LMFDB - Embedded newform 1617.4.a.u.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1617,4,Mod(1,1617)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1617.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [8,-4,-24,30,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$95.4060884793$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 39x^{6} + 130x^{5} + 495x^{4} - 1290x^{3} - 2045x^{2} + 3952x + 1488 $ x^8 - 4x^7 - 39x^6 + 130x^5 + 495x^4 - 1290x^3 - 2045x^2 + 3952*x + 1488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.64724$ 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)

$f(q)$	$=$	\(q+3.64724 q^{2} -3.00000 q^{3} +5.30235 q^{4} -3.24850 q^{5} -10.9417 q^{6} -9.83896 q^{8} +9.00000 q^{9} -11.8481 q^{10} +11.0000 q^{11} -15.9071 q^{12} +40.9617 q^{13} +9.74551 q^{15} -78.3039 q^{16} +95.2103 q^{17} +32.8252 q^{18} -82.0022 q^{19} -17.2247 q^{20} +40.1196 q^{22} -56.3312 q^{23} +29.5169 q^{24} -114.447 q^{25} +149.397 q^{26} -27.0000 q^{27} +12.0038 q^{29} +35.5442 q^{30} +203.204 q^{31} -206.881 q^{32} -33.0000 q^{33} +347.255 q^{34} +47.7212 q^{36} -148.963 q^{37} -299.082 q^{38} -122.885 q^{39} +31.9619 q^{40} -30.5005 q^{41} +269.054 q^{43} +58.3259 q^{44} -29.2365 q^{45} -205.453 q^{46} +189.893 q^{47} +234.912 q^{48} -417.416 q^{50} -285.631 q^{51} +217.194 q^{52} -453.909 q^{53} -98.4755 q^{54} -35.7335 q^{55} +246.007 q^{57} +43.7807 q^{58} -376.189 q^{59} +51.6742 q^{60} -238.381 q^{61} +741.134 q^{62} -128.115 q^{64} -133.064 q^{65} -120.359 q^{66} +347.523 q^{67} +504.839 q^{68} +168.994 q^{69} -769.107 q^{71} -88.5506 q^{72} +600.823 q^{73} -543.302 q^{74} +343.342 q^{75} -434.805 q^{76} -448.192 q^{78} -1232.34 q^{79} +254.370 q^{80} +81.0000 q^{81} -111.243 q^{82} -351.925 q^{83} -309.291 q^{85} +981.305 q^{86} -36.0114 q^{87} -108.229 q^{88} -757.486 q^{89} -106.633 q^{90} -298.688 q^{92} -609.612 q^{93} +692.585 q^{94} +266.384 q^{95} +620.644 q^{96} -818.178 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} - 24 q^{3} + 30 q^{4} + 20 q^{5} + 12 q^{6} - 78 q^{8} + 72 q^{9} + 94 q^{10} + 88 q^{11} - 90 q^{12} - 94 q^{13} - 60 q^{15} - 10 q^{16} + 144 q^{17} - 36 q^{18} - 8 q^{19} - 144 q^{20} - 44 q^{22}+ \cdots + 792 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 - 24 * q^3 + 30 * q^4 + 20 * q^5 + 12 * q^6 - 78 * q^8 + 72 * q^9 + 94 * q^10 + 88 * q^11 - 90 * q^12 - 94 * q^13 - 60 * q^15 - 10 * q^16 + 144 * q^17 - 36 * q^18 - 8 * q^19 - 144 * q^20 - 44 * q^22 - 286 * q^23 + 234 * q^24 - 206 * q^25 + 132 * q^26 - 216 * q^27 + 20 * q^29 - 282 * q^30 + 192 * q^31 - 948 * q^32 - 264 * q^33 - 486 * q^34 + 270 * q^36 - 666 * q^37 + 200 * q^38 + 282 * q^39 + 486 * q^40 + 60 * q^41 - 246 * q^43 + 330 * q^44 + 180 * q^45 + 454 * q^46 + 922 * q^47 + 30 * q^48 + 1486 * q^50 - 432 * q^51 - 1714 * q^52 - 1064 * q^53 + 108 * q^54 + 220 * q^55 + 24 * q^57 - 1832 * q^58 + 50 * q^59 + 432 * q^60 + 1174 * q^61 - 694 * q^62 + 1602 * q^64 - 1840 * q^65 + 132 * q^66 - 766 * q^67 + 2806 * q^68 + 858 * q^69 - 786 * q^71 - 702 * q^72 - 398 * q^73 - 1504 * q^74 + 618 * q^75 - 1518 * q^76 - 396 * q^78 - 1886 * q^79 - 504 * q^80 + 648 * q^81 + 2738 * q^82 - 782 * q^83 - 1642 * q^85 - 356 * q^86 - 60 * q^87 - 858 * q^88 + 2380 * q^89 + 846 * q^90 - 2256 * q^92 - 576 * q^93 - 2588 * q^94 + 58 * q^95 + 2844 * q^96 - 252 * q^97 + 792 * 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 $

$2$	3.64724	1.28949	0.644747	−	0.764396i	$-0.276963\pi$
$2$	3.64724	1.28949	0.644747	+	0.764396i	$0.276963\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	5.30235	0.662794
$5$	−3.24850	−0.290555	−0.145277	−	0.989391i	$-0.546407\pi$
$5$	−3.24850	−0.290555	−0.145277	+	0.989391i	$0.546407\pi$
$6$	−10.9417	−0.744490
$7$	0	0
$7$	0	0
$8$	−9.83896	−0.434825
$9$	9.00000	0.333333
$10$	−11.8481	−0.374669
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−15.9071	−0.382664
$13$	40.9617	0.873903	0.436952	−	0.899485i	$-0.356058\pi$
$13$	40.9617	0.873903	0.436952	+	0.899485i	$0.356058\pi$
$14$	0	0
$15$	9.74551	0.167752
$16$	−78.3039	−1.22350
$17$	95.2103	1.35835	0.679173	−	0.733978i	$-0.262339\pi$
$17$	95.2103	1.35835	0.679173	+	0.733978i	$0.262339\pi$
$18$	32.8252	0.429831
$19$	−82.0022	−0.990136	−0.495068	−	0.868854i	$-0.664857\pi$
$19$	−82.0022	−0.990136	−0.495068	+	0.868854i	$0.664857\pi$
$20$	−17.2247	−0.192578
$21$	0	0
$22$	40.1196	0.388797
$23$	−56.3312	−0.510690	−0.255345	−	0.966850i	$-0.582189\pi$
$23$	−56.3312	−0.510690	−0.255345	+	0.966850i	$0.582189\pi$
$24$	29.5169	0.251046
$25$	−114.447	−0.915578
$26$	149.397	1.12689
$27$	−27.0000	−0.192450
$28$	0	0
$29$	12.0038	0.0768637	0.0384318	−	0.999261i	$-0.487764\pi$
$29$	12.0038	0.0768637	0.0384318	+	0.999261i	$0.487764\pi$
$30$	35.5442	0.216315
$31$	203.204	1.17731	0.588653	−	0.808385i	$-0.299658\pi$
$31$	203.204	1.17731	0.588653	+	0.808385i	$0.299658\pi$
$32$	−206.881	−1.14287
$33$	−33.0000	−0.174078
$34$	347.255	1.75158
$35$	0	0
$36$	47.7212	0.220931
$37$	−148.963	−0.661873	−0.330936	−	0.943653i	$-0.607365\pi$
$37$	−148.963	−0.661873	−0.330936	+	0.943653i	$0.607365\pi$
$38$	−299.082	−1.27677
$39$	−122.885	−0.504548
$40$	31.9619	0.126340
$41$	−30.5005	−0.116180	−0.0580900	−	0.998311i	$-0.518501\pi$
$41$	−30.5005	−0.116180	−0.0580900	+	0.998311i	$0.518501\pi$
$42$	0	0
$43$	269.054	0.954195	0.477098	−	0.878850i	$-0.341689\pi$
$43$	269.054	0.954195	0.477098	+	0.878850i	$0.341689\pi$
$44$	58.3259	0.199840
$45$	−29.2365	−0.0968517
$46$	−205.453	−0.658532
$47$	189.893	0.589335	0.294667	−	0.955600i	$-0.404791\pi$
$47$	189.893	0.589335	0.294667	+	0.955600i	$0.404791\pi$
