LMFDB - Embedded newform 1617.4.a.h.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:	$1$
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)

$f(q)$	$=$	\(q+0.561553 q^{2} -3.00000 q^{3} -7.68466 q^{4} -18.6847 q^{5} -1.68466 q^{6} -8.80776 q^{8} +9.00000 q^{9} -10.4924 q^{10} -11.0000 q^{11} +23.0540 q^{12} -36.4384 q^{13} +56.0540 q^{15} +56.5312 q^{16} -41.1231 q^{17} +5.05398 q^{18} +23.6998 q^{19} +143.585 q^{20} -6.17708 q^{22} -140.047 q^{23} +26.4233 q^{24} +224.116 q^{25} -20.4621 q^{26} -27.0000 q^{27} +278.455 q^{29} +31.4773 q^{30} -191.447 q^{31} +102.207 q^{32} +33.0000 q^{33} -23.0928 q^{34} -69.1619 q^{36} +196.115 q^{37} +13.3087 q^{38} +109.315 q^{39} +164.570 q^{40} +322.695 q^{41} -3.67615 q^{43} +84.5312 q^{44} -168.162 q^{45} -78.6440 q^{46} +397.596 q^{47} -169.594 q^{48} +125.853 q^{50} +123.369 q^{51} +280.017 q^{52} +597.508 q^{53} -15.1619 q^{54} +205.531 q^{55} -71.0994 q^{57} +156.367 q^{58} -668.779 q^{59} -430.756 q^{60} +667.788 q^{61} -107.508 q^{62} -394.855 q^{64} +680.840 q^{65} +18.5312 q^{66} -730.762 q^{67} +316.017 q^{68} +420.142 q^{69} -31.2651 q^{71} -79.2699 q^{72} +434.408 q^{73} +110.129 q^{74} -672.349 q^{75} -182.125 q^{76} +61.3863 q^{78} -782.004 q^{79} -1056.27 q^{80} +81.0000 q^{81} +181.210 q^{82} +426.705 q^{83} +768.371 q^{85} -2.06435 q^{86} -835.366 q^{87} +96.8854 q^{88} +899.693 q^{89} -94.4318 q^{90} +1076.22 q^{92} +574.341 q^{93} +223.271 q^{94} -442.823 q^{95} -306.622 q^{96} +942.716 q^{97} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 6 q^{3} - 3 q^{4} - 25 q^{5} + 9 q^{6} + 3 q^{8} + 18 q^{9} + 12 q^{10} - 22 q^{11} + 9 q^{12} - 77 q^{13} + 75 q^{15} - 23 q^{16} - 74 q^{17} - 27 q^{18} + 101 q^{19} + 114 q^{20} + 33 q^{22}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 - 6 * q^3 - 3 * q^4 - 25 * q^5 + 9 * q^6 + 3 * q^8 + 18 * q^9 + 12 * q^10 - 22 * q^11 + 9 * q^12 - 77 * q^13 + 75 * q^15 - 23 * q^16 - 74 * q^17 - 27 * q^18 + 101 * q^19 + 114 * q^20 + 33 * q^22 + 58 * q^23 - 9 * q^24 + 139 * q^25 + 124 * q^26 - 54 * q^27 + 91 * q^29 - 36 * q^30 - 152 * q^31 + 291 * q^32 + 66 * q^33 + 94 * q^34 - 27 * q^36 + 619 * q^37 - 262 * q^38 + 231 * q^39 + 90 * q^40 - 138 * q^41 - 230 * q^43 + 33 * q^44 - 225 * q^45 - 784 * q^46 - 149 * q^47 + 69 * q^48 + 429 * q^50 + 222 * q^51 + 90 * q^52 + 1228 * q^53 + 81 * q^54 + 275 * q^55 - 303 * q^57 + 824 * q^58 - 649 * q^59 - 342 * q^60 + 412 * q^61 - 248 * q^62 - 431 * q^64 + 937 * q^65 - 99 * q^66 - 1243 * q^67 + 162 * q^68 - 174 * q^69 + 960 * q^71 + 27 * q^72 + 741 * q^73 - 1396 * q^74 - 417 * q^75 + 180 * q^76 - 372 * q^78 - 492 * q^79 - 554 * q^80 + 162 * q^81 + 1822 * q^82 + 1744 * q^83 + 976 * q^85 + 804 * q^86 - 273 * q^87 - 33 * q^88 + 1552 * q^89 + 108 * q^90 + 2004 * q^92 + 456 * q^93 + 2170 * q^94 - 931 * q^95 - 873 * q^96 - 440 * 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
$5$	−18.6847	−1.67121	−0.835603	−	0.549333i	$-0.814882\pi$
$5$	−18.6847	−1.67121	−0.835603	+	0.549333i	$0.814882\pi$
$6$	−1.68466	−0.114626
$7$	0	0
$7$	0	0
$8$	−8.80776	−0.389252
$9$	9.00000	0.333333
$10$	−10.4924	−0.331800
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	23.0540	0.554592
$13$	−36.4384	−0.777401	−0.388700	−	0.921364i	$-0.627076\pi$
$13$	−36.4384	−0.777401	−0.388700	+	0.921364i	$0.627076\pi$
$14$	0	0
$15$	56.0540	0.964872
$16$	56.5312	0.883301
$17$	−41.1231	−0.586695	−0.293348	−	0.956006i	$-0.594769\pi$
$17$	−41.1231	−0.586695	−0.293348	+	0.956006i	$0.594769\pi$
$18$	5.05398	0.0661796
$19$	23.6998	0.286164	0.143082	−	0.989711i	$-0.454299\pi$
$19$	23.6998	0.286164	0.143082	+	0.989711i	$0.454299\pi$
$20$	143.585	1.60533
$21$	0	0
$22$	−6.17708	−0.0598617
$23$	−140.047	−1.26965	−0.634824	−	0.772657i	$-0.718927\pi$
$23$	−140.047	−1.26965	−0.634824	+	0.772657i	$0.718927\pi$
$24$	26.4233	0.224735
$25$	224.116	1.79293
$26$	−20.4621	−0.154344
$27$	−27.0000	−0.192450
$28$	0	0
$29$	278.455	1.78303	0.891515	−	0.452991i	$-0.149643\pi$
$29$	278.455	1.78303	0.891515	+	0.452991i	$0.149643\pi$
$30$	31.4773	0.191565
$31$	−191.447	−1.10919	−0.554595	−	0.832120i	$-0.687127\pi$
$31$	−191.447	−1.10919	−0.554595	+	0.832120i	$0.687127\pi$
$32$	102.207	0.564621
$33$	33.0000	0.174078
$34$	−23.0928	−0.116482
$35$	0	0
$36$	−69.1619	−0.320194
$37$	196.115	0.871379	0.435690	−	0.900097i	$-0.356504\pi$
$37$	196.115	0.871379	0.435690	+	0.900097i	$0.356504\pi$
$38$	13.3087	0.0568146
$39$	109.315	0.448832
$40$	164.570	0.650520
$41$	322.695	1.22918	0.614591	−	0.788846i	$-0.289321\pi$
$41$	322.695	1.22918	0.614591	+	0.788846i	$0.289321\pi$
$42$	0	0
$43$	−3.67615	−0.0130374	−0.00651869	−	0.999979i	$-0.502075\pi$
$43$	−3.67615	−0.0130374	−0.00651869	+	0.999979i	$0.502075\pi$
$44$	84.5312	0.289626
$45$	−168.162	−0.557069
$46$	−78.6440	−0.252074
$47$	397.596	1.23394	0.616971	−	0.786986i	$-0.288360\pi$
$47$	397.596	1.23394	0.616971	+	0.786986i	$0.288360\pi$
$48$	−169.594	−0.509974
$49$	0	0
$50$	125.853	0.355967
$51$	123.369	0.338729
$52$	280.017	0.746757
$53$	597.508	1.54857	0.774283	−	0.632840i	$-0.218111\pi$
$53$	597.508	1.54857	0.774283	+	0.632840i	$0.218111\pi$
$54$	−15.1619	−0.0382088
$55$	205.531	0.503888
$56$	0	0
$57$	−71.0994	−0.165217
$58$	156.367	0.354001
$59$	−668.779	−1.47572	−0.737861	−	0.674952i	$-0.764164\pi$
$59$	−668.779	−1.47572	−0.737861	+	0.674952i	$0.764164\pi$
$60$	−430.756	−0.926839
$61$	667.788	1.40166	0.700832	−	0.713327i	$-0.252812\pi$
$61$	667.788	1.40166	0.700832	+	0.713327i	$0.252812\pi$
