LMFDB - Embedded newform 847.4.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,4,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("847.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	847.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:	$49.9746177749$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.20317$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.89098 q^{2} +6.57251 q^{3} +15.9217 q^{4} +15.5514 q^{5} +32.1460 q^{6} +7.00000 q^{7} +38.7449 q^{8} +16.1978 q^{9} +76.0614 q^{10} +104.646 q^{12} -74.3459 q^{13} +34.2369 q^{14} +102.211 q^{15} +62.1271 q^{16} +94.0836 q^{17} +79.2234 q^{18} -135.682 q^{19} +247.604 q^{20} +46.0075 q^{21} +81.1793 q^{23} +254.651 q^{24} +116.845 q^{25} -363.625 q^{26} -70.9972 q^{27} +111.452 q^{28} +53.4259 q^{29} +499.914 q^{30} -9.50536 q^{31} -6.09673 q^{32} +460.161 q^{34} +108.860 q^{35} +257.897 q^{36} -9.14224 q^{37} -663.620 q^{38} -488.639 q^{39} +602.536 q^{40} +339.461 q^{41} +225.022 q^{42} -433.078 q^{43} +251.899 q^{45} +397.046 q^{46} -54.4784 q^{47} +408.331 q^{48} +49.0000 q^{49} +571.486 q^{50} +618.365 q^{51} -1183.71 q^{52} +123.830 q^{53} -347.246 q^{54} +271.215 q^{56} -891.773 q^{57} +261.305 q^{58} -534.396 q^{59} +1627.38 q^{60} +358.624 q^{61} -46.4905 q^{62} +113.385 q^{63} -526.836 q^{64} -1156.18 q^{65} -694.318 q^{67} +1497.97 q^{68} +533.551 q^{69} +532.430 q^{70} -278.330 q^{71} +627.585 q^{72} +886.688 q^{73} -44.7145 q^{74} +767.963 q^{75} -2160.29 q^{76} -2389.93 q^{78} +185.631 q^{79} +966.162 q^{80} -903.972 q^{81} +1660.30 q^{82} +122.624 q^{83} +732.519 q^{84} +1463.13 q^{85} -2118.18 q^{86} +351.142 q^{87} -847.086 q^{89} +1232.03 q^{90} -520.422 q^{91} +1292.51 q^{92} -62.4740 q^{93} -266.453 q^{94} -2110.04 q^{95} -40.0708 q^{96} +1002.49 q^{97} +239.658 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} - 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} + 2 q^{10} + 70 q^{12} - 58 q^{13} + 14 q^{14} + 284 q^{15} + 2 q^{16} - 4 q^{17} + 62 q^{18} - 258 q^{19} + 182 q^{20}+ \cdots + 98 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 - 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 + 2 * q^10 + 70 * q^12 - 58 * q^13 + 14 * q^14 + 284 * q^15 + 2 * q^16 - 4 * q^17 + 62 * q^18 - 258 * q^19 + 182 * q^20 + 98 * q^21 + 8 * q^23 + 498 * q^24 + 80 * q^25 - 482 * q^26 + 428 * q^27 + 182 * q^28 + 396 * q^29 + 628 * q^30 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 - 418 * q^36 + 84 * q^37 - 942 * q^38 + 412 * q^39 + 1026 * q^40 - 52 * q^41 - 98 * q^42 - 