0.247692
$221$	−4081.02	−1.24217
$222$	−3589.49	−1.08518
$223$	3003.50	0.901924	0.450962	−	0.892543i	$-0.351081\pi$
$223$	3003.50	0.901924	0.450962	+	0.892543i	$0.351081\pi$
$224$	0	0
$225$	1089.08	0.322689
$226$	−916.453	−0.269741
$227$	−1252.20	−0.366130	−0.183065	−	0.983101i	$-0.558602\pi$
$227$	−1252.20	−0.366130	−0.183065	+	0.983101i	$0.558602\pi$
$228$	188.438	0.0547350
$229$	−2862.80	−0.826110	−0.413055	−	0.910706i	$-0.635538\pi$
$229$	−2862.80	−0.826110	−0.413055	+	0.910706i	$0.635538\pi$
$230$	3296.69	0.945118
$231$	0	0
$232$	−818.638	−0.231665
$233$	4969.92	1.39738	0.698692	−	0.715423i	$-0.253766\pi$
$233$	4969.92	1.39738	0.698692	+	0.715423i	$0.253766\pi$
$234$	1696.16	0.473852
$235$	9062.27	2.51556
$236$	2804.78	0.773627
$237$	2766.11	0.758137
$238$	0	0
$239$	4602.18	1.24557	0.622783	−	0.782395i	$-0.286002\pi$
$239$	4602.18	1.24557	0.622783	+	0.782395i	$0.286002\pi$
$240$	3742.26	1.00651
$241$	4147.29	1.10851	0.554254	−	0.832348i	$-0.313004\pi$
$241$	4147.29	1.10851	0.554254	+	0.832348i	$0.313004\pi$
$242$	−430.948	−0.114473
$243$	−243.000	−0.0641500
$244$	1582.85	0.415292
$245$	0	0
$246$	3404.32	0.882323
$247$	709.501	0.182771
$248$	−897.032	−0.229684
$249$	−3663.51	−0.932392
$250$	222.972	0.0564078
$251$	−988.383	−0.248551	−0.124275	−	0.992248i	$-0.539661\pi$
$251$	−988.383	−0.248551	−0.124275	+	0.992248i	$0.539661\pi$
$252$	0	0
$253$	−649.167	−0.161315
$254$	−266.920	−0.0659373
$255$	−3628.95	−0.891191
$256$	5209.83	1.27193
$257$	2856.10	0.693224	0.346612	−	0.938009i	$-0.387332\pi$
$257$	2856.10	0.693224	0.346612	+	0.938009i	$0.387332\pi$
$258$	−611.818	−0.147636
$259$	0	0
$260$	−3888.10	−0.927423
$261$	−623.974	−0.147981
$262$	−9389.34	−2.21403
$263$	5772.45	1.35340	0.676701	−	0.736258i	$-0.263409\pi$
$263$	5772.45	1.35340	0.676701	+	0.736258i	$0.263409\pi$
$264$	−389.656	−0.0908397
$265$	4949.37	1.14731
$266$	0	0
$267$	4740.58	1.08659
$268$	−504.909	−0.115083
$269$	1626.04	0.368555	0.184278	−	0.982874i	$-0.441005\pi$
$269$	1626.04	0.368555	0.184278	+	0.982874i	$0.441005\pi$
$270$	1508.27	0.339964
$271$	1816.20	0.407109	0.203554	−	0.979064i	$-0.434751\pi$
$271$	1816.20	0.407109	0.203554	+	0.979064i	$0.434751\pi$
$272$	−6133.70	−1.36732
$273$	0	0
$274$	−4084.64	−0.900592
$275$	1331.09	0.291884
$276$	829.398	0.180884
$277$	−8091.98	−1.75523	−0.877617	−	0.479362i	$-0.840868\pi$
$277$	−8091.98	−1.75523	−0.877617	+	0.479362i	$0.840868\pi$
$278$	−3607.85	−0.778361
$279$	−683.727	−0.146716
$280$	0	0
$281$	−8041.32	−1.70713	−0.853567	−	0.520982i	$-0.825566\pi$
$281$	−8041.32	−1.70713	−0.853567	+	0.520982i	$0.825566\pi$
$282$	6173.37	1.30361
$283$	−644.465	−0.135369	−0.0676846	−	0.997707i	$-0.521561\pi$
$283$	−644.465	−0.135369	−0.0676846	+	0.997707i	$0.521561\pi$
$284$	1899.11	0.396802
$285$	630.906	0.131129
$286$	2073.08	0.428615
$287$	0	0
$288$	1699.13	0.347647
$289$	1034.97	0.210660
$290$	3872.92	0.784227
$291$	−861.290	−0.173504
$292$	624.466	0.125151
$293$	6451.96	1.28644	0.643221	−	0.765681i	$-0.277598\pi$
$293$	6451.96	1.28644	0.643221	+	0.765681i	$0.277598\pi$
$294$	0	0
$295$	9390.67	1.85338
$296$	−3966.79	−0.778936
$297$	−297.000	−0.0580259
$298$	7966.86	1.54868
$299$	3122.83	0.604006
$300$	−1700.65	−0.327290
$301$	0	0
$302$	−2267.49	−0.432051
$303$	2342.60	0.444155
$304$	1066.37	0.201185
$305$	5299.51	0.994916
$306$	−2472.10	−0.461832
$307$	−4175.92	−0.776327	−0.388164	−	0.921590i	$-0.626890\pi$
$307$	−4175.92	−0.776327	−0.388164	+	0.921590i	$0.626890\pi$
$308$	0	0
$309$	3654.65	0.672834
$310$	4243.80	0.777521
$311$	8619.22	1.57155	0.785774	−	0.618514i	$-0.212265\pi$
$311$	8619.22	1.57155	0.785774	+	0.618514i	$0.212265\pi$
$312$	1874.45	0.340127
$313$	10300.8	1.86019	0.930093	−	0.367324i	$-0.119726\pi$
$313$	10300.8	1.86019	0.930093	+	0.367324i	$0.119726\pi$
$314$	−12684.8	−2.27976
$315$	0	0
$316$	−4319.43	−0.768946
$317$	7696.26	1.36361	0.681806	−	0.731533i	$-0.261195\pi$
$317$	7696.26	1.36361	0.681806	+	0.731533i	$0.261195\pi$
$318$	3371.60	0.594559
$319$	−762.635	−0.133854