$48$	234.912	0.706387
$49$	0	0
$50$	−417.416	−1.18063
$51$	−285.631	−0.784242
$52$	217.194	0.579218
$53$	−453.909	−1.17640	−0.588200	−	0.808716i	$-0.700163\pi$
$53$	−453.909	−1.17640	−0.588200	+	0.808716i	$0.700163\pi$
$54$	−98.4755	−0.248163
$55$	−35.7335	−0.0876056
$56$	0	0
$57$	246.007	0.571655
$58$	43.7807	0.0991153
$59$	−376.189	−0.830095	−0.415048	−	0.909800i	$-0.636235\pi$
$59$	−376.189	−0.830095	−0.415048	+	0.909800i	$0.636235\pi$
$60$	51.6742	0.111185
$61$	−238.381	−0.500354	−0.250177	−	0.968200i	$-0.580489\pi$
$61$	−238.381	−0.500354	−0.250177	+	0.968200i	$0.580489\pi$
$62$	741.134	1.51813
$63$	0	0
$64$	−128.115	−0.250224
$65$	−133.064	−0.253917
$66$	−120.359	−0.224472
$67$	347.523	0.633682	0.316841	−	0.948479i	$-0.397378\pi$
$67$	347.523	0.633682	0.316841	+	0.948479i	$0.397378\pi$
$68$	504.839	0.900304
$69$	168.994	0.294847
$70$	0	0
$71$	−769.107	−1.28558	−0.642790	−	0.766042i	$-0.722223\pi$
$71$	−769.107	−1.28558	−0.642790	+	0.766042i	$0.722223\pi$
$72$	−88.5506	−0.144942
$73$	600.823	0.963302	0.481651	−	0.876363i	$-0.340037\pi$
$73$	600.823	0.963302	0.481651	+	0.876363i	$0.340037\pi$
$74$	−543.302	−0.853481
$75$	343.342	0.528609
$76$	−434.805	−0.656257
$77$	0	0
$78$	−448.192	−0.650612
$79$	−1232.34	−1.75506	−0.877528	−	0.479526i	$-0.840809\pi$
$79$	−1232.34	−1.75506	−0.877528	+	0.479526i	$0.840809\pi$
$80$	254.370	0.355493
$81$	81.0000	0.111111
$82$	−111.243	−0.149814
$83$	−351.925	−0.465407	−0.232703	−	0.972548i	$-0.574757\pi$
$83$	−351.925	−0.465407	−0.232703	+	0.972548i	$0.574757\pi$
$84$	0	0
$85$	−309.291	−0.394674
$86$	981.305	1.23043
$87$	−36.0114	−0.0443773
$88$	−108.229	−0.131105
$89$	−757.486	−0.902173	−0.451087	−	0.892480i	$-0.648963\pi$
$89$	−757.486	−0.902173	−0.451087	+	0.892480i	$0.648963\pi$
$90$	−106.633	−0.124890
$91$	0	0
$92$	−298.688	−0.338482
$93$	−609.612	−0.679718
$94$	692.585	0.759944
$95$	266.384	0.287689
$96$	620.644	0.659835
$97$	−818.178	−0.856427	−0.428214	−	0.903678i	$-0.640857\pi$
$97$	−818.178	−0.856427	−0.428214	+	0.903678i	$0.640857\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	−606.840	−0.606840
$101$	553.647	0.545445	0.272722	−	0.962093i	$-0.412076\pi$
$101$	553.647	0.545445	0.272722	+	0.962093i	$0.412076\pi$
$102$	−1041.76	−1.01127
$103$	−1533.22	−1.46673	−0.733363	−	0.679837i	$-0.762050\pi$
$103$	−1533.22	−1.46673	−0.733363	+	0.679837i	$0.762050\pi$
$104$	−403.021	−0.379995
$105$	0	0
$106$	−1655.51	−1.51696
$107$	−330.037	−0.298186	−0.149093	−	0.988823i	$-0.547635\pi$
$107$	−330.037	−0.298186	−0.149093	+	0.988823i	$0.547635\pi$
$108$	−143.164	−0.127555
$109$	−1466.41	−1.28859	−0.644296	−	0.764777i	$-0.722849\pi$
$109$	−1466.41	−1.28859	−0.644296	+	0.764777i	$0.722849\pi$
$110$	−130.329	−0.112967
$111$	446.888	0.382132
$112$	0	0
$113$	−1082.09	−0.900836	−0.450418	−	0.892818i	$-0.648725\pi$
$113$	−1082.09	−0.900836	−0.450418	+	0.892818i	$0.648725\pi$
$114$	897.245	0.737146
$115$	182.992	0.148384
$116$	63.6483	0.0509448
$117$	368.656	0.291301
$118$	−1372.05	−1.07040
$119$	0	0
$120$	−95.8857	−0.0729427
$121$	121.000	0.0909091
$122$	−869.433	−0.645203
$123$	91.5016	0.0670766
$124$	1077.46	0.780312
$125$	777.845	0.556581
$126$	0	0
$127$	−1047.31	−0.731760	−0.365880	−	0.930662i	$-0.619232\pi$
$127$	−1047.31	−0.731760	−0.365880	+	0.930662i	$0.619232\pi$
$128$	1187.79	0.820206
$129$	−807.163	−0.550905
$130$	−485.317	−0.327424
$131$	−2556.91	−1.70533	−0.852664	−	0.522459i	$-0.825015\pi$
$131$	−2556.91	−1.70533	−0.852664	+	0.522459i	$0.825015\pi$
$132$	−174.978	−0.115378
$133$	0	0
$134$	1267.50	0.817129
$135$	87.7096	0.0559173
$136$	−936.770	−0.590642
$137$	697.531	0.434993	0.217497	−	0.976061i	$-0.430211\pi$
$137$	697.531	0.434993	0.217497	+	0.976061i	$0.430211\pi$
$138$	616.360	0.380203
$139$	99.4507	0.0606856	0.0303428	−	0.999540i	$-0.490340\pi$
$139$	99.4507	0.0606856	0.0303428	+	0.999540i	$0.490340\pi$
$140$	0	0
$141$	−569.679	−0.340253
$142$	−2805.12	−1.65775
$143$	450.579	0.263492
$144$	−704.735	−0.407833
$145$	−38.9943	−0.0223331
$146$	2191.35	1.24217
$147$	0	0
$148$	−789.852	−0.438686
$149$	−3248.44	−1.78606	−0.893030	−	0.449997i	$-0.851425\pi$
$149$	−3248.44	−1.78606	−0.893030	+	0.449997i	$0.851425\pi$
$150$	1252.25	0.681638
$151$	−2472.19	−1.33235	−0.666173	−	0.745797i	$-0.732069\pi$
$151$	−2472.19	−1.33235	−0.666173	+	0.745797i	$0.732069\pi$
$152$	806.816	0.430536
$153$	856.893	0.452782
$154$	0	0
$155$	−660.109	−0.342072
$156$	−651.581	−0.334412
$157$	−617.329	−0.313811	−0.156905	−	0.987614i	$-0.550152\pi$
$157$	−617.329	−0.313811	−0.156905	+	0.987614i	$0.550152\pi$
$158$	−4494.65	−2.26313
$159$	1361.73	0.679195
$160$	672.055	0.332066
$161$	0	0
$162$	295.426	0.143277
$163$	2674.29	1.28507	0.642535	−	0.766256i	$-0.277883\pi$
$163$	2674.29	1.28507	0.642535	+	0.766256i	$0.277883\pi$
$164$	−161.725	−0.0770035
$165$	107.201	0.0505791
$166$	−1283.55	−0.600139
$167$	2747.54	1.27312	0.636560	−	0.771227i	$-0.280357\pi$
$167$	2747.54	1.27312	0.636560	+	0.771227i	$0.280357\pi$
$168$	0	0
$169$	−519.137	−0.236293
$170$	−1128.06	−0.508930
$171$	−738.020	−0.330045
$172$	1426.62	0.632435
$173$	−506.133	−0.222431	−0.111216	−	0.993796i	$-0.535474\pi$
$173$	−506.133	−0.222431	−0.111216	+	0.993796i	$0.535474\pi$
$174$	−131.342	−0.0572242
$175$	0	0
$176$	−861.343	−0.368899
$177$	1128.57	0.479256
$178$	−2762.73	−1.16335
$179$	607.459	0.253652	0.126826	−	0.991925i	$-0.459521\pi$
$179$	607.459	0.253652	0.126826	+	0.991925i	$0.459521\pi$
$180$	−155.022	−0.0641927
$181$	2546.22	1.04563	0.522815	−	0.852446i	$-0.324882\pi$
$181$	2546.22	1.04563	0.522815	+	0.852446i	$0.324882\pi$
$182$	0	0
$183$	715.144	0.288879