408 * q^43 + 826 * q^45 - 368 * q^46 + 8 * q^47 + 982 * q^48 + 196 * q^49 + 1642 * q^50 + 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 126 * q^56 - 48 * q^57 + 864 * q^58 - 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 14 * q^70 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 - 996 * q^74 - 806 * q^75 - 3042 * q^76 - 2632 * q^78 + 1324 * q^79 + 2342 * q^80 + 1444 * q^81 + 1984 * q^82 - 450 * q^83 + 490 * q^84 + 1736 * q^85 - 4380 * q^86 - 588 * q^87 - 3072 * q^89 + 218 * q^90 - 406 * q^91 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 - 24 * q^95 - 862 * q^96 - 652 * q^97 + 98 * q^98

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 $

$2$	4.89098	1.72922	0.864612	−	0.502441i	$-0.167564\pi$
$2$	4.89098	1.72922	0.864612	+	0.502441i	$0.167564\pi$
$3$	6.57251	1.26488	0.632440	−	0.774610i	$-0.282054\pi$
$3$	6.57251	1.26488	0.632440	+	0.774610i	$0.282054\pi$
$4$	15.9217	1.99021
$5$	15.5514	1.39096	0.695478	−	0.718547i	$-0.255193\pi$
$5$	15.5514	1.39096	0.695478	+	0.718547i	$0.255193\pi$
$6$	32.1460	2.18726
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	38.7449	1.71230
$9$	16.1978	0.599920
$10$	76.0614	2.40527
$11$	0	0
$11$	0	0
$12$	104.646	2.51738
$13$	−74.3459	−1.58614	−0.793071	−	0.609129i	$-0.791519\pi$
$13$	−74.3459	−1.58614	−0.793071	+	0.609129i	$0.791519\pi$
$14$	34.2369	0.653585
$15$	102.211	1.75939
$16$	62.1271	0.970737
$17$	94.0836	1.34227	0.671136	−	0.741334i	$-0.265807\pi$
$17$	94.0836	1.34227	0.671136	+	0.741334i	$0.265807\pi$
$18$	79.2234	1.03740
$19$	−135.682	−1.63830	−0.819149	−	0.573581i	$-0.805554\pi$
$19$	−135.682	−1.63830	−0.819149	+	0.573581i	$0.805554\pi$
$20$	247.604	2.76830
$21$	46.0075	0.478080
$22$	0	0
$23$	81.1793	0.735959	0.367979	−	0.929834i	$-0.380050\pi$
$23$	81.1793	0.735959	0.367979	+	0.929834i	$0.380050\pi$
$24$	254.651	2.16585
$25$	116.845	0.934758
$26$	−363.625	−2.74279
$27$	−70.9972	−0.506053
$28$	111.452	0.752230
$29$	53.4259	0.342102	0.171051	−	0.985262i	$-0.445284\pi$
$29$	53.4259	0.342102	0.171051	+	0.985262i	$0.445284\pi$
$30$	499.914	3.04238
$31$	−9.50536	−0.0550714	−0.0275357	−	0.999621i	$-0.508766\pi$
$31$	−9.50536	−0.0550714	−0.0275357	+	0.999621i	$0.508766\pi$
$32$	−6.09673	−0.0336800
$33$	0	0
$34$	460.161	2.32109
$35$	108.860	0.525732
$36$	257.897	1.19397
$37$	−9.14224	−0.0406209	−0.0203105	−	0.999794i	$-0.506465\pi$
$37$	−9.14224	−0.0406209	−0.0203105	+	0.999794i	$0.506465\pi$
$38$	−663.620	−2.83298
$39$	−488.639	−2.00628
$40$	602.536	2.38173
$41$	339.461	1.29305	0.646523	−	0.762894i	$-0.276222\pi$