$320$	−566.920	−0.0990369
$321$	2619.59	0.455486
$322$	0	0
$323$	−1034.08	−0.178135
$324$	379.457	0.0650647
$325$	−6403.25	−1.09289
$326$	−9048.48	−1.53727
$327$	−379.273	−0.0641402
$328$	3762.16	0.633325
$329$	0	0
$330$	1843.44	0.307509
$331$	6322.41	1.04988	0.524941	−	0.851138i	$-0.324087\pi$
$331$	6322.41	1.04988	0.524941	+	0.851138i	$0.324087\pi$
$332$	5720.77	0.945686
$333$	−3023.53	−0.497563
$334$	6186.75	1.01355
$335$	−1690.48	−0.275704
$336$	0	0
$337$	1576.06	0.254758	0.127379	−	0.991854i	$-0.459344\pi$
$337$	1576.06	0.254758	0.127379	+	0.991854i	$0.459344\pi$
$338$	−2147.88	−0.345648
$339$	−771.955	−0.123678
$340$	5666.80	0.903897
$341$	−835.667	−0.132709
$342$	429.784	0.0679534
$343$	0	0
$344$	−676.129	−0.105972
$345$	2776.90	0.433342
$346$	14940.6	2.32142
$347$	4989.29	0.771870	0.385935	−	0.922526i	$-0.373879\pi$
$347$	4989.29	0.771870	0.385935	+	0.922526i	$0.373879\pi$
$348$	974.369	0.150091
$349$	−4348.98	−0.667035	−0.333518	−	0.942744i	$-0.608236\pi$
$349$	−4348.98	−0.667035	−0.333518	+	0.942744i	$0.608236\pi$
$350$	0	0
$351$	1428.72	0.217264
$352$	2076.72	0.314459
$353$	3724.71	0.561604	0.280802	−	0.959766i	$-0.409399\pi$
$353$	3724.71	0.561604	0.280802	+	0.959766i	$0.409399\pi$
$354$	6397.08	0.960455
$355$	6358.40	0.950617
$356$	−7402.66	−1.10208
$357$	0	0
$358$	−4807.56	−0.709741
$359$	−5815.53	−0.854963	−0.427481	−	0.904024i	$-0.640599\pi$
$359$	−5815.53	−0.854963	−0.427481	+	0.904024i	$0.640599\pi$
$360$	1666.81	0.244023
$361$	−6679.22	−0.973789
$362$	−4266.94	−0.619518
$363$	−363.000	−0.0524864
$364$	0	0
$365$	2090.77	0.299824
$366$	3610.12	0.515585
$367$	−9298.43	−1.32255	−0.661273	−	0.750146i	$-0.729983\pi$
$367$	−9298.43	−1.32255	−0.661273	+	0.750146i	$0.729983\pi$
$368$	4693.55	0.664859
$369$	2867.56	0.404550
$370$	18766.6	2.63684
$371$	0	0
$372$	1067.68	0.148808
$373$	3697.85	0.513318	0.256659	−	0.966502i	$-0.417378\pi$
$373$	3697.85	0.513318	0.256659	+	0.966502i	$0.417378\pi$
$374$	−3021.46	−0.417743
$375$	187.815	0.0258633
$376$	6822.28	0.935724
$377$	3668.67	0.501184
$378$	0	0
$379$	2040.70	0.276580	0.138290	−	0.990392i	$-0.455839\pi$
$379$	2040.70	0.276580	0.138290	+	0.990392i	$0.455839\pi$
$380$	−985.193	−0.132998
$381$	−224.835	−0.0302327
$382$	−2724.14	−0.364866
$383$	3311.92	0.441857	0.220928	−	0.975290i	$-0.429091\pi$
$383$	3311.92	0.441857	0.220928	+	0.975290i	$0.429091\pi$
$384$	4144.83	0.550820
$385$	0	0
$386$	7192.63	0.948433
$387$	−515.352	−0.0676921
$388$	1344.95	0.175978
$389$	13095.1	1.70680	0.853401	−	0.521255i	$-0.174536\pi$
$389$	13095.1	1.70680	0.853401	+	0.521255i	$0.174536\pi$
$390$	−8867.89	−1.15139
$391$	−4551.43	−0.588685
$392$	0	0
$393$	−7908.91	−1.01514
$394$	−5342.13	−0.683077
$395$	−14461.8	−1.84216
$396$	463.781	0.0588532
$397$	−15359.1	−1.94169	−0.970843	−	0.239716i	$-0.922946\pi$
$397$	−15359.1	−1.94169	−0.970843	+	0.239716i	$0.922946\pi$
$398$	−16292.0	−2.05187
$399$	0	0
$400$	−9623.96	−1.20299
$401$	−11767.6	−1.46545	−0.732726	−	0.680524i	$-0.761752\pi$
$401$	−11767.6	−1.46545	−0.732726	+	0.680524i	$0.761752\pi$
$402$	−1151.59	−0.142875
$403$	4019.99	0.496898
$404$	−3658.10	−0.450488
$405$	1270.46	0.155875
$406$	0	0
$407$	−3695.43	−0.450063
$408$	−2731.95	−0.331500
$409$	−5071.33	−0.613108	−0.306554	−	0.951853i	$-0.599176\pi$
$409$	−5071.33	−0.613108	−0.306554	+	0.951853i	$0.599176\pi$
$410$	−17798.5	−2.14392
$411$	−3440.61	−0.412927
$412$	−5706.93	−0.682427
$413$	0	0
$414$	1891.67	0.224566
$415$	19153.6	2.26558
$416$	−9990.10	−1.17742
$417$	−3038.99	−0.356883
$418$	525.292	0.0614662
$419$	−5085.27	−0.592916	−0.296458	−	0.955046i	$-0.595806\pi$
$419$	−5085.27	−0.592916	−0.296458	+	0.955046i	$0.595806\pi$
$420$	0	0
$421$	15655.8	1.81239	0.906196	−	0.422859i	$-0.138973\pi$
$421$	15655.8	1.81239	0.906196	+	0.422859i	$0.138973\pi$
$422$	15873.7	1.83109
$423$	5200.01	0.597715
$424$	3726.00	0.426770
$425$	9332.55	1.06517
$426$	4331.45	0.492628
$427$	0	0
$428$	−4090.62	−0.461981
$429$	1746.22	0.196523
$430$	3198.72	0.358735