$184$	554.241	0.222061
$185$	483.905	0.192310
$186$	−2223.40	−0.876493
$187$	1047.31	0.409557
$188$	1006.88	0.390608
$189$	0	0
$190$	971.568	0.370973
$191$	2378.16	0.900931	0.450465	−	0.892794i	$-0.351258\pi$
$191$	2378.16	0.900931	0.450465	+	0.892794i	$0.351258\pi$
$192$	384.344	0.144467
$193$	2415.45	0.900868	0.450434	−	0.892810i	$-0.351269\pi$
$193$	2415.45	0.900868	0.450434	+	0.892810i	$0.351269\pi$
$194$	−2984.09	−1.10436
$195$	399.193	0.146599
$196$	0	0
$197$	−3381.10	−1.22281	−0.611404	−	0.791318i	$-0.709395\pi$
$197$	−3381.10	−1.22281	−0.611404	+	0.791318i	$0.709395\pi$
$198$	361.077	0.129599
$199$	−4787.68	−1.70548	−0.852738	−	0.522339i	$-0.825059\pi$
$199$	−4787.68	−1.70548	−0.852738	+	0.522339i	$0.825059\pi$
$200$	1126.04	0.398116
$201$	−1042.57	−0.365856
$202$	2019.28	0.703348
$203$	0	0
$204$	−1514.52	−0.519791
$205$	99.0811	0.0337567
$206$	−5592.02	−1.89133
$207$	−506.981	−0.170230
$208$	−3207.46	−1.06922
$209$	−902.024	−0.298537
$210$	0	0
$211$	2580.29	0.841870	0.420935	−	0.907091i	$-0.361702\pi$
$211$	2580.29	0.841870	0.420935	+	0.907091i	$0.361702\pi$
$212$	−2406.79	−0.779711
$213$	2307.32	0.742230
$214$	−1203.73	−0.384509
$215$	−874.024	−0.277246
$216$	265.652	0.0836820
$217$	0	0
$218$	−5348.34	−1.66163
$219$	−1802.47	−0.556163
$220$	−189.472	−0.0580645
$221$	3899.98	1.18706
$222$	1629.91	0.492757
$223$	−3554.80	−1.06747	−0.533737	−	0.845650i	$-0.679213\pi$
$223$	−3554.80	−1.06747	−0.533737	+	0.845650i	$0.679213\pi$
$224$	0	0
$225$	−1030.03	−0.305193
$226$	−3946.64	−1.16162
$227$	−4875.17	−1.42545	−0.712723	−	0.701446i	$-0.752538\pi$
$227$	−4875.17	−1.42545	−0.712723	+	0.701446i	$0.752538\pi$
$228$	1304.41	0.378890
$229$	6303.38	1.81895	0.909475	−	0.415759i	$-0.136484\pi$
$229$	6303.38	1.81895	0.909475	+	0.415759i	$0.136484\pi$
$230$	667.416	0.191340
$231$	0	0
$232$	−118.105	−0.0334222
$233$	3905.06	1.09798	0.548989	−	0.835829i	$-0.315013\pi$
$233$	3905.06	1.09798	0.548989	+	0.835829i	$0.315013\pi$
$234$	1344.58	0.375631
$235$	−616.868	−0.171234
$236$	−1994.69	−0.550183
$237$	3697.03	1.01328
$238$	0	0
$239$	−3569.46	−0.966063	−0.483032	−	0.875603i	$-0.660464\pi$
$239$	−3569.46	−0.966063	−0.483032	+	0.875603i	$0.660464\pi$
$240$	−763.111	−0.205244
$241$	4453.22	1.19028	0.595139	−	0.803623i	$-0.297097\pi$
$241$	4453.22	1.19028	0.595139	+	0.803623i	$0.297097\pi$
$242$	441.316	0.117227
$243$	−243.000	−0.0641500
$244$	−1263.98	−0.331632
$245$	0	0
$246$	333.728	0.0864949
$247$	−3358.95	−0.865283
$248$	−1999.32	−0.511922
$249$	1055.77	0.268703
$250$	2836.99	0.717707
$251$	5783.13	1.45429	0.727147	−	0.686482i	$-0.240846\pi$
$251$	5783.13	1.45429	0.727147	+	0.686482i	$0.240846\pi$
$252$	0	0
$253$	−619.643	−0.153979
$254$	−3819.78	−0.943600
$255$	927.873	0.227865
$256$	5357.06	1.30787
$257$	2593.12	0.629395	0.314698	−	0.949192i	$-0.398097\pi$
$257$	2593.12	0.629395	0.314698	+	0.949192i	$0.398097\pi$
$258$	−2943.92	−0.710388
$259$	0	0
$260$	−705.554	−0.168295
$261$	108.034	0.0256212
$262$	−9325.65	−2.19901
$263$	8286.95	1.94295	0.971474	−	0.237145i	$-0.0762115\pi$
$263$	8286.95	1.94295	0.971474	+	0.237145i	$0.0762115\pi$
$264$	324.686	0.0756933
$265$	1474.52	0.341809
$266$	0	0
$267$	2272.46	0.520870
$268$	1842.69	0.420001
$269$	1891.43	0.428708	0.214354	−	0.976756i	$-0.431235\pi$
$269$	1891.43	0.428708	0.214354	+	0.976756i	$0.431235\pi$
$270$	319.898	0.0721051
$271$	−1469.34	−0.329357	−0.164679	−	0.986347i	$-0.552659\pi$
$271$	−1469.34	−0.329357	−0.164679	+	0.986347i	$0.552659\pi$
$272$	−7455.33	−1.66193
$273$	0	0
$274$	2544.06	0.560921
$275$	−1258.92	−0.276057
$276$	896.064	0.195423
$277$	−6429.80	−1.39469	−0.697346	−	0.716735i	$-0.745636\pi$
$277$	−6429.80	−1.39469	−0.697346	+	0.716735i	$0.745636\pi$
$278$	362.721	0.0782537
$279$	1828.84	0.392436
$280$	0	0
$281$	−210.954	−0.0447846	−0.0223923	−	0.999749i	$-0.507128\pi$
$281$	−210.954	−0.0447846	−0.0223923	+	0.999749i	$0.507128\pi$
$282$	−2077.76	−0.438754
$283$	926.036	0.194513	0.0972564	−	0.995259i	$-0.468993\pi$
$283$	926.036	0.194513	0.0972564	+	0.995259i	$0.468993\pi$
$284$	−4078.08	−0.852075
$285$	−799.153	−0.166097
$286$	1643.37	0.339771
$287$	0	0
$288$	−1861.93	−0.380956
$289$	4152.00	0.845105
$290$	−142.222	−0.0287984
$291$	2454.54	0.494458
$292$	3185.78	0.638471
$293$	−1290.46	−0.257303	−0.128651	−	0.991690i	$-0.541065\pi$
$293$	−1290.46	−0.257303	−0.128651	+	0.991690i	$0.541065\pi$
$294$	0	0
$295$	1222.05	0.241188
$296$	1465.64	0.287799
$297$	−297.000	−0.0580259
$298$	−11847.9	−2.30311
$299$	−2307.42	−0.446294
$300$	1820.52	0.350359
$301$	0	0
$302$	−9016.68	−1.71805
$303$	−1660.94	−0.314913
$304$	6421.09	1.21143
$305$	774.382	0.145380
$306$	3125.29	0.583860
$307$	−3882.39	−0.721757	−0.360879	−	0.932613i	$-0.617523\pi$
$307$	−3882.39	−0.721757	−0.360879	+	0.932613i	$0.617523\pi$
$308$	0	0
$309$	4599.66	0.846814
$310$	−2407.58	−0.441100
$311$	−6548.48	−1.19399	−0.596994	−	0.802246i	$-0.703638\pi$
$311$	−6548.48	−1.19399	−0.596994	+	0.802246i	$0.703638\pi$
$312$	1209.06	0.219390
$313$	382.812	0.0691304	0.0345652	−	0.999402i	$-0.488995\pi$
$313$	382.812	0.0691304	0.0345652	+	0.999402i	$0.488995\pi$
$314$	−2251.55	−0.404657
$315$	0	0
$316$	−6534.32	−1.16324
$317$	8659.95	1.53436	0.767178	−	0.641434i	$-0.221660\pi$
$317$	8659.95	1.53436	0.767178	+	0.641434i	$0.221660\pi$
$318$	4966.54	0.875817
$319$	132.042	0.0231753
$320$	416.181	0.0727038
$321$	990.112	0.172158
$322$	0	0
$323$	−7807.45	−1.34495
$324$	429.491	0.0736438
$325$	−4687.96	−0.800126
$326$	9753.77	1.65709
$327$	4399.22	0.743968
$328$	300.094	0.0505180
$329$	0	0
$330$	390.986	0.0652215