$62$	−107.508	−0.220217
$63$	0	0
$64$	−394.855	−0.771201
$65$	680.840	1.29920
$66$	18.5312	0.0345612
$67$	−730.762	−1.33249	−0.666245	−	0.745733i	$-0.732099\pi$
$67$	−730.762	−1.33249	−0.666245	+	0.745733i	$0.732099\pi$
$68$	316.017	0.563569
$69$	420.142	0.733031
$70$	0	0
$71$	−31.2651	−0.0522603	−0.0261302	−	0.999659i	$-0.508318\pi$
$71$	−31.2651	−0.0522603	−0.0261302	+	0.999659i	$0.508318\pi$
$72$	−79.2699	−0.129751
$73$	434.408	0.696488	0.348244	−	0.937404i	$-0.386778\pi$
$73$	434.408	0.696488	0.348244	+	0.937404i	$0.386778\pi$
$74$	110.129	0.173003
$75$	−672.349	−1.03515
$76$	−182.125	−0.274884
$77$	0	0
$78$	61.3863	0.0891107
$79$	−782.004	−1.11370	−0.556850	−	0.830613i	$-0.687990\pi$
$79$	−782.004	−1.11370	−0.556850	+	0.830613i	$0.687990\pi$
$80$	−1056.27	−1.47618
$81$	81.0000	0.111111
$82$	181.210	0.244041
$83$	426.705	0.564300	0.282150	−	0.959370i	$-0.408952\pi$
$83$	426.705	0.564300	0.282150	+	0.959370i	$0.408952\pi$
$84$	0	0
$85$	768.371	0.980489
$86$	−2.06435	−0.00258843
$87$	−835.366	−1.02943
$88$	96.8854	0.117364
$89$	899.693	1.07154	0.535771	−	0.844363i	$-0.320021\pi$
$89$	899.693	1.07154	0.535771	+	0.844363i	$0.320021\pi$
$90$	−94.4318	−0.110600
$91$	0	0
$92$	1076.22	1.21960
$93$	574.341	0.640391
$94$	223.271	0.244986
$95$	−442.823	−0.478239
$96$	−306.622	−0.325984
$97$	942.716	0.986786	0.493393	−	0.869806i	$-0.335756\pi$
$97$	942.716	0.986786	0.493393	+	0.869806i	$0.335756\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	−1722.26	−1.72226
$101$	−366.945	−0.361509	−0.180754	−	0.983528i	$-0.557854\pi$
$101$	−366.945	−0.361509	−0.180754	+	0.983528i	$0.557854\pi$
$102$	69.2784	0.0672508
$103$	−1007.35	−0.963658	−0.481829	−	0.876265i	$-0.660027\pi$
$103$	−1007.35	−0.963658	−0.481829	+	0.876265i	$0.660027\pi$
$104$	320.941	0.302605
$105$	0	0
$106$	335.532	0.307451
$107$	−1690.80	−1.52763	−0.763814	−	0.645437i	$-0.776675\pi$
$107$	−1690.80	−1.52763	−0.763814	+	0.645437i	$0.776675\pi$
$108$	207.486	0.184864
$109$	−1808.57	−1.58926	−0.794631	−	0.607093i	$-0.792336\pi$
$109$	−1808.57	−1.58926	−0.794631	+	0.607093i	$0.792336\pi$
$110$	115.417	0.100041
$111$	−588.344	−0.503091
$112$	0	0
$113$	952.557	0.793000	0.396500	−	0.918035i	$-0.370225\pi$
$113$	952.557	0.793000	0.396500	+	0.918035i	$0.370225\pi$
$114$	−39.9261	−0.0328019
$115$	2616.74	2.12184
$116$	−2139.84	−1.71275
$117$	−327.946	−0.259134
$118$	−375.555	−0.292988
$119$	0	0
$120$	−493.710	−0.375578
$121$	121.000	0.0909091
$122$	374.998	0.278285
$123$	−968.085	−0.709669
$124$	1471.20	1.06547
$125$	−1851.96	−1.32515
$126$	0	0
$127$	252.951	0.176738	0.0883691	−	0.996088i	$-0.471834\pi$
$127$	252.951	0.176738	0.0883691	+	0.996088i	$0.471834\pi$
$128$	−1039.39	−0.717735
$129$	11.0284	0.00752713
$130$	382.328	0.257941
$131$	1497.38	0.998679	0.499339	−	0.866406i	$-0.333576\pi$
$131$	1497.38	0.998679	0.499339	+	0.866406i	$0.333576\pi$
$132$	−253.594	−0.167216
$133$	0	0
$134$	−410.362	−0.264551
$135$	504.486	0.321624
$136$	362.203	0.228372
$137$	−1207.56	−0.753056	−0.376528	−	0.926405i	$-0.622882\pi$
$137$	−1207.56	−0.753056	−0.376528	+	0.926405i	$0.622882\pi$
$138$	235.932	0.145535
$139$	−212.519	−0.129681	−0.0648404	−	0.997896i	$-0.520654\pi$
$139$	−212.519	−0.129681	−0.0648404	+	0.997896i	$0.520654\pi$
$140$	0	0
$141$	−1192.79	−0.712417
$142$	−17.5570	−0.0103757
$143$	400.823	0.234395
$144$	508.781	0.294434
$145$	−5202.85	−2.97981
$146$	243.943	0.138280
$147$	0	0
$148$	−1507.07	−0.837032
$149$	879.656	0.483653	0.241826	−	0.970320i	$-0.422254\pi$
$149$	879.656	0.483653	0.241826	+	0.970320i	$0.422254\pi$
$150$	−377.560	−0.205517
$151$	−1215.18	−0.654899	−0.327450	−	0.944869i	$-0.606189\pi$
$151$	−1215.18	−0.654899	−0.327450	+	0.944869i	$0.606189\pi$
$152$	−208.742	−0.111390
$153$	−370.108	−0.195565
$154$	0	0
$155$	3577.12	1.85369
$156$	−840.051	−0.431140
$157$	−1226.57	−0.623507	−0.311754	−	0.950163i	$-0.600916\pi$
$157$	−1226.57	−0.623507	−0.311754	+	0.950163i	$0.600916\pi$
$158$	−439.136	−0.221113
$159$	−1792.52	−0.894065
$160$	−1909.71	−0.943599
$161$	0	0
$162$	45.4858	0.0220599
$163$	−441.224	−0.212021	−0.106010	−	0.994365i	$-0.533808\pi$
$163$	−441.224	−0.212021	−0.106010	+	0.994365i	$0.533808\pi$
$164$	−2479.80	−1.18073
$165$	−616.594	−0.290920
$166$	239.617	0.112036
$167$	1793.53	0.831064	0.415532	−	0.909579i	$-0.363595\pi$
$167$	1793.53	0.831064	0.415532	+	0.909579i	$0.363595\pi$
$168$	0	0
$169$	−869.240	−0.395648
$170$	431.481	0.194665
$171$	213.298	0.0953879
$172$	28.2499	0.0125235
$173$	−3825.04	−1.68099	−0.840497	−	0.541816i	$-0.817737\pi$
$173$	−3825.04	−1.68099	−0.840497	+	0.541816i	$0.817737\pi$
$174$	−469.102	−0.204383
$175$	0	0
$176$	−621.844	−0.266325
$177$	2006.34	0.852009
$178$	505.225	0.212743
$179$	1315.65	0.549365	0.274683	−	0.961535i	$-0.411427\pi$
$179$	1315.65	0.549365	0.274683	+	0.961535i	$0.411427\pi$
$180$	1292.27	0.535111
$181$	−674.682	−0.277065	−0.138532	−	0.990358i	$-0.544238\pi$
$181$	−674.682	−0.277065	−0.138532	+	0.990358i	$0.544238\pi$
$182$	0	0
$183$	−2003.36	−0.809251
$184$	1233.50	0.494213
$185$	−3664.33	−1.45626
$186$	322.523	0.127143
$187$	452.354	0.176895
$188$	−3055.39	−1.18530
$189$	0	0
$190$	−248.668	−0.0949490
$191$	1915.69	0.725730	0.362865	−	0.931842i	$-0.381799\pi$
$191$	1915.69	0.725730	0.362865	+	0.931842i	$0.381799\pi$
$192$	1184.57	0.445253
$193$	−2394.71	−0.893134	−0.446567	−	0.894750i	$-0.647354\pi$
$193$	−2394.71	−0.893134	−0.446567	+	0.894750i	$0.647354\pi$
$194$	529.385	0.195915
$195$	−2042.52	−0.750092
