LMFDB - Embedded newform 1617.4.a.i.1.2

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			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	\(q+0.561553 q^{2} -3.00000 q^{3} -7.68466 q^{4} +3.31534 q^{5} -1.68466 q^{6} -8.80776 q^{8} +9.00000 q^{9} +1.86174 q^{10} +11.0000 q^{11} +23.0540 q^{12} +41.9157 q^{13} -9.94602 q^{15} +56.5312 q^{16} +68.8769 q^{17} +5.05398 q^{18} +114.408 q^{19} -25.4773 q^{20} +6.17708 q^{22} -124.985 q^{23} +26.4233 q^{24} -114.009 q^{25} +23.5379 q^{26} -27.0000 q^{27} -147.670 q^{29} -5.58522 q^{30} +55.9697 q^{31} +102.207 q^{32} -33.0000 q^{33} +38.6780 q^{34} -69.1619 q^{36} +162.948 q^{37} +64.2462 q^{38} -125.747 q^{39} -29.2007 q^{40} -258.617 q^{41} -106.739 q^{43} -84.5312 q^{44} +29.8381 q^{45} -70.1856 q^{46} -110.779 q^{47} -169.594 q^{48} -64.0218 q^{50} -206.631 q^{51} -322.108 q^{52} +10.4451 q^{53} -15.1619 q^{54} +36.4688 q^{55} -343.224 q^{57} -82.9242 q^{58} +182.283 q^{59} +76.4318 q^{60} -189.879 q^{61} +31.4299 q^{62} -394.855 q^{64} +138.965 q^{65} -18.5312 q^{66} +580.779 q^{67} -529.295 q^{68} +374.955 q^{69} -1161.39 q^{71} -79.2699 q^{72} +79.6998 q^{73} +91.5038 q^{74} +342.026 q^{75} -879.187 q^{76} -70.6137 q^{78} +1090.04 q^{79} +187.420 q^{80} +81.0000 q^{81} -145.227 q^{82} +874.830 q^{83} +228.350 q^{85} -59.9394 q^{86} +443.009 q^{87} -96.8854 q^{88} +844.193 q^{89} +16.7557 q^{90} +960.466 q^{92} -167.909 q^{93} -62.2084 q^{94} +379.302 q^{95} -306.622 q^{96} -925.097 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.561553	0.198539	0.0992695	−	0.995061i	$-0.468349\pi$
$2$	0.561553	0.198539	0.0992695	+	0.995061i	$0.468349\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.68466	−0.960582
$5$	3.31534	0.296533	0.148267	−	0.988947i	$-0.452631\pi$
$5$	3.31534	0.296533	0.148267	+	0.988947i	$0.452631\pi$
$6$	−1.68466	−0.114626
$7$	0	0
$7$	0	0
$8$	−8.80776	−0.389252
$9$	9.00000	0.333333
$10$	1.86174	0.0588734
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	23.0540	0.554592
$13$	41.9157	0.894256	0.447128	−	0.894470i	$-0.352447\pi$
$13$	41.9157	0.894256	0.447128	+	0.894470i	$0.352447\pi$
$14$	0	0
$15$	−9.94602	−0.171204
$16$	56.5312	0.883301
$17$	68.8769	0.982653	0.491326	−	0.870975i	$-0.336512\pi$
$17$	68.8769	0.982653	0.491326	+	0.870975i	$0.336512\pi$
$18$	5.05398	0.0661796
$19$	114.408	1.38142	0.690711	−	0.723131i	$-0.257298\pi$
$19$	114.408	1.38142	0.690711	+	0.723131i	$0.257298\pi$
$20$	−25.4773	−0.284845
$21$	0	0
$22$	6.17708	0.0598617
$23$	−124.985	−1.13309	−0.566547	−	0.824030i	$-0.691721\pi$
$23$	−124.985	−1.13309	−0.566547	+	0.824030i	$0.691721\pi$
$24$	26.4233	0.224735
$25$	−114.009	−0.912068
$26$	23.5379	0.177545
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−147.670	−0.945570	−0.472785	−	0.881178i	$-0.656751\pi$
$29$	−147.670	−0.945570	−0.472785	+	0.881178i	$0.656751\pi$
$30$	−5.58522	−0.0339906
$31$	55.9697	0.324273	0.162136	−	0.986768i	$-0.448162\pi$
$31$	55.9697	0.324273	0.162136	+	0.986768i	$0.448162\pi$
$32$	102.207	0.564621
$33$	−33.0000	−0.174078
$34$	38.6780	0.195095
$35$	0	0
$36$	−69.1619	−0.320194
$37$	162.948	0.724013	0.362006	−	0.932176i	$-0.382092\pi$
$37$	162.948	0.724013	0.362006	+	0.932176i	$0.382092\pi$
$38$	64.2462	0.274266
$39$	−125.747	−0.516299
$40$	−29.2007	−0.115426
$41$	−258.617	−0.985104	−0.492552	−	0.870283i	$-0.663936\pi$
$41$	−258.617	−0.985104	−0.492552	+	0.870283i	$0.663936\pi$
$42$	0	0
$43$	−106.739	−0.378546	−0.189273	−	0.981924i	$-0.560613\pi$
$43$	−106.739	−0.378546	−0.189273	+	0.981924i	$0.560613\pi$
$44$	−84.5312	−0.289626
$45$	29.8381	0.0988444
$46$	−70.1856	−0.224963
$47$	−110.779	−0.343805	−0.171902	−	0.985114i	$-0.554991\pi$
$47$	−110.779	−0.343805	−0.171902	+	0.985114i	$0.554991\pi$
$48$	−169.594	−0.509974
$49$	0	0
$50$	−64.0218	−0.181081
$51$	−206.631	−0.567335
$52$	−322.108	−0.859006
$53$	10.4451	0.0270706	0.0135353	−	0.999908i	$-0.495691\pi$
$53$	10.4451	0.0270706	0.0135353	+	0.999908i	$0.495691\pi$
$54$	−15.1619	−0.0382088
$55$	36.4688	0.0894081
$56$	0	0
$57$	−343.224	−0.797565
$58$	−82.9242	−0.187732
$59$	182.283	0.402225	0.201112	−	0.979568i	$-0.435544\pi$
$59$	182.283	0.402225	0.201112	+	0.979568i	$0.435544\pi$
$60$	76.4318	0.164455
$61$	−189.879	−0.398549	−0.199274	−	0.979944i	$-0.563859\pi$
$61$	−189.879	−0.398549	−0.199274	+	0.979944i	$0.563859\pi$
$62$	31.4299	0.0643807
$63$	0	0
$64$	−394.855	−0.771201
$65$	138.965	0.265177
$66$	−18.5312	−0.0345612
$67$	580.779	1.05901	0.529504	−	0.848308i	$-0.322378\pi$
$67$	580.779	1.05901	0.529504	+	0.848308i	$0.322378\pi$
$68$	−529.295	−0.943919
$69$	374.955	0.654192
$70$	0	0
$71$	−1161.39	−1.94129	−0.970645	−	0.240515i	$-0.922684\pi$
$71$	−1161.39	−1.94129	−0.970645	+	0.240515i	$0.922684\pi$
$72$	−79.2699	−0.129751
$73$	79.6998	0.127783	0.0638915	−	0.997957i	$-0.479649\pi$
$73$	79.6998	0.127783	0.0638915	+	0.997957i	$0.479649\pi$
$74$	91.5038	0.143745
$75$	342.026	0.526583
$76$	−879.187	−1.32697
$77$	0	0
$78$	−70.6137	−0.102505
$79$	1090.04	1.55239	0.776195	−	0.630493i	$-0.217147\pi$
$79$	1090.04	1.55239	0.776195	+	0.630493i	$0.217147\pi$
$80$	187.420	0.261928
$81$	81.0000	0.111111
$82$	−145.227	−0.195581
$83$	874.830	1.15693	0.578464	−	0.815708i	$-0.303652\pi$