$331$	−10596.6	−1.75964	−0.879821	−	0.475304i	$-0.842338\pi$
$331$	−10596.6	−1.75964	−0.879821	+	0.475304i	$0.842338\pi$
$332$	−1866.03	−0.308469
$333$	−1340.66	−0.220624
$334$	10020.9	1.64168
$335$	−1128.93	−0.184119
$336$	0	0
$337$	−3561.41	−0.575675	−0.287838	−	0.957679i	$-0.592936\pi$
$337$	−3561.41	−0.575675	−0.287838	+	0.957679i	$0.592936\pi$
$338$	−1893.42	−0.304699
$339$	3246.27	0.520098
$340$	−1639.97	−0.261588
$341$	2235.24	0.354971
$342$	−2691.73	−0.425592
$343$	0	0
$344$	−2647.21	−0.414907
$345$	−548.977	−0.0856693
$346$	−1845.99	−0.286824
$347$	−195.225	−0.0302024	−0.0151012	−	0.999886i	$-0.504807\pi$
$347$	−195.225	−0.0302024	−0.0151012	+	0.999886i	$0.504807\pi$
$348$	−190.945	−0.0294130
$349$	−614.409	−0.0942366	−0.0471183	−	0.998889i	$-0.515004\pi$
$349$	−614.409	−0.0942366	−0.0471183	+	0.998889i	$0.515004\pi$
$350$	0	0
$351$	−1105.97	−0.168183
$352$	−2275.69	−0.344588
$353$	−1545.91	−0.233090	−0.116545	−	0.993185i	$-0.537182\pi$
$353$	−1545.91	−0.233090	−0.116545	+	0.993185i	$0.537182\pi$
$354$	4116.15	0.617997
$355$	2498.45	0.373532
$356$	−4016.46	−0.597955
$357$	0	0
$358$	2215.55	0.327082
$359$	2724.00	0.400466	0.200233	−	0.979748i	$-0.435830\pi$
$359$	2724.00	0.400466	0.200233	+	0.979748i	$0.435830\pi$
$360$	287.657	0.0421135
$361$	−134.643	−0.0196301
$362$	9286.68	1.34833
$363$	−363.000	−0.0524864
$364$	0	0
$365$	−1951.78	−0.279892
$366$	2608.30	0.372508
$367$	−9559.12	−1.35962	−0.679812	−	0.733386i	$-0.737939\pi$
$367$	−9559.12	−1.35962	−0.679812	+	0.733386i	$0.737939\pi$
$368$	4410.95	0.624828
$369$	−274.505	−0.0387267
$370$	1764.92	0.247983
$371$	0	0
$372$	−3232.38	−0.450514
$373$	7269.54	1.00912	0.504561	−	0.863376i	$-0.331654\pi$
$373$	7269.54	1.00912	0.504561	+	0.863376i	$0.331654\pi$
$374$	3819.80	0.528121
$375$	−2333.54	−0.321342
$376$	−1868.35	−0.256257
$377$	491.696	0.0671714
$378$	0	0
$379$	12989.3	1.76047	0.880234	−	0.474540i	$-0.157385\pi$
$379$	12989.3	1.76047	0.880234	+	0.474540i	$0.157385\pi$
$380$	1412.46	0.190679
$381$	3141.92	0.422482
$382$	8673.73	1.16174
$383$	−2308.92	−0.308043	−0.154021	−	0.988068i	$-0.549222\pi$
$383$	−2308.92	−0.308043	−0.154021	+	0.988068i	$0.549222\pi$
$384$	−3563.36	−0.473546
$385$	0	0
$386$	8809.71	1.16166
$387$	2421.49	0.318065
$388$	−4338.27	−0.567635
$389$	−5270.18	−0.686912	−0.343456	−	0.939169i	$-0.611598\pi$
$389$	−5270.18	−0.686912	−0.343456	+	0.939169i	$0.611598\pi$
$390$	1455.95	0.189038
$391$	−5363.31	−0.693694
$392$	0	0
$393$	7670.72	0.984572
$394$	−12331.7	−1.57680
$395$	4003.27	0.509940
$396$	524.933	0.0666133
$397$	12746.8	1.61145	0.805724	−	0.592291i	$-0.201776\pi$
$397$	12746.8	1.61145	0.805724	+	0.592291i	$0.201776\pi$
$398$	−17461.8	−2.19920
$399$	0	0
$400$	8961.66	1.12021
$401$	−6003.06	−0.747578	−0.373789	−	0.927514i	$-0.621942\pi$
$401$	−6003.06	−0.747578	−0.373789	+	0.927514i	$0.621942\pi$
$402$	−3802.50	−0.471769
$403$	8323.59	1.02885
$404$	2935.63	0.361518
$405$	−263.129	−0.0322839
$406$	0	0
$407$	−1638.59	−0.199562
$408$	2810.31	0.341008
$409$	−8220.38	−0.993818	−0.496909	−	0.867803i	$-0.665532\pi$
$409$	−8220.38	−0.993818	−0.496909	+	0.867803i	$0.665532\pi$
$410$	361.373	0.0435291
$411$	−2092.59	−0.251144
$412$	−8129.68	−0.972137
$413$	0	0
$414$	−1849.08	−0.219511
$415$	1143.23	0.135226
$416$	−8474.22	−0.998756
$417$	−298.352	−0.0350369
$418$	−3289.90	−0.384962
$419$	−2847.79	−0.332037	−0.166019	−	0.986123i	$-0.553091\pi$
$419$	−2847.79	−0.332037	−0.166019	+	0.986123i	$0.553091\pi$
$420$	0	0
$421$	−1481.78	−0.171538	−0.0857688	−	0.996315i	$-0.527335\pi$
$421$	−1481.78	−0.171538	−0.0857688	+	0.996315i	$0.527335\pi$
$422$	9410.94	1.08559
$423$	1709.04	0.196445
$424$	4465.99	0.511528
$425$	−10896.6	−1.24367
$426$	8415.35	0.957101
$427$	0	0
$428$	−1749.98	−0.197636
$429$	−1351.74	−0.152127
$430$	−3187.77	−0.357507
$431$	978.752	0.109385	0.0546924	−	0.998503i	$-0.482582\pi$
$431$	978.752	0.109385	0.0546924	+	0.998503i	$0.482582\pi$
$432$	2114.20	0.235462
$433$	15417.0	1.71108	0.855538	−	0.517740i	$-0.173226\pi$
$433$	15417.0	1.71108	0.855538	+	0.517740i	$0.173226\pi$
$434$	0	0
$435$	116.983	0.0128940
$436$	−7775.42	−0.854071
$437$	4619.28	0.505653
$438$	−6574.04	−0.717168
$439$	−5719.23	−0.621785	−0.310893	−	0.950445i	$-0.600628\pi$
$439$	−5719.23	−0.621785	−0.310893	+	0.950445i	$0.600628\pi$
$440$	351.581	0.0380931
$441$	0	0
$442$	14224.2	1.53071
$443$	−9355.29	−1.00335	−0.501674	−	0.865057i	$-0.667282\pi$
$443$	−9355.29	−1.00335	−0.501674	+	0.865057i	$0.667282\pi$
$444$	2369.56	0.253275
$445$	2460.70	0.262131
$446$	−12965.2	−1.37650
$447$	9745.33	1.03118
$448$	0	0
$449$	8588.44	0.902703	0.451351	−	0.892346i	$-0.350942\pi$
$449$	8588.44	0.902703	0.451351	+	0.892346i	$0.350942\pi$
$450$	−3756.75	−0.393544
$451$	−335.506	−0.0350296
$452$	−5737.62	−0.597069
$453$	7416.58	0.769231
$454$	−17780.9	−1.83810
$455$	0	0
$456$	−2420.45	−0.248570
$457$	13105.9	1.34151	0.670753	−	0.741681i	$-0.265971\pi$
$457$	13105.9	1.34151	0.670753	+	0.741681i	$0.265971\pi$
$458$	22990.0	2.34552
$459$	−2570.68	−0.261414
$460$	970.289	0.0983478
$461$	3602.92	0.364002	0.182001	−	0.983298i	$-0.441743\pi$
$461$	3602.92	0.364002	0.182001	+	0.983298i	$0.441743\pi$
$462$	0	0
$463$	−12636.3	−1.26838	−0.634190	−	0.773177i	$-0.718667\pi$
$463$	−12636.3	−1.26838	−0.634190	+	0.773177i	$0.718667\pi$
$464$	−939.943	−0.0940426
$465$	1980.33	0.197496
$466$	14242.7	1.41584
$467$	−14027.3	−1.38995	−0.694976	−	0.719033i	$-0.744585\pi$
$467$	−14027.3	−1.38995	−0.694976	+	0.719033i	$0.744585\pi$
$468$	1954.74	0.193073
$469$	0	0
$470$	−2249.87	−0.220805
$471$	1851.99	0.181179
$472$	3701.31	0.360946