$41$	339.461	1.29305	0.646523	+	0.762894i	$0.276222\pi$
$42$	225.022	0.826706
$43$	−433.078	−1.53590	−0.767950	−	0.640509i	$-0.778723\pi$
$43$	−433.078	−1.53590	−0.767950	+	0.640509i	$0.778723\pi$
$44$	0	0
$45$	251.899	0.834463
$46$	397.046	1.27264
$47$	−54.4784	−0.169074	−0.0845371	−	0.996420i	$-0.526941\pi$
$47$	−54.4784	−0.169074	−0.0845371	+	0.996420i	$0.526941\pi$
$48$	408.331	1.22786
$49$	49.0000	0.142857
$50$	571.486	1.61641
$51$	618.365	1.69781
$52$	−1183.71	−3.15676
$53$	123.830	0.320932	0.160466	−	0.987041i	$-0.448700\pi$
$53$	123.830	0.320932	0.160466	+	0.987041i	$0.448700\pi$
$54$	−347.246	−0.875078
$55$	0	0
$56$	271.215	0.647189
$57$	−891.773	−2.07225
$58$	261.305	0.591570
$59$	−534.396	−1.17919	−0.589596	−	0.807698i	$-0.700713\pi$
$59$	−534.396	−1.17919	−0.589596	+	0.807698i	$0.700713\pi$
$60$	1627.38	3.50157
$61$	358.624	0.752740	0.376370	−	0.926469i	$-0.377172\pi$
$61$	358.624	0.752740	0.376370	+	0.926469i	$0.377172\pi$
$62$	−46.4905	−0.0952307
$63$	113.385	0.226749
$64$	−526.836	−1.02898
$65$	−1156.18	−2.20625
$66$	0	0
$67$	−694.318	−1.26604	−0.633019	−	0.774137i	$-0.718184\pi$
$67$	−694.318	−1.26604	−0.633019	+	0.774137i	$0.718184\pi$
$68$	1497.97	2.67141
$69$	533.551	0.930899
$70$	532.430	0.909108
$71$	−278.330	−0.465236	−0.232618	−	0.972568i	$-0.574729\pi$
$71$	−278.330	−0.465236	−0.232618	+	0.972568i	$0.574729\pi$
$72$	627.585	1.02724
$73$	886.688	1.42163	0.710815	−	0.703379i	$-0.248326\pi$
$73$	886.688	1.42163	0.710815	+	0.703379i	$0.248326\pi$
$74$	−44.7145	−0.0702427
$75$	767.963	1.18236
$76$	−2160.29	−3.26056
$77$	0	0
$78$	−2389.93	−3.46931
$79$	185.631	0.264369	0.132184	−	0.991225i	$-0.457801\pi$
$79$	185.631	0.264369	0.132184	+	0.991225i	$0.457801\pi$
$80$	966.162	1.35025
$81$	−903.972	−1.24002
$82$	1660.30	2.23597
$83$	122.624	0.162166	0.0810830	−	0.996707i	$-0.474162\pi$
$83$	122.624	0.162166	0.0810830	+	0.996707i	$0.474162\pi$
$84$	732.519	0.951480
$85$	1463.13	1.86704
$86$	−2118.18	−2.65592
$87$	351.142	0.432717
$88$	0	0
$89$	−847.086	−1.00889	−0.504443	−	0.863445i	$-0.668302\pi$
$89$	−847.086	−1.00889	−0.504443	+	0.863445i	$0.668302\pi$
$90$	1232.03	1.44297
$91$	−520.422	−0.599506
$92$	1292.51	1.46471
$93$	−62.4740	−0.0696586
$94$	−266.453	−0.292367
$95$	−2110.04	−2.27880
$96$	−40.0708	−0.0426011
$97$	1002.49	1.04935	0.524676	−	0.851302i	$-0.324186\pi$
$97$	1002.49	1.04935	0.524676	+	0.851302i	$0.324186\pi$
$98$	239.658	0.247032
$99$	0	0
$100$	1860.37	1.86037
$101$	−1124.79	−1.10812	−0.554062	−	0.832476i	$-0.686923\pi$
$101$	−1124.79	−1.10812	−0.554062	+	0.832476i	$0.686923\pi$