$196$	0	0
$197$	1356.81	0.490703	0.245351	−	0.969434i	$-0.421097\pi$
$197$	1356.81	0.490703	0.245351	+	0.969434i	$0.421097\pi$
$198$	−55.5937	−0.0199539
$199$	−2121.45	−0.755707	−0.377854	−	0.925865i	$-0.623338\pi$
$199$	−2121.45	−0.755707	−0.377854	+	0.925865i	$0.623338\pi$
$200$	−1973.96	−0.697902
$201$	2192.29	0.769313
$202$	−206.059	−0.0717736
$203$	0	0
$204$	−948.051	−0.325377
$205$	−6029.45	−2.05422
$206$	−565.678	−0.191324
$207$	−1260.43	−0.423216
$208$	−2059.91	−0.686678
$209$	−260.698	−0.0862816
$210$	0	0
$211$	4492.12	1.46564	0.732821	−	0.680421i	$-0.238203\pi$
$211$	4492.12	1.46564	0.732821	+	0.680421i	$0.238203\pi$
$212$	−4591.64	−1.48752
$213$	93.7953	0.0301725
$214$	−949.476	−0.303294
$215$	68.6876	0.0217882
$216$	237.810	0.0749116
$217$	0	0
$218$	−1015.61	−0.315530
$219$	−1303.22	−0.402118
$220$	−1579.44	−0.484026
$221$	1498.46	0.456097
$222$	−330.386	−0.0998832
$223$	−5142.35	−1.54420	−0.772101	−	0.635499i	$-0.780794\pi$
$223$	−5142.35	−1.54420	−0.772101	+	0.635499i	$0.780794\pi$
$224$	0	0
$225$	2017.05	0.597644
$226$	534.911	0.157441
$227$	−2987.86	−0.873618	−0.436809	−	0.899554i	$-0.643891\pi$
$227$	−2987.86	−0.873618	−0.436809	+	0.899554i	$0.643891\pi$
$228$	546.375	0.158704
$229$	5121.38	1.47786	0.738931	−	0.673781i	$-0.235331\pi$
$229$	5121.38	1.47786	0.738931	+	0.673781i	$0.235331\pi$
$230$	1469.44	0.421268
$231$	0	0
$232$	−2452.57	−0.694048
$233$	1103.58	0.310291	0.155146	−	0.987892i	$-0.450415\pi$
$233$	1103.58	0.310291	0.155146	+	0.987892i	$0.450415\pi$
$234$	−184.159	−0.0514481
$235$	−7428.94	−2.06217
$236$	5139.34	1.41755
$237$	2346.01	0.642995
$238$	0	0
$239$	−1798.18	−0.486671	−0.243336	−	0.969942i	$-0.578242\pi$
$239$	−1798.18	−0.486671	−0.243336	+	0.969942i	$0.578242\pi$
$240$	3168.80	0.852272
$241$	7342.09	1.96243	0.981215	−	0.192919i	$-0.0617956\pi$
$241$	7342.09	1.96243	0.981215	+	0.192919i	$0.0617956\pi$
$242$	67.9479	0.0180490
$243$	−243.000	−0.0641500
$244$	−5131.72	−1.34641
$245$	0	0
$246$	−543.631	−0.140897
$247$	−863.584	−0.222464
$248$	1686.22	0.431754
$249$	−1280.11	−0.325799
$250$	−1039.97	−0.263094
$251$	−1799.05	−0.452411	−0.226206	−	0.974080i	$-0.572632\pi$
$251$	−1799.05	−0.452411	−0.226206	+	0.974080i	$0.572632\pi$
$252$	0	0
$253$	1540.52	0.382813
$254$	142.045	0.0350894
$255$	−2305.11	−0.566086
$256$	2575.17	0.628703
$257$	5282.71	1.28220	0.641102	−	0.767455i	$-0.278477\pi$
$257$	5282.71	1.28220	0.641102	+	0.767455i	$0.278477\pi$
$258$	6.19305	0.00149443
$259$	0	0
$260$	−5232.02	−1.24799
$261$	2506.10	0.594343
$262$	840.859	0.198277
$263$	3275.55	0.767981	0.383990	−	0.923337i	$-0.374550\pi$
$263$	3275.55	0.767981	0.383990	+	0.923337i	$0.374550\pi$
$264$	−290.656	−0.0677601
$265$	−11164.2	−2.58797
$266$	0	0
$267$	−2699.08	−0.618655
$268$	5615.66	1.27997
$269$	−3682.35	−0.834636	−0.417318	−	0.908761i	$-0.637030\pi$
$269$	−3682.35	−0.834636	−0.417318	+	0.908761i	$0.637030\pi$
$270$	283.295	0.0638548
$271$	7301.19	1.63659	0.818295	−	0.574799i	$-0.194920\pi$
$271$	7301.19	1.63659	0.818295	+	0.574799i	$0.194920\pi$
$272$	−2324.74	−0.518228
$273$	0	0
$274$	−678.108	−0.149511
$275$	−2465.28	−0.540589
$276$	−3228.65	−0.704137
$277$	9035.64	1.95993	0.979963	−	0.199182i	$-0.0638283\pi$
$277$	9035.64	1.95993	0.979963	+	0.199182i	$0.0638283\pi$
$278$	−119.341	−0.0257467
$279$	−1723.02	−0.369730
$280$	0	0
$281$	1651.82	0.350674	0.175337	−	0.984509i	$-0.443899\pi$
$281$	1651.82	0.350674	0.175337	+	0.984509i	$0.443899\pi$
$282$	−669.813	−0.141442
$283$	−3091.62	−0.649391	−0.324695	−	0.945819i	$-0.605262\pi$
$283$	−3091.62	−0.649391	−0.324695	+	0.945819i	$0.605262\pi$
$284$	240.262	0.0502004
$285$	1328.47	0.276111
$286$	225.083	0.0465365
$287$	0	0
$288$	919.867	0.188207
$289$	−3221.89	−0.655789
$290$	−2921.67	−0.591609
$291$	−2828.15	−0.569721
$292$	−3338.28	−0.669034
$293$	−2231.84	−0.445001	−0.222500	−	0.974933i	$-0.571422\pi$
$293$	−2231.84	−0.445001	−0.222500	+	0.974933i	$0.571422\pi$
$294$	0	0
$295$	12495.9	2.46624
$296$	−1727.33	−0.339186
$297$	297.000	0.0580259
$298$	493.973	0.0960239
$299$	5103.11	0.987024
$300$	5166.78	0.994346
$301$	0	0
$302$	−682.387	−0.130023
$303$	1100.83	0.208717
$304$	1339.78	0.252769
$305$	−12477.4	−2.34247
$306$	−207.835	−0.0388273
$307$	9667.03	1.79715	0.898577	−	0.438816i	$-0.144602\pi$
$307$	9667.03	1.79715	0.898577	+	0.438816i	$0.144602\pi$
$308$	0	0
$309$	3022.04	0.556368
$310$	2008.74	0.368029
$311$	7170.97	1.30749	0.653743	−	0.756717i	$-0.273198\pi$
$311$	7170.97	1.30749	0.653743	+	0.756717i	$0.273198\pi$
$312$	−962.824	−0.174709
$313$	−295.578	−0.0533771	−0.0266886	−	0.999644i	$-0.508496\pi$
$313$	−295.578	−0.0533771	−0.0266886	+	0.999644i	$0.508496\pi$
$314$	−688.782	−0.123790
$315$	0	0
$316$	6009.43	1.06980
$317$	7189.81	1.27388	0.636940	−	0.770914i	$-0.280200\pi$
$317$	7189.81	1.27388	0.636940	+	0.770914i	$0.280200\pi$
$318$	−1006.60	−0.177507
$319$	−3063.01	−0.537604
$320$	7377.73	1.28884
$321$	5072.41	0.881976
$322$	0	0
$323$	−974.610	−0.167891
$324$	−622.457	−0.106731
$325$	−8166.46	−1.39383
$326$	−247.771	−0.0420943
$327$	5425.71	0.917561
$328$	−2842.22	−0.478462
$329$	0	0
$330$	−346.250	−0.0577589
$331$	−2878.87	−0.478058	−0.239029	−	0.971012i	$-0.576829\pi$
$331$	−2878.87	−0.478058	−0.239029	+	0.971012i	$0.576829\pi$
$332$	−3279.08	−0.542057
$333$	1765.03	0.290460
$334$	1007.16	0.164999
$335$	13654.0	2.22687
$336$	0	0
$337$	−11639.7	−1.88146	−0.940731	−	0.339153i	$-0.889860\pi$
$337$	−11639.7	−1.88146	−0.940731	+	0.339153i	$0.889860\pi$