$473$	2959.60	0.287701
$474$	13483.9	1.30662
$475$	9384.92	0.906547
$476$	0	0
$477$	−4085.18	−0.392133
$478$	−13018.7	−1.24573
$479$	−10283.7	−0.980946	−0.490473	−	0.871456i	$-0.663176\pi$
$479$	−10283.7	−0.980946	−0.490473	+	0.871456i	$0.663176\pi$
$480$	−2016.16	−0.191718
$481$	−6101.76	−0.578413
$482$	16241.9	1.53486
$483$	0	0
$484$	641.585	0.0602540
$485$	2657.86	0.248839
$486$	−886.279	−0.0827211
$487$	5602.07	0.521260	0.260630	−	0.965439i	$-0.416070\pi$
$487$	5602.07	0.521260	0.260630	+	0.965439i	$0.416070\pi$
$488$	2345.42	0.217566
$489$	−8022.87	−0.741936
$490$	0	0
$491$	17867.1	1.64222	0.821109	−	0.570772i	$-0.193356\pi$
$491$	17867.1	1.64222	0.821109	+	0.570772i	$0.193356\pi$
$492$	485.174	0.0444580
$493$	1142.88	0.104408
$494$	−12250.9	−1.11578
$495$	−321.602	−0.0292019
$496$	−15911.7	−1.44043
$497$	0	0
$498$	3850.66	0.346490
$499$	−4310.61	−0.386713	−0.193356	−	0.981129i	$-0.561937\pi$
$499$	−4310.61	−0.386713	−0.193356	+	0.981129i	$0.561937\pi$
$500$	4124.41	0.368899
$501$	−8242.62	−0.735036
$502$	21092.5	1.87530
$503$	21922.9	1.94333	0.971664	−	0.236368i	$-0.0759570\pi$
$503$	21922.9	1.94333	0.971664	+	0.236368i	$0.0759570\pi$
$504$	0	0
$505$	−1798.52	−0.158482
$506$	−2259.99	−0.198555
$507$	1557.41	0.136424
$508$	−5553.19	−0.485006
$509$	−414.595	−0.0361033	−0.0180517	−	0.999837i	$-0.505746\pi$
$509$	−414.595	−0.0361033	−0.0180517	+	0.999837i	$0.505746\pi$
$510$	3384.17	0.293831
$511$	0	0
$512$	10036.2	0.866290
$513$	2214.06	0.190552
$514$	9457.74	0.811601
$515$	4980.67	0.426164
$516$	−4279.86	−0.365137
$517$	2088.82	0.177691
$518$	0	0
$519$	1518.40	0.128421
$520$	1309.21	0.110409
$521$	−12637.0	−1.06264	−0.531322	−	0.847170i	$-0.678304\pi$
$521$	−12637.0	−1.06264	−0.531322	+	0.847170i	$0.678304\pi$
$522$	394.026	0.0330384
$523$	8622.02	0.720870	0.360435	−	0.932784i	$-0.382628\pi$
$523$	8622.02	0.720870	0.360435	+	0.932784i	$0.382628\pi$
$524$	−13557.6	−1.13028
$525$	0	0
$526$	30224.5	2.50542
$527$	19347.1	1.59919
$528$	2584.03	0.212984
$529$	−8993.79	−0.739196
$530$	5377.95	0.440760
$531$	−3385.70	−0.276698
$532$	0	0
$533$	−1249.35	−0.101530
$534$	8288.20	0.671658
$535$	1072.13	0.0866395
$536$	−3419.26	−0.275540
$537$	−1822.38	−0.146446
$538$	6898.49	0.552816
$539$	0	0
$540$	465.067	0.0370617
$541$	6030.56	0.479250	0.239625	−	0.970866i	$-0.422976\pi$
$541$	6030.56	0.479250	0.239625	+	0.970866i	$0.422976\pi$
$542$	−5359.02	−0.424704
$543$	−7638.67	−0.603695
$544$	−19697.2	−1.55241
$545$	4763.63	0.374407
$546$	0	0
$547$	−4459.63	−0.348593	−0.174296	−	0.984693i	$-0.555765\pi$
$547$	−4459.63	−0.348593	−0.174296	+	0.984693i	$0.555765\pi$
$548$	3698.56	0.288311
$549$	−2145.43	−0.166785
$550$	−4591.58	−0.355974
$551$	−984.337	−0.0761055
$552$	−1662.72	−0.128207
$553$	0	0
$554$	−23451.0	−1.79845
$555$	−1451.72	−0.111031
$556$	527.323	0.0402221
$557$	−8263.96	−0.628645	−0.314323	−	0.949316i	$-0.601777\pi$
$557$	−8263.96	−0.628645	−0.314323	+	0.949316i	$0.601777\pi$
$558$	6670.20	0.506043
$559$	11020.9	0.833874
$560$	0	0
$561$	−3141.94	−0.236458
$562$	−769.400	−0.0577494
$563$	11358.0	0.850235	0.425118	−	0.905138i	$-0.360233\pi$
$563$	11358.0	0.850235	0.425118	+	0.905138i	$0.360233\pi$
$564$	−3020.64	−0.225517
$565$	3515.17	0.261742
$566$	3377.48	0.250823
$567$	0	0
$568$	7567.21	0.559002
$569$	−9877.17	−0.727720	−0.363860	−	0.931454i	$-0.618541\pi$
$569$	−9877.17	−0.727720	−0.363860	+	0.931454i	$0.618541\pi$
$570$	−2914.70	−0.214182
$571$	5265.18	0.385886	0.192943	−	0.981210i	$-0.438197\pi$
$571$	5265.18	0.385886	0.192943	+	0.981210i	$0.438197\pi$
$572$	2389.13	0.174641
$573$	−7134.49	−0.520153
$574$	0	0
$575$	6446.95	0.467576
$576$	−1153.03	−0.0834080
$577$	−19193.1	−1.38478	−0.692390	−	0.721524i	$-0.743442\pi$
$577$	−19193.1	−1.38478	−0.692390	+	0.721524i	$0.743442\pi$
$578$	15143.3	1.08976
$579$	−7246.34	−0.520116
$580$	−206.762	−0.0148023
$581$	0	0
$582$	8952.28	0.637601
$583$	−4993.00	−0.354698
$584$	−5911.48	−0.418868
$585$	−1197.58	−0.0846390
$586$	−4706.63	−0.331790
$587$	21860.1	1.53708	0.768538	−	0.639805i	$-0.220985\pi$
$587$	21860.1	1.53708	0.768538	+	0.639805i	$0.220985\pi$
$588$	0	0
$589$	−16663.2	−1.16569
$590$	4457.11	0.311011
$591$	10143.3	0.705989
$592$	11664.3	0.809800
$593$	5715.59	0.395803	0.197901	−	0.980222i	$-0.436587\pi$
$593$	5715.59	0.395803	0.197901	+	0.980222i	$0.436587\pi$
$594$	−1083.23	−0.0748240
$595$	0	0
$596$	−17224.4	−1.18379
$597$	14363.0	0.984657
$598$	−8415.73	−0.575493
$599$	14681.1	1.00143	0.500714	−	0.865613i	$-0.333071\pi$
$599$	14681.1	1.00143	0.500714	+	0.865613i	$0.333071\pi$
$600$	−3378.12	−0.229852
$601$	−12450.1	−0.845011	−0.422505	−	0.906360i	$-0.638849\pi$
$601$	−12450.1	−0.845011	−0.422505	+	0.906360i	$0.638849\pi$
$602$	0	0
$603$	3127.71	0.211227
$604$	−13108.4	−0.883072
$605$	−393.069	−0.0264141
$606$	−6057.85	−0.406078
$607$	−9521.00	−0.636648	−0.318324	−	0.947982i	$-0.603120\pi$
$607$	−9521.00	−0.636648	−0.318324	+	0.947982i	$0.603120\pi$
$608$	16964.7	1.13160
$609$	0	0
$610$	2824.36	0.187467
$611$	7778.34	0.515021
$612$	4543.55	0.300101
$613$	−29.5388	−0.00194627	−0.000973133	−	1.00000i	$-0.500310\pi$
$613$	−29.5388	−0.00194627	−0.000973133		1.00000i	$0.500310\pi$
$614$	−14160.0	−0.930702
$615$	−297.243	−0.0194894
$616$	0	0
$617$	−13875.2	−0.905342	−0.452671	−	0.891678i	$-0.649529\pi$
$617$	−13875.2	−0.905342	−0.452671	+	0.891678i	$0.649529\pi$
$618$	16776.1	1.09196
$619$	10367.2	0.673169	0.336584	−	0.941653i	$-0.390728\pi$
$619$	10367.2	0.673169	0.336584	+	0.941653i	$0.390728\pi$
$620$	−3500.13	−0.226724
$621$	1520.94	0.0982823
$622$	−23883.9	−1.53964
$623$	0	0
$624$	9622.39	0.617314