$83$	874.830	1.15693	0.578464	+	0.815708i	$0.303652\pi$
$84$	0	0
$85$	228.350	0.291389
$86$	−59.9394	−0.0751562
$87$	443.009	0.545925
$88$	−96.8854	−0.117364
$89$	844.193	1.00544	0.502721	−	0.864449i	$-0.332332\pi$
$89$	844.193	1.00544	0.502721	+	0.864449i	$0.332332\pi$
$90$	16.7557	0.0196245
$91$	0	0
$92$	960.466	1.08843
$93$	−167.909	−0.187219
$94$	−62.2084	−0.0682586
$95$	379.302	0.409638
$96$	−306.622	−0.325984
$97$	−925.097	−0.968344	−0.484172	−	0.874973i	$-0.660879\pi$
$97$	−925.097	−0.968344	−0.484172	+	0.874973i	$0.660879\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	876.116	0.876116
$101$	884.867	0.871758	0.435879	−	0.900005i	$-0.356438\pi$
$101$	884.867	0.871758	0.435879	+	0.900005i	$0.356438\pi$
$102$	−116.034	−0.112638
$103$	−1069.78	−1.02339	−0.511694	−	0.859168i	$-0.670982\pi$
$103$	−1069.78	−1.02339	−0.511694	+	0.859168i	$0.670982\pi$
$104$	−369.184	−0.348091
$105$	0	0
$106$	5.86547	0.00537457
$107$	178.196	0.160999	0.0804993	−	0.996755i	$-0.474349\pi$
$107$	178.196	0.160999	0.0804993	+	0.996755i	$0.474349\pi$
$108$	207.486	0.184864
$109$	1973.58	1.73426	0.867130	−	0.498082i	$-0.165962\pi$
$109$	1973.58	1.73426	0.867130	+	0.498082i	$0.165962\pi$
$110$	20.4791	0.0177510
$111$	−488.844	−0.418009
$112$	0	0
$113$	1642.68	1.36753	0.683763	−	0.729704i	$-0.260342\pi$
$113$	1642.68	1.36753	0.683763	+	0.729704i	$0.260342\pi$
$114$	−192.739	−0.158348
$115$	−414.367	−0.336000
$116$	1134.79	0.908298
$117$	377.241	0.298085
$118$	102.362	0.0798572
$119$	0	0
$120$	87.6022	0.0666413
$121$	121.000	0.0909091
$122$	−106.627	−0.0791275
$123$	775.852	0.568750
$124$	−430.108	−0.311491
$125$	−792.395	−0.566992
$126$	0	0
$127$	−1796.94	−1.25554	−0.627768	−	0.778400i	$-0.716031\pi$
$127$	−1796.94	−1.25554	−0.627768	+	0.778400i	$0.716031\pi$
$128$	−1039.39	−0.717735
$129$	320.216	0.218554
$130$	78.0361	0.0526479
$131$	−934.305	−0.623134	−0.311567	−	0.950224i	$-0.600854\pi$
$131$	−934.305	−0.623134	−0.311567	+	0.950224i	$0.600854\pi$
$132$	253.594	0.167216
$133$	0	0
$134$	326.138	0.210254
$135$	−89.5142	−0.0570678
$136$	−606.651	−0.382499
$137$	−1590.87	−0.992097	−0.496048	−	0.868295i	$-0.665216\pi$
$137$	−1590.87	−0.992097	−0.496048	+	0.868295i	$0.665216\pi$
$138$	210.557	0.129882
$139$	1549.00	0.945213	0.472607	−	0.881274i	$-0.343313\pi$
$139$	1549.00	0.945213	0.472607	+	0.881274i	$0.343313\pi$
$140$	0	0
$141$	332.338	0.198496
$142$	−652.182	−0.385422
$143$	461.073	0.269628
$144$	508.781	0.294434
$145$	−489.575	−0.280393
$146$	44.7557	0.0253699
$147$	0	0
$148$	−1252.20	−0.695474
$149$	−468.094	−0.257367	−0.128684	−	0.991686i	$-0.541075\pi$
$149$	−468.094	−0.257367	−0.128684	+	0.991686i	$0.541075\pi$
$150$	192.065	0.104547
$151$	1865.34	1.00529	0.502647	−	0.864492i	$-0.332360\pi$
$151$	1865.34	1.00529	0.502647	+	0.864492i	$0.332360\pi$
$152$	−1007.68	−0.537721
$153$	619.892	0.327551
$154$	0	0
$155$	185.559	0.0961576
$156$	966.324	0.495948
$157$	−1501.59	−0.763310	−0.381655	−	0.924305i	$-0.624646\pi$
$157$	−1501.59	−0.763310	−0.381655	+	0.924305i	$0.624646\pi$
$158$	612.114	0.308210
$159$	−31.3353	−0.0156292
$160$	338.852	0.167429
$161$	0	0
$162$	45.4858	0.0220599
$163$	524.401	0.251989	0.125995	−	0.992031i	$-0.459788\pi$
$163$	524.401	0.251989	0.125995	+	0.992031i	$0.459788\pi$
$164$	1987.39	0.946273
$165$	−109.406	−0.0516198
$166$	491.263	0.229695
$167$	3293.09	1.52591	0.762956	−	0.646450i	$-0.223747\pi$
$167$	3293.09	1.52591	0.762956	+	0.646450i	$0.223747\pi$
$168$	0	0
$169$	−440.073	−0.200306
$170$	128.231	0.0578521
$171$	1029.67	0.460474
$172$	820.250	0.363625
$173$	3036.96	1.33466	0.667330	−	0.744762i	$-0.267437\pi$
$173$	3036.96	1.33466	0.667330	+	0.744762i	$0.267437\pi$
$174$	248.773	0.108387
$175$	0	0
$176$	621.844	0.266325
$177$	−546.849	−0.232224
$178$	474.059	0.199619
$179$	690.152	0.288181	0.144090	−	0.989565i	$-0.453974\pi$
$179$	690.152	0.288181	0.144090	+	0.989565i	$0.453974\pi$
$180$	−229.295	−0.0949482
$181$	2533.94	1.04059	0.520294	−	0.853987i	$-0.325822\pi$
$181$	2533.94	1.04059	0.520294	+	0.853987i	$0.325822\pi$
$182$	0	0
$183$	569.636	0.230102
$184$	1100.84	0.441059
$185$	540.228	0.214694
$186$	−94.2898	−0.0371702
$187$	757.646	0.296281
$188$	851.301	0.330253
$189$	0	0
$190$	212.998	0.0813290
$191$	1845.13	0.698998	0.349499	−	0.936937i	$-0.386352\pi$
$191$	1845.13	0.698998	0.349499	+	0.936937i	$0.386352\pi$
$192$	1184.57	0.445253
$193$	−2654.48	−0.990019	−0.495009	−	0.868888i	$-0.664835\pi$
$193$	−2654.48	−0.990019	−0.495009	+	0.868888i	$0.664835\pi$
$194$	−519.491	−0.192254
$195$	−416.895	−0.153100
$196$	0	0
$197$	164.057	0.0593328	0.0296664	−	0.999560i	$-0.490555\pi$
$197$	164.057	0.0593328	0.0296664	+	0.999560i	$0.490555\pi$
$198$	55.5937	0.0199539
$199$	−2888.41	−1.02891	−0.514457	−	0.857516i	$-0.672007\pi$
$199$	−2888.41	−1.02891	−0.514457	+	0.857516i	$0.672007\pi$
$200$	1004.16	0.355024
$201$	−1742.34	−0.611418
$202$	496.900	0.173078
$203$	0	0
$204$	1587.89	0.544972
$205$	−857.405	−0.292116
$206$	−600.740	−0.203182
$207$	−1124.86	−0.377698
$208$	2369.55	0.789897
$209$	1258.49	0.416515
$210$	0	0
$211$	630.956	0.205862	0.102931	−	0.994689i	$-0.467178\pi$
$211$	630.956	0.205862	0.102931	+	0.994689i	$0.467178\pi$