$338$	−488.124	−0.0785516
$339$	−2857.67	−0.457839
$340$	−5904.67	−0.941840
$341$	2105.92	0.334433
$342$	119.778	0.0189382
$343$	0	0
$344$	32.3786	0.00507482
$345$	−7850.21	−1.22505
$346$	−2147.96	−0.333743
$347$	−4752.91	−0.735302	−0.367651	−	0.929964i	$-0.619838\pi$
$347$	−4752.91	−0.735302	−0.367651	+	0.929964i	$0.619838\pi$
$348$	6419.51	0.988855
$349$	7420.08	1.13807	0.569037	−	0.822312i	$-0.307316\pi$
$349$	7420.08	1.13807	0.569037	+	0.822312i	$0.307316\pi$
$350$	0	0
$351$	983.838	0.149611
$352$	−1124.28	−0.170240
$353$	517.851	0.0780805	0.0390403	−	0.999238i	$-0.487570\pi$
$353$	517.851	0.0780805	0.0390403	+	0.999238i	$0.487570\pi$
$354$	1126.66	0.169157
$355$	584.178	0.0873378
$356$	−6913.83	−1.02930
$357$	0	0
$358$	738.808	0.109070
$359$	−9874.35	−1.45167	−0.725833	−	0.687871i	$-0.758546\pi$
$359$	−9874.35	−1.45167	−0.725833	+	0.687871i	$0.758546\pi$
$360$	1481.13	0.216840
$361$	−6297.32	−0.918110
$362$	−378.869	−0.0550081
$363$	−363.000	−0.0524864
$364$	0	0
$365$	−8116.77	−1.16398
$366$	−1124.99	−0.160668
$367$	−2098.51	−0.298477	−0.149239	−	0.988801i	$-0.547682\pi$
$367$	−2098.51	−0.298477	−0.149239	+	0.988801i	$0.547682\pi$
$368$	−7917.05	−1.12148
$369$	2904.26	0.409728
$370$	−2057.72	−0.289123
$371$	0	0
$372$	−4413.61	−0.615148
$373$	−19.4150	−0.00269510	−0.00134755	−	0.999999i	$-0.500429\pi$
$373$	−19.4150	−0.00269510	−0.00134755	+	0.999999i	$0.500429\pi$
$374$	254.021	0.0351206
$375$	5555.87	0.765077
$376$	−3501.93	−0.480314
$377$	−10146.5	−1.38613
$378$	0	0
$379$	12541.3	1.69974	0.849869	−	0.526994i	$-0.176681\pi$
$379$	12541.3	1.69974	0.849869	+	0.526994i	$0.176681\pi$
$380$	3402.94	0.459388
$381$	−758.852	−0.102040
$382$	1075.76	0.144086
$383$	−9131.73	−1.21830	−0.609151	−	0.793054i	$-0.708490\pi$
$383$	−9131.73	−1.21830	−0.609151	+	0.793054i	$0.708490\pi$
$384$	3118.17	0.414384
$385$	0	0
$386$	−1344.76	−0.177322
$387$	−33.0853	−0.00434579
$388$	−7244.45	−0.947890
$389$	9542.57	1.24377	0.621886	−	0.783108i	$-0.286367\pi$
$389$	9542.57	1.24377	0.621886	+	0.783108i	$0.286367\pi$
$390$	−1146.98	−0.148922
$391$	5759.18	0.744896
$392$	0	0
$393$	−4492.15	−0.576587
$394$	761.919	0.0974236
$395$	14611.5	1.86122
$396$	760.781	0.0965422
$397$	−5322.85	−0.672912	−0.336456	−	0.941699i	$-0.609228\pi$
$397$	−5322.85	−0.672912	−0.336456	+	0.941699i	$0.609228\pi$
$398$	−1191.31	−0.150037
$399$	0	0
$400$	12669.6	1.58370
$401$	5875.92	0.731744	0.365872	−	0.930665i	$-0.380771\pi$
$401$	5875.92	0.731744	0.365872	+	0.930665i	$0.380771\pi$
$402$	1231.08	0.152739
$403$	6976.03	0.862285
$404$	2819.85	0.347259
$405$	−1513.46	−0.185690
$406$	0	0
$407$	−2157.26	−0.262731
$408$	−1086.61	−0.131851
$409$	6658.52	0.804994	0.402497	−	0.915421i	$-0.368142\pi$
$409$	6658.52	0.804994	0.402497	+	0.915421i	$0.368142\pi$
$410$	−3385.85	−0.407842
$411$	3622.68	0.434777
$412$	7741.11	0.925673
$413$	0	0
$414$	−707.796	−0.0840248
$415$	−7972.83	−0.943062
$416$	−3724.28	−0.438937
$417$	637.557	0.0748712
$418$	−146.396	−0.0171303
$419$	−10913.7	−1.27248	−0.636238	−	0.771493i	$-0.719510\pi$
$419$	−10913.7	−1.27248	−0.636238	+	0.771493i	$0.719510\pi$
$420$	0	0
$421$	3936.75	0.455737	0.227868	−	0.973692i	$-0.426824\pi$
$421$	3936.75	0.455737	0.227868	+	0.973692i	$0.426824\pi$
$422$	2522.56	0.290987
$423$	3578.36	0.411314
$424$	−5262.71	−0.602782
$425$	−9216.36	−1.05190
$426$	52.6710	0.00599042
$427$	0	0
$428$	12993.2	1.46741
$429$	−1202.47	−0.135328
$430$	38.5717	0.00432580
$431$	13769.3	1.53885	0.769424	−	0.638738i	$-0.220543\pi$
$431$	13769.3	1.53885	0.769424	+	0.638738i	$0.220543\pi$
$432$	−1526.34	−0.169991
$433$	−2858.51	−0.317254	−0.158627	−	0.987339i	$-0.550707\pi$
$433$	−2858.51	−0.317254	−0.158627	+	0.987339i	$0.550707\pi$
$434$	0	0
$435$	15608.5	1.72040
$436$	13898.2	1.52662
$437$	−3319.10	−0.363327
$438$	−731.829	−0.0798360
$439$	−8408.98	−0.914211	−0.457106	−	0.889412i	$-0.651114\pi$
$439$	−8408.98	−0.914211	−0.457106	+	0.889412i	$0.651114\pi$
$440$	−1810.27	−0.196139
$441$	0	0
$442$	841.466	0.0905530
$443$	−3537.39	−0.379383	−0.189692	−	0.981844i	$-0.560749\pi$
$443$	−3537.39	−0.379383	−0.189692	+	0.981844i	$0.560749\pi$
$444$	4521.22	0.483260
$445$	−16810.5	−1.79077
$446$	−2887.70	−0.306584
$447$	−2638.97	−0.279237
$448$	0	0
$449$	473.820	0.0498016	0.0249008	−	0.999690i	$-0.492073\pi$
$449$	473.820	0.0498016	0.0249008	+	0.999690i	$0.492073\pi$
$450$	1132.68	0.118656
$451$	−3549.65	−0.370613
$452$	−7320.07	−0.761742
$453$	3645.53	0.378106
$454$	−1677.84	−0.173447
$455$	0	0
$456$	626.227	0.0643109
$457$	535.534	0.0548167	0.0274083	−	0.999624i	$-0.491275\pi$
$457$	535.534	0.0548167	0.0274083	+	0.999624i	$0.491275\pi$
$458$	2875.93	0.293413
$459$	1110.32	0.112910
$460$	−20108.7	−2.03820
$461$	−1075.05	−0.108612	−0.0543058	−	0.998524i	$-0.517295\pi$
$461$	−1075.05	−0.108612	−0.0543058	+	0.998524i	$0.517295\pi$
$462$	0	0
$463$	−11373.8	−1.14165	−0.570826	−	0.821071i	$-0.693377\pi$
$463$	−11373.8	−1.14165	−0.570826	+	0.821071i	$0.693377\pi$
$464$	15741.4	1.57495
$465$	−10731.4	−1.07023
$466$	619.718	0.0616049
$467$	15171.1	1.50328	0.751642	−	0.659572i	$-0.229262\pi$
$467$	15171.1	1.50328	0.751642	+	0.659572i	$0.229262\pi$
$468$	2520.15	0.248919
$469$	0	0
$470$	−4171.74	−0.409421
$471$	3679.70	0.359982
$472$	5890.45	0.574428
$473$	40.4376	0.00393092
$474$	1317.41	0.127660
$475$	5311.52	0.513072
$476$	0	0
$477$	5377.57	0.516189
$478$	−1009.77	−0.0966231
$479$	3045.24	0.290482	0.145241	−	0.989396i	$-0.453604\pi$