$431$	5775.20	0.645433	0.322717	−	0.946496i	$-0.395404\pi$
$431$	5775.20	0.645433	0.322717	+	0.946496i	$0.395404\pi$
$432$	2147.34	0.239153
$433$	6756.97	0.749929	0.374964	−	0.927039i	$-0.377655\pi$
$433$	6756.97	0.749929	0.374964	+	0.927039i	$0.377655\pi$
$434$	0	0
$435$	3262.28	0.359573
$436$	592.255	0.0650547
$437$	791.283	0.0866183
$438$	1424.27	0.155375
$439$	9068.21	0.985882	0.492941	−	0.870063i	$-0.335922\pi$
$439$	9068.21	0.985882	0.492941	+	0.870063i	$0.335922\pi$
$440$	2037.21	0.220727
$441$	0	0
$442$	14534.8	1.56414
$443$	−7095.86	−0.761026	−0.380513	−	0.924776i	$-0.624252\pi$
$443$	−7095.86	−0.761026	−0.380513	+	0.924776i	$0.624252\pi$
$444$	4721.40	0.504657
$445$	−24784.8	−2.64025
$446$	−10697.1	−1.13570
$447$	6710.72	0.710081
$448$	0	0
$449$	630.680	0.0662887	0.0331443	−	0.999451i	$-0.489448\pi$
$449$	630.680	0.0662887	0.0331443	+	0.999451i	$0.489448\pi$
$450$	−3878.80	−0.406330
$451$	3504.79	0.365929
$452$	1205.45	0.125441
$453$	−1909.97	−0.198098
$454$	4459.78	0.461030
$455$	0	0
$456$	474.960	0.0487764
$457$	12432.5	1.27258	0.636290	−	0.771450i	$-0.280468\pi$
$457$	12432.5	1.27258	0.636290	+	0.771450i	$0.280468\pi$
$458$	10196.0	1.04024
$459$	−2082.32	−0.211753
$460$	−4336.27	−0.439521
$461$	−13057.3	−1.31918	−0.659588	−	0.751627i	$-0.729269\pi$
$461$	−13057.3	−1.31918	−0.659588	+	0.751627i	$0.729269\pi$
$462$	0	0
$463$	8608.12	0.864046	0.432023	−	0.901863i	$-0.357800\pi$
$463$	8608.12	0.864046	0.432023	+	0.901863i	$0.357800\pi$
$464$	5513.94	0.551677
$465$	3574.68	0.356498
$466$	−17700.6	−1.75958
$467$	−14997.6	−1.48610	−0.743049	−	0.669237i	$-0.766621\pi$
$467$	−14997.6	−1.48610	−0.743049	+	0.669237i	$0.766621\pi$
$468$	−2231.03	−0.220362
$469$	0	0
$470$	−32275.8	−3.16760
$471$	−10684.8	−1.04528
$472$	7069.51	0.689408
$473$	−629.875	−0.0612298
$474$	−9851.66	−0.954645
$475$	−1622.50	−0.156727
$476$	0	0
$477$	2839.99	0.272609
$478$	−16390.9	−1.56841
$479$	20698.8	1.97443	0.987216	−	0.159386i	$-0.0509515\pi$
$479$	20698.8	1.97443	0.987216	+	0.159386i	$0.0509515\pi$
$480$	−8883.44	−0.844733
$481$	17776.9	1.68515
$482$	−14770.8	−1.39583
$483$	0	0
$484$	566.844	0.0532348
$485$	4503.01	0.421590
$486$	865.457	0.0807777
$487$	15390.1	1.43202	0.716008	−	0.698092i	$-0.245968\pi$
$487$	15390.1	1.43202	0.716008	+	0.698092i	$0.245968\pi$
$488$	3989.59	0.370083
$489$	−7621.80	−0.704846
$490$	0	0
$491$	10771.4	0.990030	0.495015	−	0.868884i	$-0.335163\pi$
$491$	10771.4	0.990030	0.495015	+	0.868884i	$0.335163\pi$
$492$	−4477.84	−0.410318
$493$	−5346.98	−0.488471
$494$	−2526.93	−0.230145
$495$	1552.78	0.140995
$496$	6041.96	0.546960
$497$	0	0
$498$	13047.8	1.17407
$499$	−5999.79	−0.538251	−0.269126	−	0.963105i	$-0.586735\pi$
$499$	−5999.79	−0.538251	−0.269126	+	0.963105i	$0.586735\pi$
$500$	−293.284	−0.0262321
$501$	5211.28	0.464717
$502$	3520.18	0.312975
$503$	−17610.5	−1.56106	−0.780528	−	0.625121i	$-0.785050\pi$
$503$	−17610.5	−1.56106	−0.780528	+	0.625121i	$0.785050\pi$
$504$	0	0
$505$	−12247.6	−1.07923
$506$	2312.04	0.203128
$507$	−1809.22	−0.158482
$508$	351.092	0.0306637
$509$	5920.42	0.515556	0.257778	−	0.966204i	$-0.417010\pi$
$509$	5920.42	0.515556	0.257778	+	0.966204i	$0.417010\pi$
$510$	12924.7	1.12219
$511$	0	0
$512$	−7502.22	−0.647567
$513$	362.020	0.0311570
$514$	−10172.1	−0.872907
$515$	−19107.3	−1.63489
$516$	804.750	0.0686572
$517$	6355.57	0.540653
$518$	0	0
$519$	12584.9	1.06438
$520$	−9800.03	−0.826461
$521$	15886.2	1.33587	0.667935	−	0.744220i	$-0.267179\pi$
$521$	15886.2	1.33587	0.667935	+	0.744220i	$0.267179\pi$
$522$	2222.32	0.186338
$523$	−1728.58	−0.144523	−0.0722613	−	0.997386i	$-0.523022\pi$
$523$	−1728.58	−0.144523	−0.0722613	+	0.997386i	$0.523022\pi$
$524$	12350.2	1.02962
$525$	0	0
$526$	−20558.9	−1.70420
$527$	−5859.02	−0.484294
$528$	2624.53	0.216322
$529$	−8684.21	−0.713751
$530$	−17627.4	−1.44469
$531$	5388.45	0.440375
$532$	0	0
$533$	−16859.9	−1.37013
$534$	−16883.8	−1.36823
$535$	−13695.8	−1.10677
$536$	−1272.63	−0.102555
$537$	−4049.55	−0.325420