$102$	3024.41	2.93590
$103$	966.118	0.924218	0.462109	−	0.886823i	$-0.347093\pi$
$103$	966.118	0.924218	0.462109	+	0.886823i	$0.347093\pi$
$104$	−2880.53	−2.71595
$105$	715.480	0.664988
$106$	605.652	0.554963
$107$	144.202	0.130286	0.0651428	−	0.997876i	$-0.479250\pi$
$107$	144.202	0.130286	0.0651428	+	0.997876i	$0.479250\pi$
$108$	−1130.40	−1.00715
$109$	1875.32	1.64792	0.823961	−	0.566647i	$-0.191760\pi$
$109$	1875.32	1.64792	0.823961	+	0.566647i	$0.191760\pi$
$110$	0	0
$111$	−60.0874	−0.0513806
$112$	434.890	0.366904
$113$	1207.46	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$113$	1207.46	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$114$	−4361.64	−3.58338
$115$	1262.45	1.02369
$116$	850.632	0.680855
$117$	−1204.24	−0.951559
$118$	−2613.72	−2.03909
$119$	658.585	0.507331
$120$	3960.18	3.01261
$121$	0	0
$122$	1754.03	1.30166
$123$	2231.11	1.63555
$124$	−151.342	−0.109604
$125$	−126.825	−0.0907483
$126$	554.564	0.392099
$127$	−1143.56	−0.799009	−0.399504	−	0.916731i	$-0.630818\pi$
$127$	−1143.56	−0.799009	−0.399504	+	0.916731i	$0.630818\pi$
$128$	−2527.97	−1.74565
$129$	−2846.41	−1.94273
$130$	−5654.86	−3.81511
$131$	−2478.40	−1.65297	−0.826484	−	0.562961i	$-0.809662\pi$
$131$	−2478.40	−1.65297	−0.826484	+	0.562961i	$0.809662\pi$
$132$	0	0
$133$	−949.776	−0.619218
$134$	−3395.90	−2.18926
$135$	−1104.10	−0.703897
$136$	3645.26	2.29837
$137$	−835.661	−0.521134	−0.260567	−	0.965456i	$-0.583909\pi$
$137$	−835.661	−0.521134	−0.260567	+	0.965456i	$0.583909\pi$
$138$	2609.59	1.60973
$139$	1726.99	1.05382	0.526912	−	0.849920i	$-0.323350\pi$
$139$	1726.99	1.05382	0.526912	+	0.849920i	$0.323350\pi$
$140$	1733.23	1.04632
$141$	−358.060	−0.213859
$142$	−1361.31	−0.804497
$143$	0	0
$144$	1006.33	0.582365
$145$	830.846	0.475848
$146$	4336.78	2.45832
$147$	322.053	0.180697
$148$	−145.560	−0.0808443
$149$	3454.95	1.89960	0.949799	−	0.312859i	$-0.101287\pi$
$149$	3454.95	1.89960	0.949799	+	0.312859i	$0.101287\pi$
$150$	3756.09	2.04456
$151$	2468.17	1.33018	0.665089	−	0.746764i	$-0.268394\pi$
$151$	2468.17	1.33018	0.665089	+	0.746764i	$0.268394\pi$
$152$	−5257.00	−2.80526
$153$	1523.95	0.805256
$154$	0	0
$155$	−147.821	−0.0766018
$156$	−7779.97	−3.99292
$157$	−1561.27	−0.793649	−0.396824	−	0.917895i	$-0.629888\pi$
$157$	−1561.27	−0.793649	−0.396824	+	0.917895i	$0.629888\pi$
$158$	907.919	0.457153
$159$	813.875	0.405940
$160$	−94.8125	−0.0468474
$161$	568.255	0.278166
$162$	−4421.31	−2.14426
$163$	3458.59	1.66195	0.830973	−	0.556312i	$-0.187784\pi$