$212$	−80.2670	−0.0260036
$213$	3484.17	1.12080
$214$	100.066	0.0319645
$215$	−353.875	−0.112252
$216$	237.810	0.0749116
$217$	0	0
$218$	1108.27	0.344318
$219$	−239.099	−0.0737755
$220$	−280.250	−0.0858839
$221$	2887.02	0.878743
$222$	−274.512	−0.0829910
$223$	6252.50	1.87757	0.938786	−	0.344501i	$-0.111952\pi$
$223$	6252.50	1.87757	0.938786	+	0.344501i	$0.111952\pi$
$224$	0	0
$225$	−1026.08	−0.304023
$226$	922.453	0.271507
$227$	3316.20	0.969621	0.484810	−	0.874619i	$-0.338889\pi$
$227$	3316.20	0.969621	0.484810	+	0.874619i	$0.338889\pi$
$228$	2637.56	0.766126
$229$	−3077.20	−0.887979	−0.443989	−	0.896032i	$-0.646437\pi$
$229$	−3077.20	−0.887979	−0.443989	+	0.896032i	$0.646437\pi$
$230$	−232.689	−0.0667090
$231$	0	0
$232$	1300.64	0.368065
$233$	1358.08	0.381849	0.190924	−	0.981605i	$-0.438851\pi$
$233$	1358.08	0.381849	0.190924	+	0.981605i	$0.438851\pi$
$234$	211.841	0.0591815
$235$	−367.271	−0.101950
$236$	−1400.78	−0.386370
$237$	−3270.11	−0.896273
$238$	0	0
$239$	−4151.18	−1.12350	−0.561752	−	0.827306i	$-0.689872\pi$
$239$	−4151.18	−1.12350	−0.561752	+	0.827306i	$0.689872\pi$
$240$	−562.261	−0.151224
$241$	3491.71	0.933282	0.466641	−	0.884447i	$-0.345464\pi$
$241$	3491.71	0.933282	0.466641	+	0.884447i	$0.345464\pi$
$242$	67.9479	0.0180490
$243$	−243.000	−0.0641500
$244$	1459.15	0.382839
$245$	0	0
$246$	435.682	0.112919
$247$	4795.50	1.23535
$248$	−492.968	−0.126224
$249$	−2624.49	−0.667953
$250$	−444.972	−0.112570
$251$	879.383	0.221140	0.110570	−	0.993868i	$-0.464732\pi$
$251$	879.383	0.221140	0.110570	+	0.993868i	$0.464732\pi$
$252$	0	0
$253$	−1374.83	−0.341640
$254$	−1009.08	−0.249273
$255$	−685.051	−0.168234
$256$	2575.17	0.628703
$257$	−5691.10	−1.38133	−0.690663	−	0.723176i	$-0.742681\pi$
$257$	−5691.10	−1.38133	−0.690663	+	0.723176i	$0.742681\pi$
$258$	179.818	0.0433914
$259$	0	0
$260$	−1067.90	−0.254724
$261$	−1329.03	−0.315190
$262$	−524.661	−0.123716
$263$	2024.55	0.474673	0.237337	−	0.971427i	$-0.423726\pi$
$263$	2024.55	0.474673	0.237337	+	0.971427i	$0.423726\pi$
$264$	290.656	0.0677601
$265$	34.6290	0.00802734
$266$	0	0
$267$	−2532.58	−0.580492
$268$	−4463.09	−1.01726
$269$	3431.96	0.777882	0.388941	−	0.921263i	$-0.372841\pi$
$269$	3431.96	0.777882	0.388941	+	0.921263i	$0.372841\pi$
$270$	−50.2670	−0.0113302
$271$	2974.80	0.666812	0.333406	−	0.942783i	$-0.391802\pi$
$271$	2974.80	0.666812	0.333406	+	0.942783i	$0.391802\pi$
$272$	3893.70	0.867978
$273$	0	0
$274$	−893.358	−0.196970
$275$	−1254.09	−0.274999
$276$	−2881.40	−0.628405
$277$	7781.98	1.68799	0.843996	−	0.536350i	$-0.180197\pi$
$277$	7781.98	1.68799	0.843996	+	0.536350i	$0.180197\pi$
$278$	869.846	0.187662
$279$	503.727	0.108091
$280$	0	0
$281$	−2627.68	−0.557845	−0.278922	−	0.960314i	$-0.589977\pi$
$281$	−2627.68	−0.557845	−0.278922	+	0.960314i	$0.589977\pi$
$282$	186.625	0.0394091
$283$	2501.46	0.525430	0.262715	−	0.964873i	$-0.415382\pi$
$283$	2501.46	0.525430	0.262715	+	0.964873i	$0.415382\pi$
$284$	8924.89	1.86477
$285$	−1137.91	−0.236504
$286$	258.917	0.0535317
$287$	0	0
$288$	919.867	0.188207
$289$	−168.973	−0.0343931
$290$	−274.922	−0.0556689
$291$	2775.29	0.559073
$292$	−612.466	−0.122746
$293$	4646.04	0.926364	0.463182	−	0.886263i	$-0.346708\pi$
$293$	4646.04	0.926364	0.463182	+	0.886263i	$0.346708\pi$
$294$	0	0
$295$	604.331	0.119273
$296$	−1435.21	−0.281823
$297$	−297.000	−0.0580259
$298$	−262.859	−0.0510974
$299$	−5238.83	−1.01328
$300$	−2628.35	−0.505826
$301$	0	0
$302$	1047.49	0.199590
$303$	−2654.60	−0.503310
$304$	6467.63	1.22021
$305$	−629.513	−0.118183
$306$	348.102	0.0650316
$307$	4325.92	0.804213	0.402107	−	0.915593i	$-0.368278\pi$
$307$	4325.92	0.804213	0.402107	+	0.915593i	$0.368278\pi$
$308$	0	0
$309$	3209.35	0.590853
$310$	104.201	0.0190910
$311$	−1845.22	−0.336440	−0.168220	−	0.985749i	$-0.553802\pi$
$311$	−1845.22	−0.336440	−0.168220	+	0.985749i	$0.553802\pi$
$312$	1107.55	0.200970
$313$	5287.15	0.954784	0.477392	−	0.878690i	$-0.341582\pi$
$313$	5287.15	0.954784	0.477392	+	0.878690i	$0.341582\pi$
$314$	−843.220	−0.151547
$315$	0	0
$316$	−8376.57	−1.49120
$317$	9353.74	1.65728	0.828641	−	0.559780i	$-0.189114\pi$
$317$	9353.74	1.65728	0.828641	+	0.559780i	$0.189114\pi$
$318$	−17.5964	−0.00310301
$319$	−1624.36	−0.285100
$320$	−1309.08	−0.228687
$321$	−534.588	−0.0929526
$322$	0	0
$323$	7880.08	1.35746
$324$	−622.457	−0.106731
$325$	−4778.75	−0.815622
$326$	294.479	0.0500296
$327$	−5920.73	−1.00128
$328$	2277.84	0.383453
$329$	0	0
$330$	−61.4374	−0.0102485
$331$	11385.6	1.89066	0.945330	−	0.326115i	$-0.105740\pi$
$331$	11385.6	1.89066	0.945330	+	0.326115i	$0.105740\pi$
$332$	−6722.77	−1.11132
$333$	1466.53	0.241338
$334$	1849.25	0.302953
$335$	1925.48	0.314031
$336$	0	0
$337$	−10686.1	−1.72732	−0.863660	−	0.504075i	$-0.831833\pi$
$337$	−10686.1	−1.72732	−0.863660	+	0.504075i	$0.831833\pi$
$338$	−247.124	−0.0397686
$339$	−4928.05	−0.789542
$340$	−1754.80	−0.279903
$341$	615.667	0.0977719
$342$	578.216	0.0914220
$343$	0	0
$344$	940.129	0.147350
$345$	1243.10	0.193990
$346$	1705.42	0.264982
$347$	6242.71	0.965782	0.482891	−	0.875680i	$-0.339587\pi$
$347$	6242.71	0.965782	0.482891	+	0.875680i	$0.339587\pi$
$348$	−3404.37	−0.524406