$538$	−5791.23	−0.464085
$539$	0	0
$540$	−1983.89	−0.158098
$541$	1941.99	0.154331	0.0771653	−	0.997018i	$-0.475413\pi$
$541$	1941.99	0.154331	0.0771653	+	0.997018i	$0.475413\pi$
$542$	−6468.51	−0.512631
$543$	−3594.17	−0.284053
$544$	14560.3	1.14755
$545$	1982.92	0.155852
$546$	0	0
$547$	−1450.56	−0.113384	−0.0566922	−	0.998392i	$-0.518055\pi$
$547$	−1450.56	−0.113384	−0.0566922	+	0.998392i	$0.518055\pi$
$548$	5372.70	0.418814
$549$	3040.91	0.236399
$550$	−4740.76	−0.367539
$551$	929.593	0.0718729
$552$	2090.51	0.161192
$553$	0	0
$554$	28820.0	2.21019
$555$	15807.7	1.20901
$556$	4745.55	0.361972
$557$	20821.8	1.58393	0.791965	−	0.610567i	$-0.209058\pi$
$557$	20821.8	1.58393	0.791965	+	0.610567i	$0.209058\pi$
$558$	2435.13	0.184744
$559$	3030.03	0.229260
$560$	0	0
$561$	−2545.06	−0.191538
$562$	28639.6	2.14962
$563$	3694.11	0.276533	0.138267	−	0.990395i	$-0.455847\pi$
$563$	3694.11	0.276533	0.138267	+	0.990395i	$0.455847\pi$
$564$	−8120.10	−0.606237
$565$	4035.95	0.300520
$566$	2295.30	0.170457
$567$	0	0
$568$	4786.75	0.353605
$569$	−16649.2	−1.22666	−0.613332	−	0.789825i	$-0.710171\pi$
$569$	−16649.2	−1.22666	−0.613332	+	0.789825i	$0.710171\pi$
$570$	−2247.01	−0.165117
$571$	546.005	0.0400168	0.0200084	−	0.999800i	$-0.493631\pi$
$571$	546.005	0.0400168	0.0200084	+	0.999800i	$0.493631\pi$
$572$	−2726.81	−0.199325
$573$	−2294.62	−0.167293
$574$	0	0
$575$	−7141.34	−0.517938
$576$	−325.304	−0.0235318
$577$	−7910.36	−0.570732	−0.285366	−	0.958419i	$-0.592115\pi$
$577$	−7910.36	−0.570732	−0.285366	+	0.958419i	$0.592115\pi$
$578$	−3686.11	−0.265263
$579$	6058.56	0.434862
$580$	−5094.22	−0.364700
$581$	0	0
$582$	3067.53	0.218476
$583$	3471.10	0.246584
$584$	1573.98	0.111527
$585$	−7469.68	−0.527920
$586$	−22979.0	−1.61989
$587$	744.718	0.0523642	0.0261821	−	0.999657i	$-0.491665\pi$
$587$	744.718	0.0523642	0.0261821	+	0.999657i	$0.491665\pi$
$588$	0	0
$589$	1018.61	0.0712584
$590$	−33445.4	−2.33377
$591$	−4499.83	−0.313195
$592$	26718.4	1.85493
$593$	8167.66	0.565608	0.282804	−	0.959178i	$-0.408735\pi$
$593$	8167.66	0.565608	0.282804	+	0.959178i	$0.408735\pi$
$594$	1057.78	0.0730661
$595$	0	0
$596$	−10479.1	−0.720205
$597$	−13723.2	−0.940795
$598$	−11122.1	−0.760564
$599$	12115.4	0.826413	0.413206	−	0.910637i	$-0.364409\pi$
$599$	12115.4	0.826413	0.413206	+	0.910637i	$0.364409\pi$
$600$	−4286.52	−0.291661
$601$	−13606.0	−0.923458	−0.461729	−	0.887021i	$-0.652771\pi$
$601$	−13606.0	−0.923458	−0.461729	+	0.887021i	$0.652771\pi$
$602$	0	0
$603$	−970.014	−0.0655092
$604$	2982.52	0.200922
$605$	1897.84	0.127534
$606$	−8343.30	−0.559280
$607$	11404.4	0.762585	0.381293	−	0.924454i	$-0.375479\pi$
$607$	11404.4	0.762585	0.381293	+	0.924454i	$0.375479\pi$
$608$	−2531.36	−0.168849
$609$	0	0
$610$	−18874.5	−1.25280
$611$	−30573.6	−2.02435
$612$	3251.66	0.214772
$613$	15266.2	1.00587	0.502934	−	0.864325i	$-0.332254\pi$
$613$	15266.2	1.00587	0.502934	+	0.864325i	$0.332254\pi$
$614$	14872.8	0.977550
$615$	−14992.2	−0.982999
$616$	0	0
$617$	23638.9	1.54241	0.771205	−	0.636586i	$-0.219654\pi$
$617$	23638.9	1.54241	0.771205	+	0.636586i	$0.219654\pi$
$618$	−13016.2	−0.847232
$619$	−27681.1	−1.79741	−0.898706	−	0.438551i	$-0.855492\pi$
$619$	−27681.1	−1.79741	−0.898706	+	0.438551i	$0.855492\pi$
$620$	−5582.05	−0.361581
$621$	1593.41	0.102965
$622$	−30697.8	−1.97889
$623$	0	0
$624$	−12625.4	−0.809966
$625$	−16108.0	−1.03091
$626$	−36687.0	−2.34235
$627$	442.469	0.0281826
$628$	16684.8	1.06019
$629$	−25909.3	−1.64241
$630$	0	0
$631$	18208.7	1.14877	0.574386	−	0.818585i	$-0.305241\pi$
$631$	18208.7	1.14877	0.574386	+	0.818585i	$0.305241\pi$
$632$	−10887.2	−0.685237
$633$	13370.9	0.839564
$634$	−27410.6	−1.71706
$635$	1175.49	0.0734610
$636$	−4434.80	−0.276496
$637$	0	0
$638$	2716.17	0.168549
$639$	3648.51	0.225873
$640$	−21670.1	−1.33841
$641$	−23373.8	−1.44026	−0.720131	−	0.693838i	$-0.755918\pi$
$641$	−23373.8	−1.44026	−0.720131	+	0.693838i	$0.755918\pi$
$642$	−9329.80	−0.573548