$163$	3458.59	1.66195	0.830973	+	0.556312i	$0.187784\pi$
$164$	5404.80	2.57344
$165$	0	0
$166$	599.754	0.280421
$167$	972.527	0.450637	0.225319	−	0.974285i	$-0.427658\pi$
$167$	972.527	0.450637	0.225319	+	0.974285i	$0.427658\pi$
$168$	1782.56	0.818616
$169$	3330.32	1.51585
$170$	7156.13	3.22853
$171$	−2197.76	−0.982848
$172$	−6895.34	−3.05677
$173$	1154.18	0.507230	0.253615	−	0.967305i	$-0.418380\pi$
$173$	1154.18	0.507230	0.253615	+	0.967305i	$0.418380\pi$
$174$	1717.43	0.748265
$175$	817.914	0.353305
$176$	0	0
$177$	−3512.32	−1.49154
$178$	−4143.08	−1.74459
$179$	259.234	0.108246	0.0541231	−	0.998534i	$-0.482764\pi$
$179$	259.234	0.108246	0.0541231	+	0.998534i	$0.482764\pi$
$180$	4010.66	1.66076
$181$	−2121.49	−0.871209	−0.435604	−	0.900138i	$-0.643465\pi$
$181$	−2121.49	−0.871209	−0.435604	+	0.900138i	$0.643465\pi$
$182$	−2545.37	−1.03668
$183$	2357.06	0.952126
$184$	3145.29	1.26018
$185$	−142.174	−0.0565019
$186$	−305.559	−0.120455
$187$	0	0
$188$	−867.389	−0.336494
$189$	−496.981	−0.191270
$190$	−10320.2	−3.94055
$191$	2918.27	1.10554	0.552772	−	0.833332i	$-0.313570\pi$
$191$	2918.27	1.10554	0.552772	+	0.833332i	$0.313570\pi$
$192$	−3462.63	−1.30153
$193$	−3757.48	−1.40139	−0.700697	−	0.713459i	$-0.747128\pi$
$193$	−3757.48	−1.40139	−0.700697	+	0.713459i	$0.747128\pi$
$194$	4903.15	1.81456
$195$	−7599.00	−2.79065
$196$	780.164	0.284316
$197$	−1608.39	−0.581689	−0.290845	−	0.956770i	$-0.593936\pi$
$197$	−1608.39	−0.581689	−0.290845	+	0.956770i	$0.593936\pi$
$198$	0	0
$199$	2865.53	1.02076	0.510382	−	0.859948i	$-0.329504\pi$
$199$	2865.53	1.02076	0.510382	+	0.859948i	$0.329504\pi$
$200$	4527.14	1.60059
$201$	−4563.41	−1.60138
$202$	−5501.31	−1.91619
$203$	373.981	0.129302
$204$	9845.43	3.37901
$205$	5279.08	1.79857
$206$	4725.27	1.59818
$207$	1314.93	0.441517
$208$	−4618.90	−1.53973
$209$	0	0
$210$	3499.40	1.14991
$211$	−821.996	−0.268192	−0.134096	−	0.990968i	$-0.542813\pi$
$211$	−821.996	−0.268192	−0.134096	+	0.990968i	$0.542813\pi$
$212$	1971.59	0.638723
$213$	−1829.33	−0.588467
$214$	705.290	0.225293
$215$	−6734.95	−2.13637
$216$	−2750.78	−0.866514
$217$	−66.5375	−0.0208150
$218$	9172.18	2.84963
$219$	5827.76	1.79819
$220$	0	0
$221$	−6994.73	−2.12903
$222$	−293.887	−0.0888485
$223$	109.532	0.0328916	0.0164458	−	0.999865i	$-0.494765\pi$
$223$	109.532	0.0328916	0.0164458	+	0.999865i	$0.494765\pi$
$224$	−42.6771	−0.0127298
$225$	1892.63	0.560780
$226$	5905.67	1.73823
$227$	−3023.03	−0.883900	−0.441950	−	0.897040i	$-0.645713\pi$
$227$	−3023.03	−0.883900	−0.441950	+	0.897040i	$0.645713\pi$