$479$	3045.24	0.290482	0.145241	+	0.989396i	$0.453604\pi$
$480$	5729.13	0.544787
$481$	−7146.11	−0.677411
$482$	4122.97	0.389619
$483$	0	0
$484$	−929.844	−0.0873257
$485$	−17614.3	−1.64912
$486$	−136.457	−0.0127363
$487$	−16956.1	−1.57773	−0.788866	−	0.614565i	$-0.789331\pi$
$487$	−16956.1	−1.57773	−0.788866	+	0.614565i	$0.789331\pi$
$488$	−5881.72	−0.545600
$489$	1323.67	0.122410
$490$	0	0
$491$	−13667.0	−1.25618	−0.628088	−	0.778142i	$-0.716162\pi$
$491$	−13667.0	−1.25618	−0.628088	+	0.778142i	$0.716162\pi$
$492$	7439.40	0.681696
$493$	−11451.0	−1.04610
$494$	−484.948	−0.0441677
$495$	1849.78	0.167963
$496$	−10822.7	−0.979748
$497$	0	0
$498$	−718.851	−0.0646837
$499$	−10939.4	−0.981394	−0.490697	−	0.871330i	$-0.663258\pi$
$499$	−10939.4	−0.981394	−0.490697	+	0.871330i	$0.663258\pi$
$500$	14231.7	1.27292
$501$	−5380.60	−0.479815
$502$	−1010.26	−0.0898213
$503$	6194.02	0.549061	0.274530	−	0.961578i	$-0.411478\pi$
$503$	6194.02	0.549061	0.274530	+	0.961578i	$0.411478\pi$
$504$	0	0
$505$	6856.24	0.604156
$506$	865.084	0.0760033
$507$	2607.72	0.228428
$508$	−1943.84	−0.169772
$509$	−16742.7	−1.45797	−0.728987	−	0.684528i	$-0.760008\pi$
$509$	−16742.7	−1.45797	−0.728987	+	0.684528i	$0.760008\pi$
$510$	−1294.44	−0.112390
$511$	0	0
$512$	9761.22	0.842557
$513$	−639.895	−0.0550722
$514$	2966.52	0.254567
$515$	18821.9	1.61047
$516$	−84.7498	−0.00723043
$517$	−4373.55	−0.372048
$518$	0	0
$519$	11475.1	0.970523
$520$	−5996.68	−0.505715
$521$	−21184.3	−1.78138	−0.890691	−	0.454609i	$-0.849779\pi$
$521$	−21184.3	−1.78138	−0.890691	+	0.454609i	$0.849779\pi$
$522$	1407.31	0.118000
$523$	−7737.42	−0.646910	−0.323455	−	0.946244i	$-0.604844\pi$
$523$	−7737.42	−0.646910	−0.323455	+	0.946244i	$0.604844\pi$
$524$	−11506.9	−0.959313
$525$	0	0
$526$	1839.39	0.152474
$527$	7872.89	0.650756
$528$	1865.53	0.153763
$529$	7446.25	0.612004
$530$	−6269.30	−0.513813
$531$	−6019.01	−0.491908
$532$	0	0
$533$	−11758.5	−0.955567
$534$	−1515.68	−0.122827
$535$	31592.1	2.55298
$536$	6436.38	0.518674
$537$	−3946.95	−0.317176
$538$	−2067.84	−0.165708
$539$	0	0
$540$	−3876.80	−0.308946
$541$	−12732.7	−1.01187	−0.505937	−	0.862571i	$-0.668853\pi$
$541$	−12732.7	−1.01187	−0.505937	+	0.862571i	$0.668853\pi$
$542$	4100.00	0.324927
$543$	2024.05	0.159963
$544$	−4203.09	−0.331261
$545$	33792.5	2.65599
$546$	0	0
$547$	4429.06	0.346202	0.173101	−	0.984904i	$-0.444621\pi$
$547$	4429.06	0.346202	0.173101	+	0.984904i	$0.444621\pi$
$548$	9279.68	0.723372
$549$	6010.09	0.467221
$550$	−1384.39	−0.107328
$551$	6599.34	0.510239
$552$	−3700.51	−0.285334
$553$	0	0
$554$	5073.99	0.389121
$555$	10993.0	0.840769
$556$	1633.14	0.124569
$557$	6080.68	0.462562	0.231281	−	0.972887i	$-0.425708\pi$
$557$	6080.68	0.462562	0.231281	+	0.972887i	$0.425708\pi$
$558$	−967.568	−0.0734058
$559$	133.953	0.0101353
$560$	0	0
$561$	−1357.06	−0.102131
$562$	927.584	0.0696223
$563$	12141.8	0.908907	0.454453	−	0.890771i	$-0.349835\pi$
$563$	12141.8	0.908907	0.454453	+	0.890771i	$0.349835\pi$
$564$	9166.16	0.684335
$565$	−17798.2	−1.32527
$566$	−1736.11	−0.128929
$567$	0	0
$568$	275.376	0.0203424
$569$	22369.0	1.64808	0.824039	−	0.566533i	$-0.191716\pi$
$569$	22369.0	1.64808	0.824039	+	0.566533i	$0.191716\pi$
$570$	746.005	0.0548188
$571$	−14445.2	−1.05869	−0.529346	−	0.848406i	$-0.677563\pi$
$571$	−14445.2	−1.05869	−0.529346	+	0.848406i	$0.677563\pi$
$572$	−3080.19	−0.225156
$573$	−5747.07	−0.419000
$574$	0	0
$575$	−31386.9	−2.27639
$576$	−3553.70	−0.257067
$577$	−1782.41	−0.128601	−0.0643004	−	0.997931i	$-0.520482\pi$
$577$	−1782.41	−0.128601	−0.0643004	+	0.997931i	$0.520482\pi$
$578$	−1809.26	−0.130200
$579$	7184.12	0.515651
$580$	39982.1	2.86235
$581$	0	0
$582$	−1588.15	−0.113112
$583$	−6572.58	−0.466910
$584$	−3826.16	−0.271109
$585$	6127.56	0.433066
$586$	−1253.29	−0.0883499
$587$	−23646.2	−1.66266	−0.831330	−	0.555779i	$-0.812420\pi$
$587$	−23646.2	−1.66266	−0.831330	+	0.555779i	$0.812420\pi$
$588$	0	0
$589$	−4537.26	−0.317410
$590$	7017.12	0.489644
$591$	−4070.42	−0.283307
$592$	11086.6	0.769690
$593$	2871.03	0.198818	0.0994089	−	0.995047i	$-0.468305\pi$
$593$	2871.03	0.198818	0.0994089	+	0.995047i	$0.468305\pi$
$594$	166.781	0.0115204
$595$	0	0
$596$	−6759.86	−0.464588
$597$	6364.36	0.436308
$598$	2865.66	0.195963
$599$	6685.11	0.456004	0.228002	−	0.973661i	$-0.426781\pi$
$599$	6685.11	0.456004	0.228002	+	0.973661i	$0.426781\pi$
$600$	5921.89	0.402934
$601$	−8875.42	−0.602389	−0.301195	−	0.953563i	$-0.597385\pi$
$601$	−8875.42	−0.602389	−0.301195	+	0.953563i	$0.597385\pi$
$602$	0	0
$603$	−6576.86	−0.444163
$604$	9338.23	0.629085
$605$	−2260.84	−0.151928
$606$	618.177	0.0414385
$607$	−379.477	−0.0253748	−0.0126874	−	0.999920i	$-0.504039\pi$
$607$	−379.477	−0.0253748	−0.0126874	+	0.999920i	$0.504039\pi$
$608$	2422.30	0.161574
$609$	0	0
$610$	−7006.71	−0.465071
$611$	−14487.8	−0.959267
$612$	2844.15	0.187856
$613$	17480.5	1.15177	0.575883	−	0.817532i	$-0.304658\pi$
$613$	17480.5	1.15177	0.575883	+	0.817532i	$0.304658\pi$
$614$	5428.55	0.356805
$615$	18088.3	1.18600
$616$	0	0
$617$	−9901.31	−0.646048	−0.323024	−	0.946391i	$-0.604699\pi$
$617$	−9901.31	−0.646048	−0.323024	+	0.946391i	$0.604699\pi$
$618$	1697.03	0.110461
$619$	152.433	0.00989788	0.00494894	−	0.999988i	$-0.498425\pi$
$619$	152.433	0.00989788	0.00494894	+	0.999988i	$0.498425\pi$
$620$	−27489.0	−1.78062
$621$	3781.28	0.244344
$622$	4026.88	0.259587
$623$	0	0
$624$	6179.73	0.396454
$625$	6588.63	0.421672
$626$	−165.983	−0.0105974