$643$	−6742.72	−0.413541	−0.206770	−	0.978389i	$-0.566295\pi$
$643$	−6742.72	−0.413541	−0.206770	+	0.978389i	$0.566295\pi$
$644$	0	0
$645$	2694.37	0.164482
$646$	3682.92	0.224307
$647$	17592.4	1.06898	0.534490	−	0.845175i	$-0.320504\pi$
$647$	17592.4	1.06898	0.534490	+	0.845175i	$0.320504\pi$
$648$	956.429	0.0579816
$649$	6585.89	0.398334
$650$	22805.5	1.37616
$651$	0	0
$652$	11901.8	0.714896
$653$	20074.0	1.20300	0.601498	−	0.798875i	$-0.294571\pi$
$653$	20074.0	1.20300	0.601498	+	0.798875i	$0.294571\pi$
$654$	1350.80	0.0807653
$655$	41349.5	2.46666
$656$	−25340.0	−1.50817
$657$	1199.70	0.0712402
$658$	0	0
$659$	−13778.4	−0.814460	−0.407230	−	0.913326i	$-0.633505\pi$
$659$	−13778.4	−0.814460	−0.407230	+	0.913326i	$0.633505\pi$
$660$	−2424.75	−0.143005
$661$	7724.16	0.454516	0.227258	−	0.973835i	$-0.427024\pi$
$661$	7724.16	0.454516	0.227258	+	0.973835i	$0.427024\pi$
$662$	−22517.6	−1.32201
$663$	12243.1	0.717167
$664$	14419.3	0.842737
$665$	0	0
$666$	10768.5	0.626531
$667$	4091.55	0.237519
$668$	−8137.69	−0.471343
$669$	−9010.49	−0.520726
$670$	6020.74	0.347166
$671$	3716.67	0.213831
$672$	0	0
$673$	−18081.2	−1.03563	−0.517816	−	0.855492i	$-0.673255\pi$
$673$	−18081.2	−1.03563	−0.517816	+	0.855492i	$0.673255\pi$
$674$	−5613.21	−0.320791
$675$	−3267.23	−0.186305
$676$	2825.19	0.160741
$677$	23199.8	1.31705	0.658523	−	0.752561i	$-0.271182\pi$
$677$	23199.8	1.31705	0.658523	+	0.752561i	$0.271182\pi$
$678$	2749.36	0.155735
$679$	0	0
$680$	14283.3	0.805497
$681$	3756.60	0.211385
$682$	2976.27	0.167107
$683$	13339.6	0.747329	0.373665	−	0.927564i	$-0.378101\pi$
$683$	13339.6	0.747329	0.373665	+	0.927564i	$0.378101\pi$
$684$	−565.313	−0.0316013
$685$	17988.3	1.00335
$686$	0	0
$687$	8588.40	0.476955
$688$	4554.07	0.252358
$689$	−16697.8	−0.923274
$690$	−9890.07	−0.545664
$691$	17221.4	0.948097	0.474048	−	0.880499i	$-0.342792\pi$
$691$	17221.4	0.948097	0.474048	+	0.880499i	$0.342792\pi$
$692$	−19652.0	−1.07956
$693$	0	0
$694$	−17769.6	−0.971938
$695$	15888.5	0.867175
$696$	2455.91	0.133752
$697$	24572.8	1.33538
$698$	15489.1	0.839930
$699$	−14909.8	−0.806780
$700$	0	0
$701$	−27196.5	−1.46533	−0.732665	−	0.680589i	$-0.761724\pi$
$701$	−27196.5	−1.46533	−0.732665	+	0.680589i	$0.761724\pi$
$702$	−5088.48	−0.273579
$703$	4504.44	0.241661
$704$	−397.594	−0.0212853
$705$	−27186.8	−1.45236
$706$	−13265.8	−0.707172
$707$	0	0
$708$	−8414.35	−0.446654
$709$	22682.8	1.20151	0.600755	−	0.799434i	$-0.294867\pi$
$709$	22682.8	1.20151	0.600755	+	0.799434i	$0.294867\pi$
$710$	−22645.8	−1.19702
$711$	−8298.34	−0.437710
$712$	−18658.5	−0.982105
$713$	4483.36	0.235488
$714$	0	0
$715$	−9129.61	−0.477522
$716$	6323.58	0.330060
$717$	−13806.5	−0.719127
$718$	20712.3	1.07657
$719$	−10878.2	−0.564241	−0.282121	−	0.959379i	$-0.591038\pi$
$719$	−10878.2	−0.564241	−0.282121	+	0.959379i	$0.591038\pi$
$720$	−11226.8	−0.581108
$721$	0	0
$722$	23788.4	1.22619
$723$	−12441.9	−0.639997
$724$	5612.49	0.288103
$725$	−8389.58	−0.429767
$726$	1292.84	0.0660908
$727$	2793.35	0.142503	0.0712514	−	0.997458i	$-0.477301\pi$
$727$	2793.35	0.142503	0.0712514	+	0.997458i	$0.477301\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−7446.38	−0.377538
$731$	−4416.17	−0.223445
$732$	−4748.54	−0.239769
$733$	7597.31	0.382828	0.191414	−	0.981509i	$-0.438693\pi$
$733$	7597.31	0.382828	0.191414	+	0.981509i	$0.438693\pi$
$734$	33116.9	1.66535
$735$	0	0
$736$	−11141.6	−0.557997
$737$	−1185.57	−0.0592553
$738$	−10213.0	−0.509409
$739$	−14304.7	−0.712052	−0.356026	−	0.934476i	$-0.615868\pi$
$739$	−14304.7	−0.712052	−0.356026	+	0.934476i	$0.615868\pi$
$740$	−24684.5	−1.22625
$741$	−2128.50	−0.105523
$742$	0	0
$743$	31212.2	1.54114	0.770569	−	0.637357i	$-0.219972\pi$
$743$	31212.2	1.54114	0.770569	+	0.637357i	$0.219972\pi$
$744$	2691.10	0.132608
$745$	−35085.1	−1.72539
$746$	−13170.1	−0.646369
$747$	10990.5	0.538317
$748$	3974.25	0.194269
$749$	0	0
$750$	−668.915	−0.0325671
$751$	30743.1	1.49378	0.746891	−	0.664946i	$-0.231546\pi$
$751$	30743.1	1.49378	0.746891	+	0.664946i	$0.231546\pi$