$625$	11779.1	0.753860
$626$	1396.21	0.0891433
$627$	2706.07	0.172361
$628$	−3273.30	−0.207992
$629$	−14182.8	−0.899053
$630$	0	0
$631$	5081.22	0.320571	0.160285	−	0.987071i	$-0.448759\pi$
$631$	5081.22	0.320571	0.160285	+	0.987071i	$0.448759\pi$
$632$	12125.0	0.763141
$633$	−7740.88	−0.486054
$634$	31584.9	1.97854
$635$	3402.18	0.212616
$636$	7220.36	0.450166
$637$	0	0
$638$	481.587	0.0298844
$639$	−6921.96	−0.428527
$640$	−3858.53	−0.238315
$641$	9720.65	0.598974	0.299487	−	0.954100i	$-0.403184\pi$
$641$	9720.65	0.598974	0.299487	+	0.954100i	$0.403184\pi$
$642$	3611.18	0.221997
$643$	22740.9	1.39473	0.697365	−	0.716716i	$-0.254355\pi$
$643$	22740.9	1.39473	0.697365	+	0.716716i	$0.254355\pi$
$644$	0	0
$645$	2622.07	0.160068
$646$	−28475.6	−1.73430
$647$	27862.7	1.69304	0.846520	−	0.532357i	$-0.178694\pi$
$647$	27862.7	1.69304	0.846520	+	0.532357i	$0.178694\pi$
$648$	−796.956	−0.0483138
$649$	−4138.08	−0.250283
$650$	−17098.1	−1.03176
$651$	0	0
$652$	14180.0	0.851737
$653$	26111.3	1.56480	0.782399	−	0.622778i	$-0.213996\pi$
$653$	26111.3	1.56480	0.782399	+	0.622778i	$0.213996\pi$
$654$	16045.0	0.959343
$655$	8306.12	0.495492
$656$	2388.31	0.142146
$657$	5407.41	0.321101
$658$	0	0
$659$	−22590.4	−1.33535	−0.667676	−	0.744452i	$-0.732711\pi$
$659$	−22590.4	−1.33535	−0.667676	+	0.744452i	$0.732711\pi$
$660$	568.416	0.0335236
$661$	30178.4	1.77580	0.887899	−	0.460039i	$-0.152164\pi$
$661$	30178.4	1.77580	0.887899	+	0.460039i	$0.152164\pi$
$662$	−38648.3	−2.26905
$663$	−11699.9	−0.685351
$664$	3462.57	0.202370
$665$	0	0
$666$	−4889.72	−0.284494
$667$	−676.188	−0.0392535
$668$	14568.4	0.843817
$669$	10664.4	0.616307
$670$	−4117.48	−0.237421
$671$	−2622.19	−0.150862
$672$	0	0
$673$	3391.09	0.194230	0.0971151	−	0.995273i	$-0.469038\pi$
$673$	3391.09	0.194230	0.0971151	+	0.995273i	$0.469038\pi$
$674$	−12989.3	−0.742330
$675$	3090.08	0.176203
$676$	−2752.65	−0.156614
$677$	3271.32	0.185712	0.0928559	−	0.995680i	$-0.470400\pi$
$677$	3271.32	0.185712	0.0928559	+	0.995680i	$0.470400\pi$
$678$	11839.9	0.670663
$679$	0	0
$680$	3043.10	0.171614
$681$	14625.5	0.822982
$682$	8152.47	0.457733
$683$	−31941.5	−1.78947	−0.894734	−	0.446600i	$-0.852635\pi$
$683$	−31941.5	−1.78947	−0.894734	+	0.446600i	$0.852635\pi$
$684$	−3913.24	−0.218752
$685$	−2265.93	−0.126389
$686$	0	0
$687$	−18910.2	−1.05017
$688$	−21068.0	−1.16746
$689$	−18592.9	−1.02806
$690$	−2002.25	−0.110470
$691$	−12811.1	−0.705292	−0.352646	−	0.935757i	$-0.614718\pi$
$691$	−12811.1	−0.705292	−0.352646	+	0.935757i	$0.614718\pi$
$692$	−2683.70	−0.147426
$693$	0	0
$694$	−712.034	−0.0389459
$695$	−323.066	−0.0176325
$696$	354.314	0.0192963
$697$	−2903.97	−0.157813
$698$	−2240.90	−0.121518
$699$	−11715.2	−0.633918
$700$	0	0
$701$	17225.0	0.928076	0.464038	−	0.885815i	$-0.346400\pi$
$701$	17225.0	0.928076	0.464038	+	0.885815i	$0.346400\pi$
$702$	−4033.73	−0.216871
$703$	12215.3	0.655344
$704$	−1409.26	−0.0754453
$705$	1850.60	0.0988621
$706$	−5638.31	−0.300568
$707$	0	0
$708$	5984.06	0.317648
$709$	−2822.10	−0.149487	−0.0747434	−	0.997203i	$-0.523814\pi$
$709$	−2822.10	−0.149487	−0.0747434	+	0.997203i	$0.523814\pi$
$710$	9112.43	0.481667
$711$	−11091.1	−0.585018
$712$	7452.88	0.392287
$713$	−11446.7	−0.601239
$714$	0	0
$715$	−1463.71	−0.0765588
$716$	3220.97	0.168119
$717$	10708.4	0.557757
$718$	9935.09	0.516399
$719$	−30929.8	−1.60429	−0.802147	−	0.597127i	$-0.796309\pi$
$719$	−30929.8	−1.60429	−0.802147	+	0.597127i	$0.796309\pi$
$720$	2289.33	0.118498
$721$	0	0
$722$	−491.076	−0.0253129
$723$	−13359.7	−0.687207
$724$	13501.0	0.693038
$725$	−1373.80	−0.0703747
$726$	−1323.95	−0.0676809
$727$	−8100.81	−0.413263	−0.206632	−	0.978419i	$-0.566250\pi$
$727$	−8100.81	−0.413263	−0.206632	+	0.978419i	$0.566250\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−7118.60	−0.360919
$731$	25616.7	1.29613
$732$	3791.95	0.191468
$733$	−31494.1	−1.58699	−0.793494	−	0.608578i	$-0.791740\pi$
$733$	−31494.1	−1.58699	−0.793494	+	0.608578i	$0.791740\pi$
$734$	−34864.4	−1.75323
$735$	0	0
$736$	11653.9	0.583652
$737$	3822.75	0.191062
$738$	−1001.18	−0.0499378
$739$	−24848.3	−1.23689	−0.618444	−	0.785829i	$-0.712237\pi$
$739$	−24848.3	−1.23689	−0.618444	+	0.785829i	$0.712237\pi$
$740$	2565.84	0.127462
$741$	10076.9	0.499571
$742$	0	0
$743$	−16842.2	−0.831600	−0.415800	−	0.909456i	$-0.636498\pi$
$743$	−16842.2	−0.831600	−0.415800	+	0.909456i	$0.636498\pi$
$744$	5997.95	0.295558
$745$	10552.6	0.518949
$746$	26513.8	1.30126
$747$	−3167.32	−0.155136
$748$	5553.23	0.271452
$749$	0	0
$750$	−8510.96	−0.414369
$751$	16080.6	0.781346	0.390673	−	0.920529i	$-0.372242\pi$
$751$	16080.6	0.781346	0.390673	+	0.920529i	$0.372242\pi$
$752$	−14869.4	−0.721050
$753$	−17349.4	−0.839637
$754$	1793.33	0.0866171
$755$	8030.93	0.387120
$756$	0	0
$757$	−19231.4	−0.923351	−0.461675	−	0.887049i	$-0.652751\pi$
$757$	−19231.4	−0.923351	−0.461675	+	0.887049i	$0.652751\pi$
$758$	47375.2	2.27011
$759$	1858.93	0.0888997
$760$	−2620.94	−0.125094
$761$	−11402.7	−0.543164	−0.271582	−	0.962415i	$-0.587547\pi$
$761$	−11402.7	−0.543164	−0.271582	+	0.962415i	$0.587547\pi$
$762$	11459.3	0.544787
$763$	0	0
$764$	12609.9	0.597132
$765$	−2783.62	−0.131558
$766$	−8421.19	−0.397219
$767$	−15409.4	−0.725423
$768$	−16071.2	−0.755102
$769$	−35970.1	−1.68675	−0.843377	−	0.537322i	$-0.819436\pi$
$769$	−35970.1	−1.68675	−0.843377	+	0.537322i	$0.819436\pi$
$770$	0	0
$771$	−7779.37	−0.363381
$772$	12807.5	0.597090
$773$	18318.2	0.852339	0.426170	−	0.904643i	$-0.359863\pi$
$773$	18318.2	0.852339	0.426170	+	0.904643i	$0.359863\pi$
$774$	8831.75	0.410143
$775$	−23256.1	−1.07792