$228$	−14198.5	−4.12422
$229$	−2278.75	−0.657571	−0.328786	−	0.944405i	$-0.606639\pi$
$229$	−2278.75	−0.657571	−0.328786	+	0.944405i	$0.606639\pi$
$230$	6174.61	1.77018
$231$	0	0
$232$	2069.98	0.585781
$233$	1864.08	0.524121	0.262061	−	0.965051i	$-0.415598\pi$
$233$	1864.08	0.524121	0.262061	+	0.965051i	$0.415598\pi$
$234$	−5889.94	−1.64546
$235$	−847.213	−0.235175
$236$	−8508.49	−2.34685
$237$	1220.06	0.334395
$238$	3221.13	0.877289
$239$	−1404.69	−0.380174	−0.190087	−	0.981767i	$-0.560877\pi$
$239$	−1404.69	−0.380174	−0.190087	+	0.981767i	$0.560877\pi$
$240$	6350.10	1.70791
$241$	−3879.32	−1.03688	−0.518442	−	0.855113i	$-0.673488\pi$
$241$	−3879.32	−1.03688	−0.518442	+	0.855113i	$0.673488\pi$
$242$	0	0
$243$	−4024.43	−1.06242
$244$	5709.91	1.49811
$245$	762.017	0.198708
$246$	10912.3	2.82823
$247$	10087.4	2.59857
$248$	−368.284	−0.0942987
$249$	805.950	0.205121
$250$	−620.297	−0.156924
$251$	−2143.54	−0.539040	−0.269520	−	0.962995i	$-0.586865\pi$
$251$	−2143.54	−0.539040	−0.269520	+	0.962995i	$0.586865\pi$
$252$	1805.28	0.451278
$253$	0	0
$254$	−5593.11	−1.38166
$255$	9616.42	2.36158
$256$	−8149.58	−1.98964
$257$	7288.62	1.76907	0.884537	−	0.466471i	$-0.154475\pi$
$257$	7288.62	1.76907	0.884537	+	0.466471i	$0.154475\pi$
$258$	−13921.7	−3.35941
$259$	−63.9957	−0.0153533
$260$	−18408.4	−4.39092
$261$	865.385	0.205234
$262$	−12121.8	−2.85835
$263$	−2670.52	−0.626127	−0.313064	−	0.949732i	$-0.601355\pi$
$263$	−2670.52	−0.626127	−0.313064	+	0.949732i	$0.601355\pi$
$264$	0	0
$265$	1925.73	0.446402
$266$	−4645.34	−1.07077
$267$	−5567.48	−1.27612
$268$	−11054.7	−2.51968
$269$	1073.80	0.243387	0.121693	−	0.992568i	$-0.461168\pi$
$269$	1073.80	0.243387	0.121693	+	0.992568i	$0.461168\pi$
$270$	−5400.15	−1.21720
$271$	−3624.88	−0.812530	−0.406265	−	0.913755i	$-0.633169\pi$
$271$	−3624.88	−0.812530	−0.406265	+	0.913755i	$0.633169\pi$
$272$	5845.14	1.30299
$273$	−3420.47	−0.758302
$274$	−4087.20	−0.901157
$275$	0	0
$276$	8495.05	1.85269
$277$	4905.35	1.06402	0.532010	−	0.846738i	$-0.321437\pi$
$277$	4905.35	1.06402	0.532010	+	0.846738i	$0.321437\pi$
$278$	8446.68	1.82230
$279$	−153.966	−0.0330384
$280$	4217.76	0.900211
$281$	2661.90	0.565109	0.282555	−	0.959251i	$-0.408818\pi$
$281$	2661.90	0.565109	0.282555	+	0.959251i	$0.408818\pi$
$282$	−1751.26	−0.369809
$283$	4367.36	0.917359	0.458680	−	0.888602i	$-0.348323\pi$
$283$	4367.36	0.917359	0.458680	+	0.888602i	$0.348323\pi$
$284$	−4431.50	−0.925919
$285$	−13868.3	−2.88241
$286$	0	0
$287$	2376.23	0.488726
$288$	−98.7539	−0.0202053
$289$	3938.72	0.801693