$752$	−45951.5	−2.22830
$753$	2965.15	0.143501
$754$	−13066.2	−0.631090
$755$	9985.75	0.481349
$756$	0	0
$757$	13176.9	0.632657	0.316328	−	0.948650i	$-0.397550\pi$
$757$	13176.9	0.632657	0.316328	+	0.948650i	$0.397550\pi$
$758$	−7268.07	−0.348270
$759$	1947.50	0.0931354
$760$	−2483.20	−0.118520
$761$	10696.4	0.509520	0.254760	−	0.967004i	$-0.418004\pi$
$761$	10696.4	0.509520	0.254760	+	0.967004i	$0.418004\pi$
$762$	800.761	0.0380689
$763$	0	0
$764$	3583.17	0.169679
$765$	10886.8	0.514529
$766$	−11795.6	−0.556385
$767$	−31681.5	−1.49147
$768$	−15629.5	−0.734350
$769$	7724.46	0.362225	0.181113	−	0.983462i	$-0.442030\pi$
$769$	7724.46	0.362225	0.181113	+	0.983462i	$0.442030\pi$
$770$	0	0
$771$	−8568.30	−0.400233
$772$	−9460.77	−0.441063
$773$	−28801.0	−1.34010	−0.670051	−	0.742315i	$-0.733728\pi$
$773$	−28801.0	−1.34010	−0.670051	+	0.742315i	$0.733728\pi$
$774$	1835.45	0.0852378
$775$	−9192.98	−0.426092
$776$	3389.97	0.156821
$777$	0	0
$778$	−46638.7	−2.14920
$779$	−4272.07	−0.196486
$780$	11664.3	0.535448
$781$	4459.29	0.204310
$782$	16210.2	0.741272
$783$	1871.92	0.0854369
$784$	0	0
$785$	55862.3	2.53989
$786$	28168.0	1.27827
$787$	11812.0	0.535007	0.267504	−	0.963557i	$-0.413801\pi$
$787$	11812.0	0.535007	0.267504	+	0.963557i	$0.413801\pi$
$788$	7026.72	0.317661
$789$	−17317.4	−0.781387
$790$	51506.6	2.31965
$791$	0	0
$792$	1168.97	0.0524463
$793$	−17879.1	−0.800637
$794$	54702.1	2.44497
$795$	−14848.1	−0.662400
$796$	21429.6	0.954209
$797$	26884.2	1.19484	0.597420	−	0.801928i	$-0.296192\pi$
$797$	26884.2	1.19484	0.597420	+	0.801928i	$0.296192\pi$
$798$	0	0
$799$	44560.1	1.97300
$800$	22845.5	1.00964
$801$	−14221.7	−0.627341
$802$	41910.9	1.84530
$803$	1466.30	0.0644392
$804$	1514.73	0.0664432
$805$	0	0
$806$	−14317.4	−0.625694
$807$	−4878.12	−0.212786
$808$	−9220.30	−0.401447
$809$	−18360.8	−0.797936	−0.398968	−	0.916965i	$-0.630632\pi$
$809$	−18360.8	−0.797936	−0.398968	+	0.916965i	$0.630632\pi$
$810$	−4524.80	−0.196278
$811$	−17033.7	−0.737528	−0.368764	−	0.929523i	$-0.620219\pi$
$811$	−17033.7	−0.737528	−0.368764	+	0.929523i	$0.620219\pi$
$812$	0	0
$813$	−5448.61	−0.235044
$814$	13161.5	0.566718
$815$	39848.4	1.71267
$816$	18401.1	0.789420
$817$	767.768	0.0328774
$818$	18061.8	0.772026
$819$	0	0
$820$	23411.1	0.997015
$821$	−29675.1	−1.26147	−0.630735	−	0.775998i	$-0.717246\pi$
$821$	−29675.1	−1.26147	−0.630735	+	0.775998i	$0.717246\pi$
$822$	12253.9	0.519957
$823$	−24677.2	−1.04519	−0.522596	−	0.852581i	$-0.675036\pi$
$823$	−24677.2	−1.04519	−0.522596	+	0.852581i	$0.675036\pi$
$824$	−14384.4	−0.608137
$825$	−3993.28	−0.168519
$826$	0	0
$827$	−9168.43	−0.385511	−0.192755	−	0.981247i	$-0.561742\pi$
$827$	−9168.43	−0.385511	−0.192755	+	0.981247i	$0.561742\pi$
$828$	−2488.19	−0.104433
$829$	−8065.03	−0.337889	−0.168945	−	0.985626i	$-0.554036\pi$
$829$	−8065.03	−0.337889	−0.168945	+	0.985626i	$0.554036\pi$
$830$	−68216.7	−2.85281
$831$	24275.9	1.01338
$832$	1912.63	0.0796979
$833$	0	0
$834$	10823.5	0.449387
$835$	−27245.7	−1.12919
$836$	−690.938	−0.0285844
$837$	2051.18	0.0847063
$838$	18111.5	0.746599
$839$	29665.7	1.22071	0.610354	−	0.792129i	$-0.291027\pi$
$839$	29665.7	1.22071	0.610354	+	0.792129i	$0.291027\pi$
$840$	0	0
$841$	−19582.3	−0.802915
$842$	−55758.9	−2.28216
$843$	24124.0	0.985615
$844$	−20879.3	−0.851535
$845$	9458.99	0.385088
$846$	−18520.1	−0.752642
$847$	0	0
$848$	−25096.5	−1.01629
$849$	1933.39	0.0781554
$850$	−33238.4	−1.34126
$851$	19826.0	0.798622
$852$	−5697.34	−0.229094
$853$	33087.0	1.32811	0.664056	−	0.747683i	$-0.268834\pi$
$853$	33087.0	1.32811	0.664056	+	0.747683i	$0.268834\pi$
$854$	0	0
$855$	−1892.72	−0.0757071
$856$	−10310.5	−0.411689
$857$	8204.78	0.327036	0.163518	−	0.986540i	$-0.447716\pi$
$857$	8204.78	0.327036	0.163518	+	0.986540i	$0.447716\pi$
$858$	−6219.25	−0.247461
$859$	−19406.0	−0.770808	−0.385404	−	0.922748i	$-0.625938\pi$
$859$	−19406.0	−0.770808	−0.385404	+	0.922748i	$0.625938\pi$
$860$	−4207.41	−0.166827
$861$	0	0