$627$	782.094	0.0498147
$628$	9425.74	0.598930
$629$	−8064.84	−0.511234
$630$	0	0
$631$	−23661.7	−1.49280	−0.746399	−	0.665499i	$-0.768219\pi$
$631$	−23661.7	−1.49280	−0.746399	+	0.665499i	$0.768219\pi$
$632$	6887.70	0.433510
$633$	−13476.4	−0.846189
$634$	4037.46	0.252915
$635$	−4726.30	−0.295366
$636$	13774.9	0.858823
$637$	0	0
$638$	−1720.04	−0.106735
$639$	−281.386	−0.0174201
$640$	19420.7	1.19948
$641$	25340.3	1.56144	0.780719	−	0.624882i	$-0.214853\pi$
$641$	25340.3	1.56144	0.780719	+	0.624882i	$0.214853\pi$
$642$	2848.43	0.175107
$643$	1774.90	0.108858	0.0544288	−	0.998518i	$-0.482666\pi$
$643$	1774.90	0.108858	0.0544288	+	0.998518i	$0.482666\pi$
$644$	0	0
$645$	−206.063	−0.0125794
$646$	−547.295	−0.0333329
$647$	23685.1	1.43919	0.719595	−	0.694394i	$-0.244328\pi$
$647$	23685.1	1.43919	0.719595	+	0.694394i	$0.244328\pi$
$648$	−713.429	−0.0432502
$649$	7356.57	0.444947
$650$	−4585.90	−0.276729
$651$	0	0
$652$	3390.66	0.203663
$653$	−1969.99	−0.118058	−0.0590288	−	0.998256i	$-0.518800\pi$
$653$	−1969.99	−0.118058	−0.0590288	+	0.998256i	$0.518800\pi$
$654$	3046.82	0.182172
$655$	−27978.1	−1.66900
$656$	18242.4	1.08574
$657$	3909.67	0.232163
$658$	0	0
$659$	15314.6	0.905270	0.452635	−	0.891696i	$-0.350484\pi$
$659$	15314.6	0.905270	0.452635	+	0.891696i	$0.350484\pi$
$660$	4738.31	0.279452
$661$	−31443.3	−1.85023	−0.925115	−	0.379687i	$-0.876032\pi$
$661$	−31443.3	−1.85023	−0.925115	+	0.379687i	$0.876032\pi$
$662$	−1616.64	−0.0949130
$663$	−4495.39	−0.263328
$664$	−3758.31	−0.219655
$665$	0	0
$666$	991.158	0.0576676
$667$	−38996.9	−2.26382
$668$	−13782.7	−0.798305
$669$	15427.0	0.891546
$670$	7667.47	0.442120
$671$	−7345.67	−0.422617
$672$	0	0
$673$	−8079.34	−0.462757	−0.231379	−	0.972864i	$-0.574324\pi$
$673$	−8079.34	−0.462757	−0.231379	+	0.972864i	$0.574324\pi$
$674$	−6536.29	−0.373543
$675$	−6051.14	−0.345050
$676$	6679.81	0.380053
$677$	15701.7	0.891383	0.445692	−	0.895187i	$-0.352958\pi$
$677$	15701.7	0.891383	0.445692	+	0.895187i	$0.352958\pi$
$678$	−1604.73	−0.0908988
$679$	0	0
$680$	−6767.63	−0.381657
$681$	8963.58	0.504384
$682$	1182.58	0.0663980
$683$	29752.9	1.66686	0.833430	−	0.552626i	$-0.186374\pi$
$683$	29752.9	1.66686	0.833430	+	0.552626i	$0.186374\pi$
$684$	−1639.12	−0.0916279
$685$	22562.8	1.25851
$686$	0	0
$687$	−15364.1	−0.853244
$688$	−207.817	−0.0115159
$689$	−21772.2	−1.20386
$690$	−4408.31	−0.243219
$691$	−3082.57	−0.169706	−0.0848528	−	0.996393i	$-0.527042\pi$
$691$	−3082.57	−0.169706	−0.0848528	+	0.996393i	$0.527042\pi$
$692$	29394.1	1.61473
$693$	0	0
$694$	−2669.01	−0.145986
$695$	3970.84	0.216723
$696$	7357.71	0.400709
$697$	−13270.2	−0.721156
$698$	4166.77	0.225952
$699$	−3310.74	−0.179147
$700$	0	0
$701$	19301.0	1.03993	0.519963	−	0.854189i	$-0.325946\pi$
$701$	19301.0	1.03993	0.519963	+	0.854189i	$0.325946\pi$
$702$	552.477	0.0297036
$703$	4647.88	0.249357
$704$	4343.41	0.232526
$705$	22286.8	1.19060
$706$	290.801	0.0155020
$707$	0	0
$708$	−15418.0	−0.818425
$709$	28531.3	1.51130	0.755652	−	0.654973i	$-0.227320\pi$
$709$	28531.3	1.51130	0.755652	+	0.654973i	$0.227320\pi$
$710$	328.047	0.0173400
$711$	−7038.03	−0.371233
$712$	−7924.29	−0.417100
$713$	26811.6	1.40828
$714$	0	0
$715$	−7489.24	−0.391723
$716$	−10110.3	−0.527711
$717$	5394.53	0.280980
$718$	−5544.97	−0.288212
$719$	−3884.65	−0.201492	−0.100746	−	0.994912i	$-0.532123\pi$
$719$	−3884.65	−0.201492	−0.100746	+	0.994912i	$0.532123\pi$
$720$	−9506.40	−0.492059
$721$	0	0
$722$	−3536.28	−0.182281
$723$	−22026.3	−1.13301
$724$	5184.70	0.266143
$725$	62406.5	3.19685
$726$	−203.844	−0.0104206
$727$	−23489.8	−1.19833	−0.599167	−	0.800624i	$-0.704501\pi$
$727$	−23489.8	−1.19833	−0.599167	+	0.800624i	$0.704501\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−4557.99	−0.231094
$731$	151.175	0.00764897
$732$	15395.2	0.777352
$733$	−30288.1	−1.52622	−0.763109	−	0.646270i	$-0.776328\pi$
$733$	−30288.1	−1.52622	−0.763109	+	0.646270i	$0.776328\pi$
$734$	−1178.42	−0.0592593
$735$	0	0
$736$	−14313.9	−0.716870
$737$	8038.39	0.401761
$738$	1630.89	0.0813469
$739$	29765.0	1.48163	0.740815	−	0.671709i	$-0.234440\pi$
$739$	29765.0	1.48163	0.740815	+	0.671709i	$0.234440\pi$
$740$	28159.2	1.39885
$741$	2590.75	0.128440
$742$	0	0
$743$	−1200.21	−0.0592617	−0.0296309	−	0.999561i	$-0.509433\pi$
$743$	−1200.21	−0.0592617	−0.0296309	+	0.999561i	$0.509433\pi$
$744$	−5058.66	−0.249273
$745$	−16436.1	−0.808284
$746$	−10.9026	−0.000535082 0
$747$	3840.34	0.188100
$748$	−3476.19	−0.169922
$749$	0	0
$750$	3119.92	0.151898
$751$	441.096	0.0214325	0.0107163	−	0.999943i	$-0.496589\pi$
$751$	441.096	0.0214325	0.0107163	+	0.999943i	$0.496589\pi$
$752$	22476.6	1.08994
$753$	5397.16	0.261200
$754$	−5697.79	−0.275200
$755$	22705.2	1.09447
$756$	0	0
$757$	−37080.3	−1.78032	−0.890162	−	0.455644i	$-0.849409\pi$
$757$	−37080.3	−1.78032	−0.890162	+	0.455644i	$0.849409\pi$
$758$	7042.58	0.337464
$759$	−4621.56	−0.221017
$760$	3900.28	0.186155
$761$	3056.78	0.145609	0.0728044	−	0.997346i	$-0.476805\pi$
$761$	3056.78	0.145609	0.0728044	+	0.997346i	$0.476805\pi$
$762$	−426.136	−0.0202589
$763$	0	0
$764$	−14721.4	−0.697123
$765$	6915.34	0.326830
$766$	−5127.95	−0.241880
$767$	24369.3	1.14723
$768$	−7725.50	−0.362982
$769$	−31834.7	−1.49283	−0.746417	−	0.665479i	$-0.768227\pi$
$769$	−31834.7	−1.49283	−0.746417	+	0.665479i	$0.768227\pi$
$770$	0	0
$771$	−15848.1	−0.740281
$772$	18402.5	0.857929
$773$	2351.46	0.109413	0.0547064	−	0.998502i	$-0.482578\pi$