$290$	4063.65	0.822848
$291$	6588.86	1.32730
$292$	14117.6	2.82935
$293$	−1992.30	−0.397240	−0.198620	−	0.980077i	$-0.563646\pi$
$293$	−1992.30	−0.397240	−0.198620	+	0.980077i	$0.563646\pi$
$294$	1575.15	0.312466
$295$	−8310.58	−1.64021
$296$	−354.215	−0.0695553
$297$	0	0
$298$	16898.1	3.28483
$299$	−6035.35	−1.16734
$300$	12227.3	2.35314
$301$	−3031.54	−0.580516
$302$	12071.8	2.30018
$303$	−7392.67	−1.40164
$304$	−8429.55	−1.59036
$305$	5577.10	1.04703
$306$	7453.62	1.39247
$307$	−7633.53	−1.41912	−0.709558	−	0.704647i	$-0.751105\pi$
$307$	−7633.53	−1.41912	−0.709558	+	0.704647i	$0.751105\pi$
$308$	0	0
$309$	6349.82	1.16902
$310$	−722.991	−0.132462
$311$	1453.52	0.265020	0.132510	−	0.991182i	$-0.457696\pi$
$311$	1453.52	0.265020	0.132510	+	0.991182i	$0.457696\pi$
$312$	−18932.3	−3.43535
$313$	1936.27	0.349663	0.174831	−	0.984598i	$-0.444062\pi$
$313$	1936.27	0.349663	0.174831	+	0.984598i	$0.444062\pi$
$314$	−7636.14	−1.37240
$315$	1763.29	0.315397
$316$	2955.56	0.526150
$317$	−1534.36	−0.271856	−0.135928	−	0.990719i	$-0.543402\pi$
$317$	−1534.36	−0.271856	−0.135928	+	0.990719i	$0.543402\pi$
$318$	3980.65	0.701961
$319$	0	0
$320$	−8193.02	−1.43126
$321$	947.770	0.164795
$322$	2779.32	0.481012
$323$	−12765.5	−2.19904
$324$	−14392.8	−2.46790
$325$	−8686.94	−1.48266
$326$	16915.9	2.87388
$327$	12325.6	2.08442
$328$	13152.4	2.21408
$329$	−381.349	−0.0639041
$330$	0	0
$331$	3807.42	0.632251	0.316125	−	0.948717i	$-0.397618\pi$
$331$	3807.42	0.632251	0.316125	+	0.948717i	$0.397618\pi$
$332$	1952.39	0.322745
$333$	−148.085	−0.0243693
$334$	4756.61	0.779252
$335$	−10797.6	−1.76100
$336$	2858.32	0.464089
$337$	4724.09	0.763614	0.381807	−	0.924242i	$-0.375302\pi$
$337$	4724.09	0.763614	0.381807	+	0.924242i	$0.375302\pi$
$338$	16288.5	2.62124
$339$	7936.05	1.27147
$340$	23295.5	3.71581
$341$	0	0
$342$	−10749.2	−1.69956
$343$	343.000	0.0539949
$344$	−16779.6	−2.62992
$345$	8297.45	1.29484
$346$	5645.08	0.877113
$347$	−6122.41	−0.947171	−0.473586	−	0.880748i	$-0.657041\pi$
$347$	−6122.41	−0.947171	−0.473586	+	0.880748i	$0.657041\pi$
$348$	5590.79	0.861200
$349$	−6778.93	−1.03974	−0.519868	−	0.854247i	$-0.674019\pi$
$349$	−6778.93	−1.03974	−0.519868	+	0.854247i	$0.674019\pi$
$350$	4000.40	0.610944
$351$	5278.35	0.802672
$352$	0	0
$353$	−3486.98	−0.525760	−0.262880	−	0.964829i	$-0.584672\pi$
$353$	−3486.98	−0.525760	−0.262880	+	0.964829i	$0.584672\pi$
$354$	−17178.7	−2.57920
$355$	−4328.42	−0.647123
$356$	−13487.1	−2.00790
$357$	4328.55	0.641713
$358$	1267.91	0.187182
$359$	−3230.16	−0.474878	−0.237439	−	0.971402i	$-0.576308\pi$