$862$	−20568.7	−0.812729
$863$	−3352.45	−0.132235	−0.0661174	−	0.997812i	$-0.521061\pi$
$863$	−3352.45	−0.132235	−0.0661174	+	0.997812i	$0.521061\pi$
$864$	−5097.40	−0.200714
$865$	−65796.6	−2.58630
$866$	−24065.3	−0.944310
$867$	−3104.92	−0.121625
$868$	0	0
$869$	−10142.4	−0.395924
$870$	−11618.8	−0.452774
$871$	5703.22	0.221867
$872$	1492.79	0.0579727
$873$	2583.87	0.100173
$874$	−2818.20	−0.109070
$875$	0	0
$876$	−1873.40	−0.0722560
$877$	−20076.9	−0.773033	−0.386516	−	0.922283i	$-0.626322\pi$
$877$	−20076.9	−0.773033	−0.386516	+	0.922283i	$0.626322\pi$
$878$	−32296.9	−1.24142
$879$	−19355.9	−0.742728
$880$	−13721.6	−0.525632
$881$	14800.2	0.565982	0.282991	−	0.959123i	$-0.408673\pi$
$881$	14800.2	0.565982	0.282991	+	0.959123i	$0.408673\pi$
$882$	0	0
$883$	−19955.4	−0.760534	−0.380267	−	0.924877i	$-0.624168\pi$
$883$	−19955.4	−0.760534	−0.380267	+	0.924877i	$0.624168\pi$
$884$	−19118.2	−0.727392
$885$	−28172.0	−1.07005
$886$	25272.3	0.958283
$887$	9205.47	0.348466	0.174233	−	0.984704i	$-0.444255\pi$
$887$	9205.47	0.348466	0.174233	+	0.984704i	$0.444255\pi$
$888$	11900.4	0.449719
$889$	0	0
$890$	88272.3	3.32460
$891$	891.000	0.0335013
$892$	14070.4	0.528151
$893$	−7746.94	−0.290304
$894$	−23900.6	−0.894133
$895$	21171.9	0.790725
$896$	0	0
$897$	−9368.49	−0.348723
$898$	−2246.20	−0.0834706
$899$	5267.02	0.195400
$900$	5101.95	0.188961
$901$	24336.6	0.899854
$902$	−12482.5	−0.460778
$903$	0	0
$904$	3038.35	0.111786
$905$	18791.1	0.690207
$906$	6802.47	0.249445
$907$	6722.78	0.246115	0.123057	−	0.992400i	$-0.460730\pi$
$907$	6722.78	0.246115	0.123057	+	0.992400i	$0.460730\pi$
$908$	−5866.13	−0.214399
$909$	−7027.81	−0.256433
$910$	0	0
$911$	51855.3	1.88589	0.942944	−	0.332952i	$-0.108045\pi$
$911$	51855.3	1.88589	0.942944	+	0.332952i	$0.108045\pi$
$912$	−3199.10	−0.116154
$913$	13432.9	0.486926
$914$	−44279.1	−1.60243
$915$	−15898.5	−0.574415
$916$	−13411.2	−0.483755
$917$	0	0
$918$	7416.31	0.266639
$919$	−55119.5	−1.97848	−0.989240	−	0.146302i	$-0.953263\pi$
$919$	−55119.5	−1.97848	−0.989240	+	0.146302i	$0.953263\pi$
$920$	−10929.7	−0.391674
$921$	12527.8	0.448213
$922$	46504.4	1.66111
$923$	−21451.5	−0.764989
$924$	0	0
$925$	−40652.6	−1.44502
$926$	−30658.3	−1.08801
$927$	−10963.9	−0.388461
$928$	−13089.1	−0.463007
$929$	−31459.7	−1.11104	−0.555522	−	0.831502i	$-0.687482\pi$
$929$	−31459.7	−1.11104	−0.555522	+	0.831502i	$0.687482\pi$
$930$	−12731.4	−0.448902
$931$	0	0
$932$	23282.4	0.818283
$933$	−25857.7	−0.907333
$934$	53414.9	1.87129
$935$	13306.1	0.465409
$936$	−5623.35	−0.196373
$937$	13745.6	0.479242	0.239621	−	0.970867i	$-0.422977\pi$
$937$	13745.6	0.479242	0.239621	+	0.970867i	$0.422977\pi$
$938$	0	0
$939$	−30902.5	−1.07398
$940$	42453.6	1.47307
$941$	29814.2	1.03285	0.516427	−	0.856331i	$-0.327262\pi$
$941$	29814.2	1.03285	0.516427	+	0.856331i	$0.327262\pi$
$942$	38054.3	1.31622
$943$	−18803.3	−0.649330
$944$	−47616.7	−1.64173
$945$	0	0
$946$	2243.33	0.0771005
$947$	32410.4	1.11214	0.556070	−	0.831135i	$-0.312309\pi$
$947$	32410.4	1.11214	0.556070	+	0.831135i	$0.312309\pi$
$948$	12958.3	0.443951
$949$	−7053.67	−0.241277
$950$	5778.61	0.197351
$951$	−23088.8	−0.787281
$952$	0	0
$953$	8931.28	0.303581	0.151790	−	0.988413i	$-0.451496\pi$
$953$	8931.28	0.303581	0.151790	+	0.988413i	$0.451496\pi$
$954$	−10114.8	−0.343269
$955$	11996.8	0.406499
$956$	21559.6	0.729381
$957$	2287.91	0.0772806
$958$	−73719.9	−2.48620
$959$	0	0
$960$	1700.76	0.0571790
$961$	−24019.6	−0.806271
$962$	−63313.4	−2.12194
$963$	−7858.76	−0.262975
$964$	19428.6	0.649122
$965$	−31675.5	−1.05665
$966$	0	0
$967$	28665.0	0.953262	0.476631	−	0.879104i	$-0.341858\pi$
$967$	28665.0	0.953262	0.476631	+	0.879104i	$0.341858\pi$
$968$	1428.74	0.0474395
$969$	3102.23	0.102846
$970$	−16037.7	−0.530866
$971$	40925.2	1.35258	0.676288	−	0.736637i	$-0.263587\pi$
$971$	40925.2	1.35258	0.676288	+	0.736637i	$0.263587\pi$
$972$	−1138.37	−0.0375651
$973$	0	0
$974$	−54812.6	−1.80319
$975$	19209.8	0.630979
$976$	−26871.9	−0.881300