$349$	−4518.02	−0.692964	−0.346482	−	0.938057i	$-0.612624\pi$
$349$	−4518.02	−0.692964	−0.346482	+	0.938057i	$0.612624\pi$
$350$	0	0
$351$	−1131.72	−0.172100
$352$	1124.28	0.170240
$353$	9270.29	1.39776	0.698878	−	0.715241i	$-0.253683\pi$
$353$	9270.29	1.39776	0.698878	+	0.715241i	$0.253683\pi$
$354$	−307.085	−0.0461056
$355$	−3850.40	−0.575657
$356$	−6487.34	−0.965809
$357$	0	0
$358$	387.557	0.0572151
$359$	8219.53	1.20838	0.604192	−	0.796839i	$-0.293496\pi$
$359$	8219.53	1.20838	0.604192	+	0.796839i	$0.293496\pi$
$360$	−262.807	−0.0384754
$361$	6230.22	0.908328
$362$	1422.94	0.206597
$363$	−363.000	−0.0524864
$364$	0	0
$365$	264.232	0.0378919
$366$	319.881	0.0456843
$367$	−9001.57	−1.28032	−0.640161	−	0.768241i	$-0.721132\pi$
$367$	−9001.57	−1.28032	−0.640161	+	0.768241i	$0.721132\pi$
$368$	−7065.55	−1.00086
$369$	−2327.56	−0.328368
$370$	303.367	0.0426251
$371$	0	0
$372$	1290.32	0.179839
$373$	1966.15	0.272931	0.136466	−	0.990645i	$-0.456426\pi$
$373$	1966.15	0.272931	0.136466	+	0.990645i	$0.456426\pi$
$374$	425.458	0.0588233
$375$	2377.18	0.327353
$376$	975.718	0.133827
$377$	−6189.67	−0.845582
$378$	0	0
$379$	1022.30	0.138554	0.0692768	−	0.997597i	$-0.477931\pi$
$379$	1022.30	0.138554	0.0692768	+	0.997597i	$0.477931\pi$
$380$	−2914.81	−0.393491
$381$	5390.83	0.724884
$382$	1036.14	0.138778
$383$	−12825.9	−1.71116	−0.855579	−	0.517672i	$-0.826799\pi$
$383$	−12825.9	−1.71116	−0.855579	+	0.517672i	$0.826799\pi$
$384$	3118.17	0.414384
$385$	0	0
$386$	−1490.63	−0.196557
$387$	−960.648	−0.126182
$388$	7109.05	0.930174
$389$	−12229.1	−1.59393	−0.796964	−	0.604027i	$-0.793562\pi$
$389$	−12229.1	−1.59393	−0.796964	+	0.604027i	$0.793562\pi$
$390$	−234.108	−0.0303963
$391$	−8608.57	−1.11344
$392$	0	0
$393$	2802.91	0.359767
$394$	92.1266	0.0117799
$395$	3613.85	0.460335
$396$	−760.781	−0.0965422
$397$	3401.07	0.429961	0.214981	−	0.976618i	$-0.431031\pi$
$397$	3401.07	0.429961	0.214981	+	0.976618i	$0.431031\pi$
$398$	−1622.00	−0.204280
$399$	0	0
$400$	−6445.04	−0.805630
$401$	−6234.40	−0.776386	−0.388193	−	0.921578i	$-0.626901\pi$
$401$	−6234.40	−0.776386	−0.388193	+	0.921578i	$0.626901\pi$
$402$	−978.415	−0.121390
$403$	2346.01	0.289983
$404$	−6799.90	−0.837396
$405$	268.543	0.0329481
$406$	0	0
$407$	1792.43	0.218298
$408$	1819.95	0.220836
$409$	11429.3	1.38177	0.690885	−	0.722964i	$-0.257221\pi$
$409$	11429.3	1.38177	0.690885	+	0.722964i	$0.257221\pi$
$410$	−481.478	−0.0579964
$411$	4772.61	0.572787
$412$	8220.93	0.983048
$413$	0	0
$414$	−631.670	−0.0749877
$415$	2900.36	0.343068
$416$	4284.10	0.504916
$417$	−4647.01	−0.545719
$418$	706.708	0.0826943
$419$	−3827.73	−0.446293	−0.223146	−	0.974785i	$-0.571633\pi$
$419$	−3827.73	−0.446293	−0.223146	+	0.974785i	$0.571633\pi$
$420$	0	0
$421$	−11626.8	−1.34597	−0.672987	−	0.739654i	$-0.734989\pi$
$421$	−11626.8	−1.34597	−0.672987	+	0.739654i	$0.734989\pi$
$422$	354.315	0.0408716
$423$	−997.014	−0.114602
$424$	−91.9979	−0.0105373
$425$	−7852.55	−0.896246
$426$	1956.55	0.222523
$427$	0	0
$428$	−1369.38	−0.154652
$429$	−1383.22	−0.155670
$430$	−198.720	−0.0222863
$431$	7469.80	0.834820	0.417410	−	0.908718i	$-0.362938\pi$
$431$	7469.80	0.834820	0.417410	+	0.908718i	$0.362938\pi$
$432$	−1526.34	−0.169991
$433$	−1546.97	−0.171692	−0.0858459	−	0.996308i	$-0.527359\pi$
$433$	−1546.97	−0.171692	−0.0858459	+	0.996308i	$0.527359\pi$
$434$	0	0
$435$	1468.72	0.161885
$436$	−15166.3	−1.66590
$437$	−14299.3	−1.56528
$438$	−134.267	−0.0146473
$439$	8082.79	0.878748	0.439374	−	0.898304i	$-0.355200\pi$
$439$	8082.79	0.878748	0.439374	+	0.898304i	$0.355200\pi$
$440$	−321.208	−0.0348023
$441$	0	0
$442$	1621.22	0.174465
$443$	−13000.1	−1.39426	−0.697128	−	0.716947i	$-0.745539\pi$
$443$	−13000.1	−1.39426	−0.697128	+	0.716947i	$0.745539\pi$
$444$	3756.60	0.401532
$445$	2798.79	0.298147
$446$	3511.11	0.372771
$447$	1404.28	0.148591
$448$	0	0
$449$	−4572.68	−0.480619	−0.240310	−	0.970696i	$-0.577249\pi$
$449$	−4572.68	−0.480619	−0.240310	+	0.970696i	$0.577249\pi$
$450$	−576.196	−0.0603603
$451$	−2844.79	−0.297020
$452$	−12623.4	−1.31362
$453$	−5596.03	−0.580407
$454$	1862.22	0.192507
$455$	0	0
$456$	3023.04	0.310454
$457$	−10846.5	−1.11024	−0.555119	−	0.831771i	$-0.687327\pi$
$457$	−10846.5	−1.11024	−0.555119	+	0.831771i	$0.687327\pi$
$458$	−1728.01	−0.176298
$459$	−1859.68	−0.189112
$460$	3184.27	0.322755
$461$	1299.33	0.131270	0.0656352	−	0.997844i	$-0.479093\pi$
$461$	1299.33	0.131270	0.0656352	+	0.997844i	$0.479093\pi$
$462$	0	0
$463$	12116.9	1.21624	0.608120	−	0.793845i	$-0.291924\pi$
$463$	12116.9	1.21624	0.608120	+	0.793845i	$0.291924\pi$
$464$	−8347.94	−0.835223
$465$	−556.676	−0.0555166
$466$	762.633	0.0758118
$467$	13546.6	1.34232	0.671160	−	0.741313i	$-0.265796\pi$
$467$	13546.6	1.34232	0.671160	+	0.741313i	$0.265796\pi$
$468$	−2898.97	−0.286335
$469$	0	0
$470$	−206.242	−0.0202409
$471$	4504.76	0.440697
$472$	−1605.51	−0.156567
$473$	−1174.12	−0.114136
$474$	−1836.34	−0.177945
$475$	−13043.5	−1.25995
$476$	0	0
$477$	94.0058	0.00902355
$478$	−2331.10	−0.223059
$479$	10663.2	1.01715	0.508573	−	0.861019i	$-0.330173\pi$
$479$	10663.2	1.01715	0.508573	+	0.861019i	$0.330173\pi$
$480$	−1016.56	−0.0966652