$773$	2351.46	0.109413	0.0547064	+	0.998502i	$0.482578\pi$
$774$	−18.5792	−0.000862809 0
$775$	−42906.4	−1.98870
$776$	−8303.22	−0.384108
$777$	0	0
$778$	5358.65	0.246937
$779$	7647.81	0.351748
$780$	15696.1	0.720525
$781$	343.916	0.0157571
$782$	3234.08	0.147891
$783$	−7518.30	−0.343144
$784$	0	0
$785$	22918.0	1.04201
$786$	−2522.58	−0.114475
$787$	41575.3	1.88310	0.941550	−	0.336874i	$-0.109370\pi$
$787$	41575.3	1.88310	0.941550	+	0.336874i	$0.109370\pi$
$788$	−10426.6	−0.471361
$789$	−9826.64	−0.443394
$790$	8205.11	0.369525
$791$	0	0
$792$	871.969	0.0391213
$793$	−24333.2	−1.08965
$794$	−2989.06	−0.133599
$795$	33492.7	1.49417
$796$	16302.6	0.725919
$797$	26199.7	1.16442	0.582208	−	0.813040i	$-0.302189\pi$
$797$	26199.7	1.16442	0.582208	+	0.813040i	$0.302189\pi$
$798$	0	0
$799$	−16350.4	−0.723948
$800$	22906.4	1.01233
$801$	8097.24	0.357181
$802$	3299.64	0.145280
$803$	−4778.49	−0.209999
$804$	−16847.0	−0.738989
$805$	0	0
$806$	3917.41	0.171197
$807$	11047.1	0.481877
$808$	3231.96	0.140718
$809$	−2338.60	−0.101633	−0.0508164	−	0.998708i	$-0.516182\pi$
$809$	−2338.60	−0.101633	−0.0508164	+	0.998708i	$0.516182\pi$
$810$	−849.886	−0.0368666
$811$	3032.48	0.131301	0.0656504	−	0.997843i	$-0.479088\pi$
$811$	3032.48	0.131301	0.0656504	+	0.997843i	$0.479088\pi$
$812$	0	0
$813$	−21903.6	−0.944886
$814$	−1211.42	−0.0521623
$815$	8244.13	0.354330
$816$	6974.22	0.299199
$817$	−87.1240	−0.00373082
$818$	3739.11	0.159823
$819$	0	0
$820$	46334.2	1.97325
$821$	−28900.3	−1.22854	−0.614268	−	0.789098i	$-0.710549\pi$
$821$	−28900.3	−1.22854	−0.614268	+	0.789098i	$0.710549\pi$
$822$	2034.32	0.0863202
$823$	−19810.2	−0.839054	−0.419527	−	0.907743i	$-0.637804\pi$
$823$	−19810.2	−0.839054	−0.419527	+	0.907743i	$0.637804\pi$
$824$	8872.47	0.375106
$825$	7395.84	0.312109
$826$	0	0
$827$	12280.7	0.516374	0.258187	−	0.966095i	$-0.416875\pi$
$827$	12280.7	0.516374	0.258187	+	0.966095i	$0.416875\pi$
$828$	9685.94	0.406534
$829$	3499.72	0.146623	0.0733114	−	0.997309i	$-0.476643\pi$
$829$	3499.72	0.146623	0.0733114	+	0.997309i	$0.476643\pi$
$830$	−4477.16	−0.187235
$831$	−27106.9	−1.13156
$832$	14387.9	0.599532
$833$	0	0
$834$	358.022	0.0148649
$835$	−33511.5	−1.38888
$836$	2003.37	0.0828806
$837$	5169.07	0.213464
$838$	−6128.60	−0.252636
$839$	11751.6	0.483562	0.241781	−	0.970331i	$-0.422268\pi$
$839$	11751.6	0.483562	0.241781	+	0.970331i	$0.422268\pi$
$840$	0	0
$841$	53148.4	2.17920
$842$	2210.69	0.0904815
$843$	−4955.46	−0.202461
$844$	−34520.4	−1.40787
$845$	16241.4	0.661210
$846$	2009.44	0.0816618
$847$	0	0
$848$	33777.8	1.36785
$849$	9274.86	0.374926
$850$	−5175.48	−0.208844
$851$	−27465.3	−1.10634
$852$	−720.785	−0.0289832
$853$	−1403.01	−0.0563166	−0.0281583	−	0.999603i	$-0.508964\pi$
$853$	−1403.01	−0.0563166	−0.0281583	+	0.999603i	$0.508964\pi$
$854$	0	0
$855$	−3985.41	−0.159413
$856$	14892.2	0.594632
$857$	−15111.0	−0.602311	−0.301156	−	0.953575i	$-0.597372\pi$
$857$	−15111.0	−0.602311	−0.301156	+	0.953575i	$0.597372\pi$
$858$	−675.250	−0.0268679
$859$	−18620.9	−0.739623	−0.369811	−	0.929107i	$-0.620578\pi$
$859$	−18620.9	−0.739623	−0.369811	+	0.929107i	$0.620578\pi$
$860$	−527.841	−0.0209293
$861$	0	0
$862$	7732.19	0.305521
$863$	−33659.0	−1.32765	−0.663827	−	0.747886i	$-0.731069\pi$
$863$	−33659.0	−1.32765	−0.663827	+	0.747886i	$0.731069\pi$
$864$	−2759.60	−0.108661
$865$	71469.5	2.80929
$866$	−1605.20	−0.0629873
$867$	9665.67	0.378620
$868$	0	0
$869$	8602.04	0.335793
$870$	8765.02	0.341565
$871$	26627.8	1.03588
$872$	15929.5	0.618623
$873$	8484.44	0.328929
$874$	−1863.85	−0.0721345
$875$	0	0
$876$	10014.8	0.386267
$877$	−2730.45	−0.105132	−0.0525659	−	0.998617i	$-0.516740\pi$
$877$	−2730.45	−0.105132	−0.0525659	+	0.998617i	$0.516740\pi$
$878$	−4722.09	−0.181507
$879$	6695.51	0.256921
$880$	11618.9	0.445084
$881$	16625.5	0.635787	0.317894	−	0.948126i	$-0.397025\pi$
$881$	16625.5	0.635787	0.317894	+	0.948126i	$0.397025\pi$
$882$	0	0
$883$	10063.4	0.383536	0.191768	−	0.981440i	$-0.438578\pi$
$883$	10063.4	0.383536	0.191768	+	0.981440i	$0.438578\pi$
$884$	−11515.2	−0.438119
$885$	−37487.7	−1.42388
$886$	−1986.43	−0.0753223
$887$	−29939.9	−1.13335	−0.566676	−	0.823940i	$-0.691771\pi$
$887$	−29939.9	−1.13335	−0.566676	+	0.823940i	$0.691771\pi$
$888$	5181.99	0.195829
$889$	0	0
$890$	−9439.96	−0.355537
$891$	−891.000	−0.0335013
$892$	39517.2	1.48333
$893$	9422.94	0.353109
$894$	−1481.92	−0.0554394
$895$	−24582.5	−0.918103
$896$	0	0
$897$	−15309.3	−0.569859
$898$	266.075	0.00988756
$899$	−53309.5	−1.97772
$900$	−15500.3	−0.574086
$901$	−24571.4	−0.908536
$902$	−1993.31	−0.0735810
$903$	0	0
$904$	−8389.90	−0.308677
$905$	12606.2	0.463032
$906$	2047.16	0.0750688
$907$	296.554	0.0108566	0.00542829	−	0.999985i	$-0.498272\pi$
$907$	296.554	0.0108566	0.00542829	+	0.999985i	$0.498272\pi$
$908$	22960.7	0.839182
$909$	−3302.50	−0.120503
$910$	0	0
$911$	−3552.96	−0.129215	−0.0646074	−	0.997911i	$-0.520580\pi$
$911$	−3552.96	−0.129215	−0.0646074	+	0.997911i	$0.520580\pi$
$912$	−4019.34	−0.145936
$913$	−4693.75	−0.170143
$914$	300.730	0.0108832
$915$	37432.2	1.35243
$916$	−39356.1	−1.41961
$917$	0	0
$918$	623.505	0.0224169
$919$	−32047.3	−1.15032	−0.575159	−	0.818042i	$-0.695060\pi$
$919$	−32047.3	−1.15032	−0.575159	+	0.818042i	$0.695060\pi$
$920$	−23047.6	−0.825931
$921$	−29001.1	−1.03759
$922$	−603.696	−0.0215636
$923$	1139.25	0.0406272
$924$	0	0
$925$	43952.5	1.56232
$926$	−6386.98	−0.226662
$927$	−9066.12	−0.321219
$928$	28460.2	1.00674