$977$	−21288.8	−0.697124	−0.348562	−	0.937286i	$-0.613330\pi$
$977$	−21288.8	−0.697124	−0.348562	+	0.937286i	$0.613330\pi$
$978$	27145.4	0.887541
$979$	−17382.1	−0.567452
$980$	0	0
$981$	1137.82	0.0370314
$982$	−38362.8	−1.24664
$983$	53128.8	1.72385	0.861925	−	0.507035i	$-0.169259\pi$
$983$	53128.8	1.72385	0.861925	+	0.507035i	$0.169259\pi$
$984$	−11286.5	−0.365650
$985$	23526.1	0.761019
$986$	19043.6	0.615082
$987$	0	0
$988$	3323.77	0.107028
$989$	3379.29	0.108650
$990$	−5530.31	−0.177540
$991$	−17978.4	−0.576290	−0.288145	−	0.957587i	$-0.593039\pi$
$991$	−17978.4	−0.576290	−0.288145	+	0.957587i	$0.593039\pi$
$992$	−14342.5	−0.459048
$993$	−18967.2	−0.606150
$994$	0	0
$995$	71748.1	2.28600
$996$	−17162.3	−0.545992
$997$	−10063.9	−0.319687	−0.159844	−	0.987142i	$-0.551099\pi$
$997$	−10063.9	−0.319687	−0.159844	+	0.987142i	$0.551099\pi$
$998$	21368.6	0.677766
$999$	9070.59	0.287268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.4.a.i.1.1		2
7.6	odd	2		231.4.a.f.1.1	✓	2
21.20	even	2		693.4.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.f.1.1	✓	2	7.6	odd	2
693.4.a.k.1.2		2	21.20	even	2
1617.4.a.i.1.1		2	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)

\(f(q)\)	\(=\)	\(q-3.56155 q^{2} -3.00000 q^{3} +4.68466 q^{4} +15.6847 q^{5} +10.6847 q^{6} +11.8078 q^{8} +9.00000 q^{9} -55.8617 q^{10} +11.0000 q^{11} -14.0540 q^{12} -52.9157 q^{13} -47.0540 q^{15} -79.5312 q^{16} +77.1231 q^{17} -32.0540 q^{18} -13.4081 q^{19} +73.4773 q^{20} -39.1771 q^{22} -59.0152 q^{23} -35.4233 q^{24} +121.009 q^{25} +188.462 q^{26} -27.0000 q^{27} -69.3305 q^{29} +167.585 q^{30} -75.9697 q^{31} +188.793 q^{32} -33.0000 q^{33} -274.678 q^{34} +42.1619 q^{36} -335.948 q^{37} +47.7538 q^{38} +158.747 q^{39} +185.201 q^{40} +318.617 q^{41} -57.2614 q^{43} +51.5312 q^{44} +141.162 q^{45} +210.186 q^{46} +577.779 q^{47} +238.594 q^{48} -430.978 q^{50} -231.369 q^{51} -247.892 q^{52} +315.555 q^{53} +96.1619 q^{54} +172.531 q^{55} +40.2244 q^{57} +246.924 q^{58} +598.717 q^{59} -220.432 q^{60} +337.879 q^{61} +270.570 q^{62} -36.1449 q^{64} -829.965 q^{65} +117.531 q^{66} -107.779 q^{67} +361.295 q^{68} +177.045 q^{69} +405.390 q^{71} +106.270 q^{72} +133.300 q^{73} +1196.50 q^{74} -363.026 q^{75} -62.8125 q^{76} -565.386 q^{78} -922.038 q^{79} -1247.42 q^{80} +81.0000 q^{81} -1134.77 q^{82} +1221.17 q^{83} +1209.65 q^{85} +203.939 q^{86} +207.991 q^{87} +129.885 q^{88} -1580.19 q^{89} -502.756 q^{90} -276.466 q^{92} +227.909 q^{93} -2057.79 q^{94} -210.302 q^{95} -566.378 q^{96} +287.097 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 6 q^{3} - 3 q^{4} + 19 q^{5} + 9 q^{6} + 3 q^{8} + 18 q^{9} - 54 q^{10} + 22 q^{11} + 9 q^{12} - 11 q^{13} - 57 q^{15} - 23 q^{16} + 146 q^{17} - 27 q^{18} + 101 q^{19} + 48 q^{20} - 33 q^{22}+ \cdots + 198 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 6 * q^3 - 3 * q^4 + 19 * q^5 + 9 * q^6 + 3 * q^8 + 18 * q^9 - 54 * q^10 + 22 * q^11 + 9 * q^12 - 11 * q^13 - 57 * q^15 - 23 * q^16 + 146 * q^17 - 27 * q^18 + 101 * q^19 + 48 * q^20 - 33 * q^22 - 184 * q^23 - 9 * q^24 + 7 * q^25 + 212 * q^26 - 54 * q^27 - 217 * q^29 + 162 * q^30 - 20 * q^31 + 291 * q^32 - 66 * q^33 - 236 * q^34 - 27 * q^36 - 173 * q^37 + 112 * q^38 + 33 * q^39 + 156 * q^40 + 60 * q^41 - 164 * q^43 - 33 * q^44 + 171 * q^45 + 140 * q^46 + 467 * q^47 + 69 * q^48 - 495 * q^50 - 438 * q^51 - 570 * q^52 + 326 * q^53 + 81 * q^54 + 209 * q^55 - 303 * q^57 + 164 * q^58 + 781 * q^59 - 144 * q^60 + 148 * q^61 + 302 * q^62 - 431 * q^64 - 691 * q^65 + 99 * q^66 + 473 * q^67 - 168 * q^68 + 552 * q^69 - 756 * q^71 + 27 * q^72 + 213 * q^73 + 1288 * q^74 - 21 * q^75 - 942 * q^76 - 636 * q^78 + 168 * q^79 - 1060 * q^80 + 162 * q^81 - 1280 * q^82 + 2096 * q^83 + 1438 * q^85 + 144 * q^86 + 651 * q^87 + 33 * q^88 - 736 * q^89 - 486 * q^90 + 684 * q^92 + 60 * q^93 - 2120 * q^94 + 169 * q^95 - 873 * q^96 - 638 * q^97 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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