$481$	6830.08	0.647453
$482$	1960.78	0.185293
$483$	0	0
$484$	−929.844	−0.0873257
$485$	−3067.01	−0.287146
$486$	−136.457	−0.0127363
$487$	8149.91	0.758332	0.379166	−	0.925329i	$-0.376211\pi$
$487$	8149.91	0.758332	0.379166	+	0.925329i	$0.376211\pi$
$488$	1672.41	0.155136
$489$	−1573.20	−0.145486
$490$	0	0
$491$	−15142.4	−1.39178	−0.695891	−	0.718147i	$-0.744990\pi$
$491$	−15142.4	−1.39178	−0.695891	+	0.718147i	$0.744990\pi$
$492$	−5962.16	−0.546331
$493$	−10171.0	−0.929167
$494$	2692.93	0.245264
$495$	328.219	0.0298027
$496$	3164.04	0.286430
$497$	0	0
$498$	−1473.79	−0.132615
$499$	10430.8	0.935764	0.467882	−	0.883791i	$-0.345017\pi$
$499$	10430.8	0.935764	0.467882	+	0.883791i	$0.345017\pi$
$500$	6089.28	0.544642
$501$	−9879.28	−0.880986
$502$	493.820	0.0439049
$503$	−2173.55	−0.192671	−0.0963356	−	0.995349i	$-0.530712\pi$
$503$	−2173.55	−0.192671	−0.0963356	+	0.995349i	$0.530712\pi$
$504$	0	0
$505$	2933.64	0.258505
$506$	−772.042	−0.0678289
$507$	1320.22	0.115647
$508$	13808.9	1.20605
$509$	13729.6	1.19559	0.597793	−	0.801651i	$-0.296045\pi$
$509$	13729.6	1.19559	0.597793	+	0.801651i	$0.296045\pi$
$510$	−384.692	−0.0334009
$511$	0	0
$512$	9761.22	0.842557
$513$	−3089.02	−0.265855
$514$	−3195.85	−0.274247
$515$	−3546.70	−0.303468
$516$	−2460.75	−0.209939
$517$	−1218.57	−0.103661
$518$	0	0
$519$	−9110.89	−0.770566
$520$	−1223.97	−0.103220
$521$	−18257.2	−1.53525	−0.767623	−	0.640902i	$-0.778561\pi$
$521$	−18257.2	−1.53525	−0.767623	+	0.640902i	$0.778561\pi$
$522$	−746.318	−0.0625775
$523$	19979.6	1.67045	0.835225	−	0.549908i	$-0.185337\pi$
$523$	19979.6	1.67045	0.835225	+	0.549908i	$0.185337\pi$
$524$	7179.81	0.598572
$525$	0	0
$526$	1136.89	0.0942411
$527$	3855.02	0.318648
$528$	−1865.53	−0.153763
$529$	3454.21	0.283900
$530$	19.4460	0.00159374
$531$	1640.55	0.134075
$532$	0	0
$533$	−10840.1	−0.880935
$534$	−1422.18	−0.115250
$535$	590.780	0.0477414
$536$	−5115.37	−0.412221
$537$	−2070.45	−0.166381
$538$	1927.23	0.154440
$539$	0	0
$540$	687.886	0.0548184
$541$	7904.01	0.628133	0.314066	−	0.949401i	$-0.398309\pi$
$541$	7904.01	0.628133	0.314066	+	0.949401i	$0.398309\pi$
$542$	1670.51	0.132388
$543$	−7601.83	−0.600784
$544$	7039.73	0.554827
$545$	6543.08	0.514265
$546$	0	0
$547$	16806.6	1.31370	0.656852	−	0.754019i	$-0.271887\pi$
$547$	16806.6	1.31370	0.656852	+	0.754019i	$0.271887\pi$
$548$	12225.3	0.952991
$549$	−1708.91	−0.132850
$550$	−704.240	−0.0545980
$551$	−16894.6	−1.30623
$552$	−3302.51	−0.254645
$553$	0	0
$554$	4369.99	0.335132
$555$	−1620.68	−0.123954
$556$	−11903.6	−0.907955
$557$	−8002.82	−0.608780	−0.304390	−	0.952548i	$-0.598453\pi$
$557$	−8002.82	−0.608780	−0.304390	+	0.952548i	$0.598453\pi$
$558$	282.869	0.0214602
$559$	−4474.03	−0.338517
$560$	0	0
$561$	−2272.94	−0.171058
$562$	−1475.58	−0.110754
$563$	12327.9	0.922840	0.461420	−	0.887182i	$-0.347340\pi$
$563$	12327.9	0.922840	0.461420	+	0.887182i	$0.347340\pi$
$564$	−2553.90	−0.190672
$565$	5446.05	0.405517
$566$	1404.70	0.104318
$567$	0	0
$568$	10229.2	0.755651
$569$	−6786.77	−0.500028	−0.250014	−	0.968242i	$-0.580435\pi$
$569$	−6786.77	−0.500028	−0.250014	+	0.968242i	$0.580435\pi$
$570$	−638.994	−0.0469553
$571$	16890.0	1.23787	0.618935	−	0.785442i	$-0.287564\pi$
$571$	16890.0	1.23787	0.618935	+	0.785442i	$0.287564\pi$
$572$	−3543.19	−0.259000
$573$	−5535.38	−0.403567
$574$	0	0
$575$	14249.3	1.03346
$576$	−3553.70	−0.257067
$577$	−1857.64	−0.134029	−0.0670144	−	0.997752i	$-0.521347\pi$
$577$	−1857.64	−0.134029	−0.0670144	+	0.997752i	$0.521347\pi$
$578$	−94.8875	−0.00682837
$579$	7963.44	0.571588
$580$	3762.22	0.269341
$581$	0	0
$582$	1558.47	0.110998
$583$	114.896	0.00816210
$584$	−701.977	−0.0497398
$585$	1250.68	0.0883922
$586$	2609.00	0.183919
$587$	−21977.7	−1.54534	−0.772672	−	0.634805i	$-0.781080\pi$
$587$	−21977.7	−1.54534	−0.772672	+	0.634805i	$0.781080\pi$
$588$	0	0
$589$	6403.39	0.447958
$590$	339.364	0.0236803
$591$	−492.171	−0.0342558
$592$	9211.65	0.639521
$593$	−17263.7	−1.19550	−0.597752	−	0.801681i	$-0.703939\pi$
$593$	−17263.7	−1.19550	−0.597752	+	0.801681i	$0.703939\pi$
$594$	−166.781	−0.0115204
$595$	0	0
$596$	3597.14	0.247223
$597$	8665.23	0.594044
$598$	−2941.88	−0.201175
$599$	−23211.4	−1.58329	−0.791646	−	0.610981i	$-0.790775\pi$
$599$	−23211.4	−1.58329	−0.791646	+	0.610981i	$0.790775\pi$
$600$	−3012.48	−0.204973
$601$	20051.0	1.36089	0.680446	−	0.732799i	$-0.261786\pi$
$601$	20051.0	1.36089	0.680446	+	0.732799i	$0.261786\pi$
$602$	0	0
$603$	5227.01	0.353002
$604$	−14334.5	−0.965668
$605$	401.156	0.0269576
$606$	−1490.70	−0.0999266
$607$	7862.63	0.525756	0.262878	−	0.964829i	$-0.415328\pi$
$607$	7862.63	0.525756	0.262878	+	0.964829i	$0.415328\pi$
$608$	11693.4	0.779981
$609$	0	0
$610$	−353.505	−0.0234639
$611$	−4643.39	−0.307449
$612$	−4763.66	−0.314640
$613$	−16366.2	−1.07835	−0.539173	−	0.842195i	$-0.681263\pi$
$613$	−16366.2	−1.07835	−0.539173	+	0.842195i	$0.681263\pi$
$614$	2429.23	0.159668
$615$	2572.21	0.168653
$616$	0	0
$617$	3749.07	0.244622	0.122311	−	0.992492i	$-0.460969\pi$
$617$	3749.07	0.244622	0.122311	+	0.992492i	$0.460969\pi$
$618$	1802.22	0.117307
$619$	−20036.9	−1.30105	−0.650525	−	0.759485i	$-0.725451\pi$