$929$	−5628.82	−0.198790	−0.0993949	−	0.995048i	$-0.531691\pi$
$929$	−5628.82	−0.198790	−0.0993949	+	0.995048i	$0.531691\pi$
$930$	−6026.23	−0.212481
$931$	0	0
$932$	−8480.63	−0.298060
$933$	−21512.9	−0.754877
$934$	8519.36	0.298460
$935$	−8452.08	−0.295629
$936$	2888.47	0.100868
$937$	41269.1	1.43885	0.719425	−	0.694570i	$-0.244405\pi$
$937$	41269.1	1.43885	0.719425	+	0.694570i	$0.244405\pi$
$938$	0	0
$939$	886.733	0.0308173
$940$	57088.9	1.98089
$941$	−2086.74	−0.0722909	−0.0361454	−	0.999347i	$-0.511508\pi$
$941$	−2086.74	−0.0722909	−0.0361454	+	0.999347i	$0.511508\pi$
$942$	2066.35	0.0714705
$943$	−45192.6	−1.56063
$944$	−37806.9	−1.30351
$945$	0	0
$946$	22.7079	0.000780440 0
$947$	−21040.7	−0.721995	−0.360998	−	0.932567i	$-0.617564\pi$
$947$	−21040.7	−0.721995	−0.360998	+	0.932567i	$0.617564\pi$
$948$	−18028.3	−0.617650
$949$	−15829.2	−0.541450
$950$	2982.70	0.101865
$951$	−21569.4	−0.735475
$952$	0	0
$953$	25644.0	0.871658	0.435829	−	0.900030i	$-0.356455\pi$
$953$	25644.0	0.871658	0.435829	+	0.900030i	$0.356455\pi$
$954$	3019.79	0.102484
$955$	−35794.0	−1.21284
$956$	13818.4	0.467488
$957$	9189.03	0.310386
$958$	1710.06	0.0576719
$959$	0	0
$960$	−22133.2	−0.744110
$961$	6860.94	0.230302
$962$	−4012.92	−0.134492
$963$	−15217.2	−0.509209
$964$	−56421.4	−1.88507
$965$	44744.3	1.49261
$966$	0	0
$967$	28262.5	0.939878	0.469939	−	0.882699i	$-0.344276\pi$
$967$	28262.5	0.939878	0.469939	+	0.882699i	$0.344276\pi$
$968$	−1065.74	−0.0353865
$969$	2923.83	0.0969318
$970$	−9891.37	−0.327415
$971$	−19180.0	−0.633899	−0.316950	−	0.948442i	$-0.602659\pi$
$971$	−19180.0	−0.633899	−0.316950	+	0.948442i	$0.602659\pi$
$972$	1867.37	0.0616214
$973$	0	0
$974$	−9521.76	−0.313241
$975$	24499.4	0.804726
$976$	37750.9	1.23809
$977$	−35269.8	−1.15495	−0.577473	−	0.816410i	$-0.695961\pi$
$977$	−35269.8	−1.15495	−0.577473	+	0.816410i	$0.695961\pi$
$978$	743.312	0.0243032
$979$	−9896.62	−0.323082
$980$	0	0
$981$	−16277.1	−0.529754
$982$	−7674.73	−0.249400
$983$	−40100.1	−1.30111	−0.650557	−	0.759458i	$-0.725464\pi$
$983$	−40100.1	−1.30111	−0.650557	+	0.759458i	$0.725464\pi$
$984$	8526.67	0.276240
$985$	−25351.5	−0.820066
$986$	−6430.32	−0.207691
$987$	0	0
$988$	6636.35	0.213695
$989$	514.835	0.0165529
$990$	1038.75	0.0333471
$991$	3191.81	0.102312	0.0511560	−	0.998691i	$-0.483709\pi$
$991$	3191.81	0.102312	0.0511560	+	0.998691i	$0.483709\pi$
$992$	−19567.3	−0.626272
$993$	8636.61	0.276007
$994$	0	0
$995$	39638.6	1.26294
$996$	9837.24	0.312957
$997$	−8275.27	−0.262869	−0.131435	−	0.991325i	$-0.541958\pi$
$997$	−8275.27	−0.262869	−0.131435	+	0.991325i	$0.541958\pi$
$998$	−6143.06	−0.194845
$999$	−5295.09	−0.167697

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.561553 q^{2} -3.00000 q^{3} -7.68466 q^{4} -18.6847 q^{5} -1.68466 q^{6} -8.80776 q^{8} +9.00000 q^{9} -10.4924 q^{10} -11.0000 q^{11} +23.0540 q^{12} -36.4384 q^{13} +56.0540 q^{15} +56.5312 q^{16} -41.1231 q^{17} +5.05398 q^{18} +23.6998 q^{19} +143.585 q^{20} -6.17708 q^{22} -140.047 q^{23} +26.4233 q^{24} +224.116 q^{25} -20.4621 q^{26} -27.0000 q^{27} +278.455 q^{29} +31.4773 q^{30} -191.447 q^{31} +102.207 q^{32} +33.0000 q^{33} -23.0928 q^{34} -69.1619 q^{36} +196.115 q^{37} +13.3087 q^{38} +109.315 q^{39} +164.570 q^{40} +322.695 q^{41} -3.67615 q^{43} +84.5312 q^{44} -168.162 q^{45} -78.6440 q^{46} +397.596 q^{47} -169.594 q^{48} +125.853 q^{50} +123.369 q^{51} +280.017 q^{52} +597.508 q^{53} -15.1619 q^{54} +205.531 q^{55} -71.0994 q^{57} +156.367 q^{58} -668.779 q^{59} -430.756 q^{60} +667.788 q^{61} -107.508 q^{62} -394.855 q^{64} +680.840 q^{65} +18.5312 q^{66} -730.762 q^{67} +316.017 q^{68} +420.142 q^{69} -31.2651 q^{71} -79.2699 q^{72} +434.408 q^{73} +110.129 q^{74} -672.349 q^{75} -182.125 q^{76} +61.3863 q^{78} -782.004 q^{79} -1056.27 q^{80} +81.0000 q^{81} +181.210 q^{82} +426.705 q^{83} +768.371 q^{85} -2.06435 q^{86} -835.366 q^{87} +96.8854 q^{88} +899.693 q^{89} -94.4318 q^{90} +1076.22 q^{92} +574.341 q^{93} +223.271 q^{94} -442.823 q^{95} -306.622 q^{96} +942.716 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 6 q^{3} - 3 q^{4} - 25 q^{5} + 9 q^{6} + 3 q^{8} + 18 q^{9} + 12 q^{10} - 22 q^{11} + 9 q^{12} - 77 q^{13} + 75 q^{15} - 23 q^{16} - 74 q^{17} - 27 q^{18} + 101 q^{19} + 114 q^{20} + 33 q^{22}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 6 * q^3 - 3 * q^4 - 25 * q^5 + 9 * q^6 + 3 * q^8 + 18 * q^9 + 12 * q^10 - 22 * q^11 + 9 * q^12 - 77 * q^13 + 75 * q^15 - 23 * q^16 - 74 * q^17 - 27 * q^18 + 101 * q^19 + 114 * q^20 + 33 * q^22 + 58 * q^23 - 9 * q^24 + 139 * q^25 + 124 * q^26 - 54 * q^27 + 91 * q^29 - 36 * q^30 - 152 * q^31 + 291 * q^32 + 66 * q^33 + 94 * q^34 - 27 * q^36 + 619 * q^37 - 262 * q^38 + 231 * q^39 + 90 * q^40 - 138 * q^41 - 230 * q^43 + 33 * q^44 - 225 * q^45 - 784 * q^46 - 149 * q^47 + 69 * q^48 + 429 * q^50 + 222 * q^51 + 90 * q^52 + 1228 * q^53 + 81 * q^54 + 275 * q^55 - 303 * q^57 + 824 * q^58 - 649 * q^59 - 342 * q^60 + 412 * q^61 - 248 * q^62 - 431 * q^64 + 937 * q^65 - 99 * q^66 - 1243 * q^67 + 162 * q^68 - 174 * q^69 + 960 * q^71 + 27 * q^72 + 741 * q^73 - 1396 * q^74 - 417 * q^75 + 180 * q^76 - 372 * q^78 - 492 * q^79 - 554 * q^80 + 162 * q^81 + 1822 * q^82 + 1744 * q^83 + 976 * q^85 + 804 * q^86 - 273 * q^87 - 33 * q^88 + 1552 * q^89 + 108 * q^90 + 2004 * q^92 + 456 * q^93 + 2170 * q^94 - 931 * q^95 - 873 * q^96 - 440 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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