$776$	8050.02	0.372396
$777$	0	0
$778$	−19221.6	−0.885769
$779$	2501.11	0.115034
$780$	2116.66	0.0971650
$781$	−8460.18	−0.387617
$782$	−19561.3	−0.894514
$783$	−324.102	−0.0147924
$784$	0	0
$785$	2005.40	0.0911792
$786$	27977.0	1.26960
$787$	−33414.1	−1.51345	−0.756724	−	0.653734i	$-0.773201\pi$
$787$	−33414.1	−1.51345	−0.756724	+	0.653734i	$0.773201\pi$
$788$	−17927.8	−0.810470
$789$	−24860.9	−1.12176
$790$	14600.9	0.657565
$791$	0	0
$792$	−974.057	−0.0437015
$793$	−9764.51	−0.437261
$794$	46490.7	2.07795
$795$	−4423.57	−0.197343
$796$	−25386.0	−1.13038
$797$	24235.5	1.07712	0.538560	−	0.842587i	$-0.318968\pi$
$797$	24235.5	1.07712	0.538560	+	0.842587i	$0.318968\pi$
$798$	0	0
$799$	18079.8	0.800521
$800$	23677.0	1.04639
$801$	−6817.38	−0.300724
$802$	−21894.6	−0.963997
$803$	6609.06	0.290447
$804$	−5528.07	−0.242487
$805$	0	0
$806$	30358.1	1.32670
$807$	−5674.28	−0.247515
$808$	−5447.31	−0.237173
$809$	−24799.7	−1.07777	−0.538883	−	0.842381i	$-0.681153\pi$
$809$	−24799.7	−1.07777	−0.538883	+	0.842381i	$0.681153\pi$
$810$	−959.694	−0.0416299
$811$	−32827.3	−1.42136	−0.710680	−	0.703515i	$-0.751613\pi$
$811$	−32827.3	−1.42136	−0.710680	+	0.703515i	$0.751613\pi$
$812$	0	0
$813$	4408.01	0.190155
$814$	−5976.32	−0.257334
$815$	−8687.44	−0.373384
$816$	22366.0	0.959518
$817$	−22063.0	−0.944783
$818$	−29981.7	−1.28152
$819$	0	0
$820$	525.363	0.0223738
$821$	12560.5	0.533942	0.266971	−	0.963705i	$-0.413977\pi$
$821$	12560.5	0.533942	0.266971	+	0.963705i	$0.413977\pi$
$822$	−7632.19	−0.323848
$823$	−3054.70	−0.129380	−0.0646902	−	0.997905i	$-0.520606\pi$
$823$	−3054.70	−0.129380	−0.0646902	+	0.997905i	$0.520606\pi$
$824$	15085.3	0.637768
$825$	3776.76	0.159382
$826$	0	0
$827$	34363.5	1.44491	0.722453	−	0.691420i	$-0.243015\pi$
$827$	34363.5	1.44491	0.722453	+	0.691420i	$0.243015\pi$
$828$	−2688.19	−0.112827
$829$	−20243.3	−0.848103	−0.424051	−	0.905638i	$-0.639392\pi$
$829$	−20243.3	−0.848103	−0.424051	+	0.905638i	$0.639392\pi$
$830$	4169.63	0.174373
$831$	19289.4	0.805225
$832$	−5247.80	−0.218671
$833$	0	0
$834$	−1088.16	−0.0451798
$835$	−8925.39	−0.369911
$836$	−4782.85	−0.197869
$837$	−5486.51	−0.226573
$838$	−10386.6	−0.428160
$839$	29046.6	1.19523	0.597616	−	0.801782i	$-0.296115\pi$
$839$	29046.6	1.19523	0.597616	+	0.801782i	$0.296115\pi$
$840$	0	0
$841$	−24244.9	−0.994092
$842$	−5404.39	−0.221197
$843$	632.862	0.0258564
$844$	13681.6	0.557987
$845$	1686.42	0.0686562
$846$	6233.27	0.253315
$847$	0	0
$848$	35542.8	1.43932
$849$	−2778.11	−0.112302
$850$	−39742.3	−1.60371
$851$	8391.24	0.338012
$852$	12234.2	0.491946
$853$	−13618.2	−0.546635	−0.273317	−	0.961924i	$-0.588121\pi$
$853$	−13618.2	−0.546635	−0.273317	+	0.961924i	$0.588121\pi$
$854$	0	0
$855$	2397.46	0.0958963
$856$	3247.23	0.129659
$857$	18214.9	0.726033	0.363016	−	0.931783i	$-0.381747\pi$
$857$	18214.9	0.726033	0.363016	+	0.931783i	$0.381747\pi$
$858$	−4930.11	−0.196167
$859$	12474.1	0.495471	0.247736	−	0.968828i	$-0.420314\pi$
$859$	12474.1	0.495471	0.247736	+	0.968828i	$0.420314\pi$
$860$	−4634.38	−0.183757
$861$	0	0
$862$	3569.74	0.141051
$863$	−32488.1	−1.28147	−0.640734	−	0.767763i	$-0.721370\pi$
$863$	−32488.1	−1.28147	−0.640734	+	0.767763i	$0.721370\pi$
$864$	5585.79	0.219945
$865$	1644.18	0.0646285
$866$	56229.7	2.20642
$867$	−12456.0	−0.487921
$868$	0	0
$869$	−13555.8	−0.529169
$870$	426.665	0.0166268
$871$	14235.1	0.553776
$872$	14427.9	0.560311
$873$	−7363.61	−0.285476
$874$	16847.6	0.652036
$875$	0	0
$876$	−9557.34	−0.368622
$877$	−31820.0	−1.22518	−0.612592	−	0.790399i	$-0.709873\pi$
$877$	−31820.0	−1.22518	−0.612592	+	0.790399i	$0.709873\pi$
$878$	−20859.4	−0.801788
$879$	3871.39	0.148554
$880$	2798.07	0.107185
$881$	32871.3	1.25705	0.628525	−	0.777789i	$-0.283659\pi$
$881$	32871.3	1.25705	0.628525	+	0.777789i	$0.283659\pi$
$882$	0	0
$883$	11692.0	0.445603	0.222802	−	0.974864i	$-0.428480\pi$
$883$	11692.0	0.445603	0.222802	+	0.974864i	$0.428480\pi$
$884$	20679.1	0.786779
$885$	−3666.15	−0.139250
$886$	−34121.0	−1.29381
$887$	6697.21	0.253518	0.126759	−	0.991934i	$-0.459543\pi$
$887$	6697.21	0.253518	0.126759	+	0.991934i	$0.459543\pi$
$888$	−4396.91	−0.166161
$889$	0	0
$890$	8974.75	0.338016
$891$	891.000	0.0335013
$892$	−18848.8	−0.707516
$893$	−15571.6	−0.583522
$894$	35543.6	1.32970
$895$	−1973.33	−0.0736998
$896$	0	0
$897$	6922.27	0.257668
$898$	31324.1	1.16403
$899$	2439.22	0.0904922
$900$	−5461.56	−0.202280
$901$	−43216.8	−1.59796
$902$	−1223.67	−0.0451705
$903$	0	0
$904$	10646.6	0.391706
$905$	−8271.41	−0.303813
$906$	27050.1	0.991918
$907$	−26086.9	−0.955019	−0.477509	−	0.878627i	$-0.658460\pi$
$907$	−26086.9	−0.955019	−0.477509	+	0.878627i	$0.658460\pi$
$908$	−25849.9	−0.944778
$909$	4982.82	0.181815
$910$	0	0
$911$	−15265.3	−0.555174	−0.277587	−	0.960701i	$-0.589535\pi$
$911$	−15265.3	−0.555174	−0.277587	+	0.960701i	$0.589535\pi$
$912$	−19263.3	−0.699419
$913$	−3871.17	−0.140325
$914$	47800.3	1.72986
$915$	−2323.15	−0.0839354
$916$	33422.8	1.20559
$917$	0	0
$918$	−9375.88	−0.337092
$919$	−1939.37	−0.0696124	−0.0348062	−	0.999394i	$-0.511081\pi$
$919$	−1939.37	−0.0696124	−0.0348062	+	0.999394i	$0.511081\pi$
$920$	−1800.45	−0.0645208
$921$	11647.2	0.416707
$922$	13140.7	0.469378
$923$	−31504.0	−1.12347
$924$	0	0
$925$	17048.4	0.605996
$926$	−46087.7	−1.63557
$927$	−13799.0	−0.488909
$928$	−2483.36	−0.0878451
$929$	−6356.47	−0.224488	−0.112244	−	0.993681i	$-0.535804\pi$
$929$	−6356.47	−0.224488	−0.112244	+	0.993681i	$0.535804\pi$
$930$	7222.73	0.254669
$931$	0	0
$932$	20706.0	0.727734
$933$	19645.4	0.689349
$934$	−51161.0	−1.79233
$935$	−3402.20	−0.118999