$619$	−20036.9	−1.30105	−0.650525	+	0.759485i	$0.725451\pi$
$620$	−1425.95	−0.0923673
$621$	3374.59	0.218064
$622$	−1036.19	−0.0667965
$623$	0	0
$624$	−7108.64	−0.456047
$625$	11624.0	0.743936
$626$	2969.01	0.189562
$627$	−3775.47	−0.240475
$628$	11539.2	0.733222
$629$	11223.3	0.711453
$630$	0	0
$631$	5839.34	0.368400	0.184200	−	0.982889i	$-0.441031\pi$
$631$	5839.34	0.368400	0.184200	+	0.982889i	$0.441031\pi$
$632$	−9600.80	−0.604271
$633$	−1892.87	−0.118854
$634$	5252.62	0.329035
$635$	−5957.49	−0.372308
$636$	240.801	0.0150132
$637$	0	0
$638$	−912.166	−0.0566035
$639$	−10452.5	−0.647097
$640$	−3445.94	−0.212832
$641$	−14682.2	−0.904702	−0.452351	−	0.891840i	$-0.649415\pi$
$641$	−14682.2	−0.904702	−0.452351	+	0.891840i	$0.649415\pi$
$642$	−300.199	−0.0184547
$643$	−30541.3	−1.87314	−0.936571	−	0.350477i	$-0.886019\pi$
$643$	−30541.3	−1.87314	−0.936571	+	0.350477i	$0.886019\pi$
$644$	0	0
$645$	1061.63	0.0648084
$646$	4425.08	0.269508
$647$	−7991.44	−0.485588	−0.242794	−	0.970078i	$-0.578064\pi$
$647$	−7991.44	−0.485588	−0.242794	+	0.970078i	$0.578064\pi$
$648$	−713.429	−0.0432502
$649$	2005.11	0.121275
$650$	−2683.52	−0.161933
$651$	0	0
$652$	−4029.84	−0.242056
$653$	18400.0	1.10268	0.551338	−	0.834282i	$-0.314117\pi$
$653$	18400.0	1.10268	0.551338	+	0.834282i	$0.314117\pi$
$654$	−3324.80	−0.198792
$655$	−3097.54	−0.184780
$656$	−14620.0	−0.870143
$657$	717.298	0.0425943
$658$	0	0
$659$	15887.4	0.939126	0.469563	−	0.882899i	$-0.344411\pi$
$659$	15887.4	0.939126	0.469563	+	0.882899i	$0.344411\pi$
$660$	840.750	0.0495851
$661$	14947.8	0.879582	0.439791	−	0.898100i	$-0.355053\pi$
$661$	14947.8	0.879582	0.439791	+	0.898100i	$0.355053\pi$
$662$	6393.61	0.375370
$663$	−8661.07	−0.507343
$664$	−7705.29	−0.450336
$665$	0	0
$666$	823.535	0.0479149
$667$	18456.4	1.07142
$668$	−25306.3	−1.46576
$669$	−18757.5	−1.08402
$670$	1081.26	0.0623473
$671$	−2088.67	−0.120167
$672$	0	0
$673$	1091.22	0.0625014	0.0312507	−	0.999512i	$-0.490051\pi$
$673$	1091.22	0.0625014	0.0312507	+	0.999512i	$0.490051\pi$
$674$	−6000.79	−0.342940
$675$	3078.23	0.175528
$676$	3381.81	0.192411
$677$	26564.2	1.50804	0.754022	−	0.656849i	$-0.228111\pi$
$677$	26564.2	1.50804	0.754022	+	0.656849i	$0.228111\pi$
$678$	−2767.36	−0.156755
$679$	0	0
$680$	−2011.26	−0.113424
$681$	−9948.60	−0.559811
$682$	345.729	0.0194115
$683$	−20461.6	−1.14633	−0.573164	−	0.819441i	$-0.694284\pi$
$683$	−20461.6	−1.14633	−0.573164	+	0.819441i	$0.694284\pi$
$684$	−7912.69	−0.442323
$685$	−5274.28	−0.294190
$686$	0	0
$687$	9231.60	0.512675
$688$	−6034.07	−0.334370
$689$	437.813	0.0242081
$690$	698.068	0.0385145
$691$	12100.6	0.666175	0.333087	−	0.942896i	$-0.391910\pi$
$691$	12100.6	0.666175	0.333087	+	0.942896i	$0.391910\pi$
$692$	−23338.0	−1.28205
$693$	0	0
$694$	3505.61	0.191745
$695$	5135.47	0.280287
$696$	−3901.91	−0.212502
$697$	−17812.8	−0.968015
$698$	−2537.11	−0.137580
$699$	−4074.24	−0.220460
$700$	0	0
$701$	27360.5	1.47417	0.737083	−	0.675802i	$-0.236203\pi$
$701$	27360.5	1.47417	0.737083	+	0.675802i	$0.236203\pi$
$702$	−635.523	−0.0341685
$703$	18642.6	1.00017
$704$	−4343.41	−0.232526
$705$	1101.81	0.0588606
$706$	5205.76	0.277509
$707$	0	0
$708$	4202.35	0.223071
$709$	19314.2	1.02308	0.511538	−	0.859261i	$-0.329076\pi$
$709$	19314.2	1.02308	0.511538	+	0.859261i	$0.329076\pi$
$710$	−2162.21	−0.114290
$711$	9810.34	0.517463
$712$	−7435.45	−0.391370
$713$	−6995.36	−0.367431
$714$	0	0
$715$	1528.61	0.0799537
$716$	−5303.58	−0.276821
$717$	12453.5	0.648655
$718$	4615.70	0.239911
$719$	11811.2	0.612635	0.306317	−	0.951929i	$-0.400903\pi$
$719$	11811.2	0.612635	0.306317	+	0.951929i	$0.400903\pi$
$720$	1686.78	0.0873093
$721$	0	0
$722$	3498.60	0.180338
$723$	−10475.1	−0.538831
$724$	−19472.5	−0.999571
$725$	16835.6	0.862424
$726$	−203.844	−0.0104206
$727$	−17995.3	−0.918034	−0.459017	−	0.888428i	$-0.651798\pi$
$727$	−17995.3	−0.918034	−0.459017	+	0.888428i	$0.651798\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	148.380	0.00752301
$731$	−7351.83	−0.371980
$732$	−4377.46	−0.221032
$733$	28542.7	1.43827	0.719133	−	0.694873i	$-0.244539\pi$
$733$	28542.7	1.43827	0.719133	+	0.694873i	$0.244539\pi$
$734$	−5054.86	−0.254194
$735$	0	0
$736$	−12774.4	−0.639769
$737$	6388.57	0.319303
$738$	−1307.05	−0.0651938
$739$	−5283.32	−0.262991	−0.131495	−	0.991317i	$-0.541978\pi$
$739$	−5283.32	−0.262991	−0.131495	+	0.991317i	$0.541978\pi$
$740$	−4151.47	−0.206231
$741$	−14386.5	−0.713227
$742$	0	0
$743$	−8423.21	−0.415905	−0.207953	−	0.978139i	$-0.566680\pi$
$743$	−8423.21	−0.415905	−0.207953	+	0.978139i	$0.566680\pi$
$744$	1478.90	0.0728753
$745$	−1551.89	−0.0763180
$746$	1104.10	0.0541874
$747$	7873.47	0.385643
$748$	−5822.25	−0.284602
$749$	0	0
$750$	1334.91	0.0649923
$751$	26417.9	1.28363	0.641814	−	0.766861i	$-0.278182\pi$
$751$	26417.9	1.28363	0.641814	+	0.766861i	$0.278182\pi$
$752$	−6262.49	−0.303683
$753$	−2638.15	−0.127675
$754$	−3475.83	−0.167881
$755$	6184.25	0.298103
$756$	0	0
$757$	−24157.9	−1.15988	−0.579942	−	0.814658i	$-0.696925\pi$
$757$	−24157.9	−1.15988	−0.579942	+	0.814658i	$0.696925\pi$
$758$	574.073	0.0275083
$759$	4124.50	0.197246
$760$	−3340.80	−0.159452
$761$	9195.59	0.438029	0.219014	−	0.975722i	$-0.429716\pi$