$936$	−3627.19	−0.126665
$937$	−11467.9	−0.399828	−0.199914	−	0.979813i	$-0.564066\pi$
$937$	−11467.9	−0.399828	−0.199914	+	0.979813i	$0.564066\pi$
$938$	0	0
$939$	−1148.44	−0.0399125
$940$	−3270.85	−0.113493
$941$	37590.1	1.30223	0.651117	−	0.758978i	$-0.274301\pi$
$941$	37590.1	1.30223	0.651117	+	0.758978i	$0.274301\pi$
$942$	6754.64	0.233629
$943$	1718.13	0.0593320
$944$	29457.1	1.01562
$945$	0	0
$946$	10794.4	0.370988
$947$	−51388.2	−1.76335	−0.881674	−	0.471858i	$-0.843583\pi$
$947$	−51388.2	−1.76335	−0.881674	+	0.471858i	$0.843583\pi$
$948$	19602.9	0.671597
$949$	24610.8	0.841833
$950$	34229.1	1.16899
$951$	−25979.8	−0.885861
$952$	0	0
$953$	18421.2	0.626151	0.313075	−	0.949728i	$-0.398641\pi$
$953$	18421.2	0.626151	0.313075	+	0.949728i	$0.398641\pi$
$954$	−14899.6	−0.505653
$955$	−7725.47	−0.261770
$956$	−18926.5	−0.640301
$957$	−396.125	−0.0133803
$958$	−37507.0	−1.26492
$959$	0	0
$960$	−1248.54	−0.0419756
$961$	11500.9	0.386052
$962$	−22254.6	−0.745860
$963$	−2970.34	−0.0993954
$964$	23612.5	0.788909
$965$	−7846.58	−0.261752
$966$	0	0
$967$	24037.9	0.799386	0.399693	−	0.916649i	$-0.369117\pi$
$967$	24037.9	0.799386	0.399693	+	0.916649i	$0.369117\pi$
$968$	−1190.51	−0.0395295
$969$	23422.4	0.776506
$970$	9693.83	0.320877
$971$	25003.2	0.826355	0.413177	−	0.910651i	$-0.364419\pi$
$971$	25003.2	0.826355	0.413177	+	0.910651i	$0.364419\pi$
$972$	−1288.47	−0.0425183
$973$	0	0
$974$	20432.1	0.672162
$975$	14063.9	0.461953
$976$	18666.2	0.612182
$977$	12020.5	0.393625	0.196812	−	0.980441i	$-0.436941\pi$
$977$	12020.5	0.393625	0.196812	+	0.980441i	$0.436941\pi$
$978$	−29261.3	−0.956721
$979$	−8332.35	−0.272015
$980$	0	0
$981$	−13197.7	−0.429530
$982$	65165.4	2.11763
$983$	39887.0	1.29420	0.647100	−	0.762405i	$-0.275982\pi$
$983$	39887.0	1.29420	0.647100	+	0.762405i	$0.275982\pi$
$984$	−900.281	−0.0291666
$985$	10983.5	0.355293
$986$	4168.37	0.134633
$987$	0	0
$988$	−17810.3	−0.573505
$989$	−15156.2	−0.487298
$990$	−1172.96	−0.0376556
$991$	28304.5	0.907289	0.453645	−	0.891183i	$-0.350124\pi$
$991$	28304.5	0.907289	0.453645	+	0.891183i	$0.350124\pi$
$992$	−42039.1	−1.34551
$993$	31789.8	1.01593
$994$	0	0
$995$	15552.8	0.495534
$996$	5598.09	0.178095
$997$	−33825.5	−1.07449	−0.537244	−	0.843427i	$-0.680535\pi$
$997$	−33825.5	−1.07449	−0.537244	+	0.843427i	$0.680535\pi$
$998$	−15721.8	−0.498664
$999$	4021.99	0.127377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.4.a.u.1.7		8
7.2	even	3		231.4.i.a.67.2	✓	16
7.4	even	3		231.4.i.a.100.2	yes	16
7.6	odd	2		1617.4.a.v.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.i.a.67.2	✓	16	7.2	even	3
231.4.i.a.100.2	yes	16	7.4	even	3
1617.4.a.u.1.7		8	1.1	even	1	trivial
1617.4.a.v.1.7		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.64724 q^{2} -3.00000 q^{3} +5.30235 q^{4} -3.24850 q^{5} -10.9417 q^{6} -9.83896 q^{8} +9.00000 q^{9} -11.8481 q^{10} +11.0000 q^{11} -15.9071 q^{12} +40.9617 q^{13} +9.74551 q^{15} -78.3039 q^{16} +95.2103 q^{17} +32.8252 q^{18} -82.0022 q^{19} -17.2247 q^{20} +40.1196 q^{22} -56.3312 q^{23} +29.5169 q^{24} -114.447 q^{25} +149.397 q^{26} -27.0000 q^{27} +12.0038 q^{29} +35.5442 q^{30} +203.204 q^{31} -206.881 q^{32} -33.0000 q^{33} +347.255 q^{34} +47.7212 q^{36} -148.963 q^{37} -299.082 q^{38} -122.885 q^{39} +31.9619 q^{40} -30.5005 q^{41} +269.054 q^{43} +58.3259 q^{44} -29.2365 q^{45} -205.453 q^{46} +189.893 q^{47} +234.912 q^{48} -417.416 q^{50} -285.631 q^{51} +217.194 q^{52} -453.909 q^{53} -98.4755 q^{54} -35.7335 q^{55} +246.007 q^{57} +43.7807 q^{58} -376.189 q^{59} +51.6742 q^{60} -238.381 q^{61} +741.134 q^{62} -128.115 q^{64} -133.064 q^{65} -120.359 q^{66} +347.523 q^{67} +504.839 q^{68} +168.994 q^{69} -769.107 q^{71} -88.5506 q^{72} +600.823 q^{73} -543.302 q^{74} +343.342 q^{75} -434.805 q^{76} -448.192 q^{78} -1232.34 q^{79} +254.370 q^{80} +81.0000 q^{81} -111.243 q^{82} -351.925 q^{83} -309.291 q^{85} +981.305 q^{86} -36.0114 q^{87} -108.229 q^{88} -757.486 q^{89} -106.633 q^{90} -298.688 q^{92} -609.612 q^{93} +692.585 q^{94} +266.384 q^{95} +620.644 q^{96} -818.178 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 24 q^{3} + 30 q^{4} + 20 q^{5} + 12 q^{6} - 78 q^{8} + 72 q^{9} + 94 q^{10} + 88 q^{11} - 90 q^{12} - 94 q^{13} - 60 q^{15} - 10 q^{16} + 144 q^{17} - 36 q^{18} - 8 q^{19} - 144 q^{20} - 44 q^{22}+ \cdots + 792 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 24 * q^3 + 30 * q^4 + 20 * q^5 + 12 * q^6 - 78 * q^8 + 72 * q^9 + 94 * q^10 + 88 * q^11 - 90 * q^12 - 94 * q^13 - 60 * q^15 - 10 * q^16 + 144 * q^17 - 36 * q^18 - 8 * q^19 - 144 * q^20 - 44 * q^22 - 286 * q^23 + 234 * q^24 - 206 * q^25 + 132 * q^26 - 216 * q^27 + 20 * q^29 - 282 * q^30 + 192 * q^31 - 948 * q^32 - 264 * q^33 - 486 * q^34 + 270 * q^36 - 666 * q^37 + 200 * q^38 + 282 * q^39 + 486 * q^40 + 60 * q^41 - 246 * q^43 + 330 * q^44 + 180 * q^45 + 454 * q^46 + 922 * q^47 + 30 * q^48 + 1486 * q^50 - 432 * q^51 - 1714 * q^52 - 1064 * q^53 + 108 * q^54 + 220 * q^55 + 24 * q^57 - 1832 * q^58 + 50 * q^59 + 432 * q^60 + 1174 * q^61 - 694 * q^62 + 1602 * q^64 - 1840 * q^65 + 132 * q^66 - 766 * q^67 + 2806 * q^68 + 858 * q^69 - 786 * q^71 - 702 * q^72 - 398 * q^73 - 1504 * q^74 + 618 * q^75 - 1518 * q^76 - 396 * q^78 - 1886 * q^79 - 504 * q^80 + 648 * q^81 + 2738 * q^82 - 782 * q^83 - 1642 * q^85 - 356 * q^86 - 60 * q^87 - 858 * q^88 + 2380 * q^89 + 846 * q^90 - 2256 * q^92 - 576 * q^93 - 2588 * q^94 + 58 * q^95 + 2844 * q^96 - 252 * q^97 + 792 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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