$761$	9195.59	0.438029	0.219014	+	0.975722i	$0.429716\pi$
$762$	3027.24	0.143918
$763$	0	0
$764$	−14179.2	−0.671445
$765$	2055.15	0.0971297
$766$	−7202.43	−0.339732
$767$	7640.53	0.359692
$768$	−7725.50	−0.362982
$769$	15430.5	0.723589	0.361794	−	0.932258i	$-0.382164\pi$
$769$	15430.5	0.723589	0.361794	+	0.932258i	$0.382164\pi$
$770$	0	0
$771$	17073.3	0.797509
$772$	20398.8	0.950994
$773$	14100.0	0.656068	0.328034	−	0.944666i	$-0.393614\pi$
$773$	14100.0	0.656068	0.328034	+	0.944666i	$0.393614\pi$
$774$	−539.454	−0.0250521
$775$	−6381.02	−0.295759
$776$	8148.03	0.376930
$777$	0	0
$778$	−6867.26	−0.316457
$779$	−29587.9	−1.36084
$780$	3203.69	0.147065
$781$	−12775.3	−0.585321
$782$	−4834.17	−0.221061
$783$	3987.08	0.181975
$784$	0	0
$785$	−4978.27	−0.226347
$786$	1573.98	0.0714277
$787$	−4428.96	−0.200604	−0.100302	−	0.994957i	$-0.531981\pi$
$787$	−4428.96	−0.200604	−0.100302	+	0.994957i	$0.531981\pi$
$788$	−1260.72	−0.0569941
$789$	−6073.65	−0.274053
$790$	2029.37	0.0913944
$791$	0	0
$792$	−871.969	−0.0391213
$793$	−7958.90	−0.356405
$794$	1909.88	0.0853640
$795$	−103.887	−0.00463459
$796$	22196.4	0.988357
$797$	−32311.2	−1.43604	−0.718019	−	0.696024i	$-0.754951\pi$
$797$	−32311.2	−1.43604	−0.718019	+	0.696024i	$0.754951\pi$
$798$	0	0
$799$	−7630.14	−0.337841
$800$	−11652.5	−0.514973
$801$	7597.74	0.335147
$802$	−3500.94	−0.154143
$803$	876.698	0.0385280
$804$	13389.3	0.587317
$805$	0	0
$806$	1317.41	0.0575729
$807$	−10295.9	−0.449110
$808$	−7793.70	−0.339334
$809$	−45404.2	−1.97321	−0.986605	−	0.163125i	$-0.947843\pi$
$809$	−45404.2	−1.97321	−0.986605	+	0.163125i	$0.947843\pi$
$810$	150.801	0.00654149
$811$	36636.7	1.58630	0.793150	−	0.609026i	$-0.208440\pi$
$811$	36636.7	1.58630	0.793150	+	0.609026i	$0.208440\pi$
$812$	0	0
$813$	−8924.39	−0.384984
$814$	1006.54	0.0433407
$815$	1738.57	0.0747231
$816$	−11681.1	−0.501127
$817$	−12211.8	−0.522932
$818$	6418.17	0.274335
$819$	0	0
$820$	6588.86	0.280601
$821$	−14967.9	−0.636279	−0.318139	−	0.948044i	$-0.603058\pi$
$821$	−14967.9	−0.636279	−0.318139	+	0.948044i	$0.603058\pi$
$822$	2680.07	0.113721
$823$	−8353.81	−0.353822	−0.176911	−	0.984227i	$-0.556610\pi$
$823$	−8353.81	−0.353822	−0.176911	+	0.984227i	$0.556610\pi$
$824$	9422.41	0.398356
$825$	3762.28	0.158771
$826$	0	0
$827$	26797.4	1.12677	0.563384	−	0.826195i	$-0.309499\pi$
$827$	26797.4	1.12677	0.563384	+	0.826195i	$0.309499\pi$
$828$	8644.19	0.362810
$829$	14645.0	0.613562	0.306781	−	0.951780i	$-0.400748\pi$
$829$	14645.0	0.613562	0.306781	+	0.951780i	$0.400748\pi$
$830$	1628.70	0.0681122
$831$	−23345.9	−0.974563
$832$	−16550.6	−0.689651
$833$	0	0
$834$	−2609.54	−0.108346
$835$	10917.7	0.452484
$836$	−9671.06	−0.400097
$837$	−1511.18	−0.0624063
$838$	−2149.47	−0.0886065
$839$	10093.3	0.415327	0.207664	−	0.978200i	$-0.433414\pi$
$839$	10093.3	0.415327	0.207664	+	0.978200i	$0.433414\pi$
$840$	0	0
$841$	−2582.72	−0.105897
$842$	−6529.06	−0.267228
$843$	7883.04	0.322072
$844$	−4848.68	−0.197747
$845$	−1458.99	−0.0593975
$846$	−559.876	−0.0227529
$847$	0	0
$848$	590.474	0.0239115
$849$	−7504.39	−0.303357
$850$	−4409.62	−0.177940
$851$	−20366.0	−0.820374
$852$	−26774.7	−1.07663
$853$	−28743.0	−1.15374	−0.576872	−	0.816835i	$-0.695727\pi$
$853$	−28743.0	−1.15374	−0.576872	+	0.816835i	$0.695727\pi$
$854$	0	0
$855$	3413.72	0.136546
$856$	−1569.51	−0.0626690
$857$	19741.2	0.786870	0.393435	−	0.919352i	$-0.371287\pi$
$857$	19741.2	0.786870	0.393435	+	0.919352i	$0.371287\pi$
$858$	−776.750	−0.0309065
$859$	41550.0	1.65037	0.825185	−	0.564863i	$-0.191071\pi$
$859$	41550.0	1.65037	0.825185	+	0.564863i	$0.191071\pi$
$860$	2719.41	0.107827
$861$	0	0
$862$	4194.69	0.165744
$863$	45844.4	1.80830	0.904150	−	0.427215i	$-0.140505\pi$
$863$	45844.4	1.80830	0.904150	+	0.427215i	$0.140505\pi$
$864$	−2759.60	−0.108661
$865$	10068.6	0.395771
$866$	−868.704	−0.0340875
$867$	506.920	0.0198569
$868$	0	0
$869$	11990.4	0.468063
$870$	824.766	0.0321405
$871$	24343.8	0.947024
$872$	−17382.8	−0.675064
$873$	−8325.87	−0.322781
$874$	−8029.80	−0.310769
$875$	0	0
$876$	1837.40	0.0708675
$877$	2558.92	0.0985277	0.0492638	−	0.998786i	$-0.484312\pi$
$877$	2558.92	0.0985277	0.0492638	+	0.998786i	$0.484312\pi$
$878$	4538.91	0.174466
$879$	−13938.1	−0.534836
$880$	2061.62	0.0789742
$881$	−1119.16	−0.0427983	−0.0213992	−	0.999771i	$-0.506812\pi$
$881$	−1119.16	−0.0427983	−0.0213992	+	0.999771i	$0.506812\pi$
$882$	0	0
$883$	−1677.64	−0.0639377	−0.0319688	−	0.999489i	$-0.510178\pi$
$883$	−1677.64	−0.0639377	−0.0319688	+	0.999489i	$0.510178\pi$
$884$	−22185.8	−0.844105
$885$	−1812.99	−0.0688622
$886$	−7300.27	−0.276814
$887$	−12127.5	−0.459076	−0.229538	−	0.973300i	$-0.573722\pi$
$887$	−12127.5	−0.459076	−0.229538	+	0.973300i	$0.573722\pi$
$888$	4305.62	0.162711
$889$	0	0
$890$	1571.67	0.0591937
$891$	891.000	0.0335013
$892$	−48048.4	−1.80356
$893$	−12674.1	−0.474940
$894$	788.578	0.0295011
$895$	2288.09	0.0854551
$896$	0	0
$897$	15716.5	0.585015
$898$	−2567.80	−0.0954216
$899$	−8265.02	−0.306623
$900$	7885.05	0.292039
$901$	719.425	0.0266010
$902$	−1597.50	−0.0589700
$903$	0	0
$904$	−14468.4	−0.532312
$905$	8400.89	0.308569
$906$	−3142.47	−0.115233
$907$	−13826.8	−0.506186	−0.253093	−	0.967442i	$-0.581448\pi$
$907$	−13826.8	−0.506186	−0.253093	+	0.967442i	$0.581448\pi$
$908$	−25483.9	−0.931401
$909$	7963.81	0.290586
$910$	0	0
$911$	−41945.3	−1.52548	−0.762739	−	0.646706i	$-0.776146\pi$
$911$	−41945.3	−1.52548	−0.762739	+	0.646706i	$0.776146\pi$
$912$	−19402.9	−0.704489
$913$	9623.13	0.348827
$914$	−6090.90	−0.220426
$915$	1888.54	0.0682330
$916$	23647.2	0.852977
$917$	0	0
$918$	−1044.31	−0.0375460
$919$	−8132.54	−0.291913	−0.145956	−	0.989291i	$-0.546626\pi$
$919$	−8132.54	−0.291913	−0.145956	+	0.989291i	$0.546626\pi$
$920$	3649.65	0.130789
$921$	−12977.8	−0.464313
$922$	729.641	0.0260623
$923$	−48680.5	−1.73601
$924$	0	0
$925$	−18577.4	−0.660349
$926$	6804.27	0.241471
$927$	−9628.06	−0.341129
$928$	−15092.9	−0.533889
$929$	48334.7	1.70701	0.853504	−	0.521086i	$-0.174473\pi$
$929$	48334.7	1.70701	0.853504	+	0.521086i	$0.174473\pi$
$930$	−312.603	−0.0110222
$931$	0	0
$932$	−10436.4	−0.366797
$933$	5535.66	0.194244
$934$	7607.15	0.266503
$935$	2511.85	0.0878571
$936$	−3322.65	−0.116030
$937$	6068.39	0.211575	0.105787	−	0.994389i	$-0.466264\pi$
$937$	6068.39	0.211575	0.105787	+	0.994389i	$0.466264\pi$
$938$	0	0
$939$	−15861.5	−0.551245
$940$	2822.35	0.0979309
$941$	−15416.2	−0.534065	−0.267032	−	0.963688i	$-0.586043\pi$
$941$	−15416.2	−0.534065	−0.267032	+	0.963688i	$0.586043\pi$
$942$	2529.66	0.0874955
$943$	32323.3	1.11621
$944$	10304.7	0.355285
$945$	0	0
$946$	−659.333	−0.0226604
$947$	−45788.4	−1.57120	−0.785598	−	0.618737i	$-0.787645\pi$
$947$	−45788.4	−1.57120	−0.785598	+	0.618737i	$0.787645\pi$
$948$	25129.7	0.860944
$949$	3340.67	0.114271
$950$	−7324.61	−0.250149
$951$	−28061.2	−0.956833
$952$	0	0
$953$	36543.7	1.24215	0.621074	−	0.783752i	$-0.286697\pi$
$953$	36543.7	1.24215	0.621074	+	0.783752i	$0.286697\pi$
$954$	52.7892	0.00179152
$955$	6117.23	0.207276
$956$	31900.4	1.07922
$957$	4873.09	0.164603
$958$	5987.94	0.201943
$959$	0	0
$960$	3927.24	0.132032
$961$	−26658.4	−0.894847
$962$	3835.45	0.128545
$963$	1603.76	0.0536662
$964$	−26832.6	−0.896494
$965$	−8800.50	−0.293573
$966$	0	0
$967$	3077.00	0.102327	0.0511633	−	0.998690i	$-0.483707\pi$
$967$	3077.00	0.102327	0.0511633	+	0.998690i	$0.483707\pi$
$968$	−1065.74	−0.0353865
$969$	−23640.2	−0.783729
$970$	−1722.29	−0.0570096
$971$	−41714.2	−1.37865	−0.689327	−	0.724451i	$-0.742094\pi$
$971$	−41714.2	−1.37865	−0.689327	+	0.724451i	$0.742094\pi$
$972$	1867.37	0.0616214
$973$	0	0
$974$	4576.61	0.150558
$975$	14336.2	0.470900
$976$	−10734.1	−0.352039
$977$	−22237.2	−0.728178	−0.364089	−	0.931364i	$-0.618620\pi$
$977$	−22237.2	−0.728178	−0.364089	+	0.931364i	$0.618620\pi$
$978$	−883.436	−0.0288846
$979$	9286.12	0.303152
$980$	0	0
$981$	17762.2	0.578086
$982$	−8503.23	−0.276323
$983$	26617.2	0.863639	0.431820	−	0.901960i	$-0.357872\pi$
$983$	26617.2	0.863639	0.431820	+	0.901960i	$0.357872\pi$
$984$	−6833.52	−0.221387
$985$	543.905	0.0175942
$986$	−5711.56	−0.184476
$987$	0	0
$988$	−36851.8	−1.18665
$989$	13340.7	0.428928
$990$	184.312	0.00591700
$991$	28839.4	0.924435	0.462217	−	0.886767i	$-0.347054\pi$
$991$	28839.4	0.924435	0.462217	+	0.886767i	$0.347054\pi$
$992$	5720.52	0.183091
$993$	−34156.8	−1.09157
$994$	0	0
$995$	−9576.07	−0.305107
$996$	20168.3	0.641624
$997$	−35924.1	−1.14115	−0.570575	−	0.821246i	$-0.693280\pi$
$997$	−35924.1	−1.14115	−0.570575	+	0.821246i	$0.693280\pi$
$998$	5857.44	0.185786
$999$	−4399.59	−0.139336

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.561553 q^{2} -3.00000 q^{3} -7.68466 q^{4} +3.31534 q^{5} -1.68466 q^{6} -8.80776 q^{8} +9.00000 q^{9} +1.86174 q^{10} +11.0000 q^{11} +23.0540 q^{12} +41.9157 q^{13} -9.94602 q^{15} +56.5312 q^{16} +68.8769 q^{17} +5.05398 q^{18} +114.408 q^{19} -25.4773 q^{20} +6.17708 q^{22} -124.985 q^{23} +26.4233 q^{24} -114.009 q^{25} +23.5379 q^{26} -27.0000 q^{27} -147.670 q^{29} -5.58522 q^{30} +55.9697 q^{31} +102.207 q^{32} -33.0000 q^{33} +38.6780 q^{34} -69.1619 q^{36} +162.948 q^{37} +64.2462 q^{38} -125.747 q^{39} -29.2007 q^{40} -258.617 q^{41} -106.739 q^{43} -84.5312 q^{44} +29.8381 q^{45} -70.1856 q^{46} -110.779 q^{47} -169.594 q^{48} -64.0218 q^{50} -206.631 q^{51} -322.108 q^{52} +10.4451 q^{53} -15.1619 q^{54} +36.4688 q^{55} -343.224 q^{57} -82.9242 q^{58} +182.283 q^{59} +76.4318 q^{60} -189.879 q^{61} +31.4299 q^{62} -394.855 q^{64} +138.965 q^{65} -18.5312 q^{66} +580.779 q^{67} -529.295 q^{68} +374.955 q^{69} -1161.39 q^{71} -79.2699 q^{72} +79.6998 q^{73} +91.5038 q^{74} +342.026 q^{75} -879.187 q^{76} -70.6137 q^{78} +1090.04 q^{79} +187.420 q^{80} +81.0000 q^{81} -145.227 q^{82} +874.830 q^{83} +228.350 q^{85} -59.9394 q^{86} +443.009 q^{87} -96.8854 q^{88} +844.193 q^{89} +16.7557 q^{90} +960.466 q^{92} -167.909 q^{93} -62.2084 q^{94} +379.302 q^{95} -306.622 q^{96} -925.097 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 6 q^{3} - 3 q^{4} + 19 q^{5} + 9 q^{6} + 3 q^{8} + 18 q^{9} - 54 q^{10} + 22 q^{11} + 9 q^{12} - 11 q^{13} - 57 q^{15} - 23 q^{16} + 146 q^{17} - 27 q^{18} + 101 q^{19} + 48 q^{20} - 33 q^{22}+ \cdots + 198 q^{99}+O(q^{100}) \) 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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