$359$	−3230.16	−0.474878	−0.237439	+	0.971402i	$0.576308\pi$
$360$	9759.79	1.42885
$361$	11550.7	1.68402
$362$	−10376.2	−1.50651
$363$	0	0
$364$	−8286.00	−1.19314
$365$	13789.2	1.97742
$366$	11528.3	1.64644
$367$	−8190.16	−1.16491	−0.582456	−	0.812862i	$-0.697908\pi$
$367$	−8190.16	−1.16491	−0.582456	+	0.812862i	$0.697908\pi$
$368$	5043.44	0.714422
$369$	5498.54	0.775725
$370$	−695.372	−0.0977045
$371$	866.812	0.121301
$372$	−994.693	−0.138636
$373$	−5008.53	−0.695260	−0.347630	−	0.937632i	$-0.613013\pi$
$373$	−5008.53	−0.695260	−0.347630	+	0.937632i	$0.613013\pi$
$374$	0	0
$375$	−833.556	−0.114786
$376$	−2110.76	−0.289506
$377$	−3972.00	−0.542622
$378$	−2430.72	−0.330748
$379$	1522.26	0.206314	0.103157	−	0.994665i	$-0.467106\pi$
$379$	1522.26	0.206314	0.103157	+	0.994665i	$0.467106\pi$
$380$	−33595.5	−4.53530
$381$	−7516.02	−1.01065
$382$	14273.2	1.91173
$383$	8520.08	1.13670	0.568349	−	0.822787i	$-0.307582\pi$
$383$	8520.08	1.13670	0.568349	+	0.822787i	$0.307582\pi$
$384$	−16615.1	−2.20804
$385$	0	0
$386$	−18377.8	−2.42332
$387$	−7014.93	−0.921418
$388$	15961.3	2.08844
$389$	1588.83	0.207088	0.103544	−	0.994625i	$-0.466982\pi$
$389$	1588.83	0.207088	0.103544	+	0.994625i	$0.466982\pi$
$390$	−37166.6	−4.82565
$391$	7637.64	0.987856
$392$	1898.50	0.244614
$393$	−16289.3	−2.09080
$394$	−7866.59	−1.00587
$395$	2886.82	0.367725
$396$	0	0
$397$	−1483.46	−0.187539	−0.0937694	−	0.995594i	$-0.529892\pi$
$397$	−1483.46	−0.187539	−0.0937694	+	0.995594i	$0.529892\pi$
$398$	14015.3	1.76513
$399$	−6242.41	−0.783236
$400$	7259.23	0.907404
$401$	10189.5	1.26892	0.634460	−	0.772956i	$-0.281222\pi$
$401$	10189.5	1.26892	0.634460	+	0.772956i	$0.281222\pi$
$402$	−22319.6	−2.76915
$403$	706.685	0.0873510
$404$	−17908.5	−2.20540
$405$	−14058.0	−1.72481
$406$	1829.14	0.223592
$407$	0	0
$408$	23958.5	2.90716
$409$	−3465.96	−0.419024	−0.209512	−	0.977806i	$-0.567188\pi$
$409$	−3465.96	−0.419024	−0.209512	+	0.977806i	$0.567188\pi$
$410$	25819.9	3.11013
$411$	−5492.39	−0.659171
$412$	15382.2	1.83939
$413$	−3740.77	−0.445693
$414$	6431.30	0.763481
$415$	1906.98	0.225566
$416$	453.267	0.0534213
$417$	11350.7	1.33296
$418$	0	0
$419$	2754.83	0.321199	0.160599	−	0.987020i	$-0.448657\pi$
$419$	2754.83	0.321199	0.160599	+	0.987020i	$0.448657\pi$
$420$	11391.7	1.32347
$421$	−3140.19	−0.363524	−0.181762	−	0.983343i	$-0.558180\pi$
$421$	−3140.19	−0.363524	−0.181762	+	0.983343i	$0.558180\pi$
$422$	−4020.37	−0.463764
$423$	−882.433	−0.101431
$424$	4797.80	0.549532
$425$	10993.2	1.25470
$426$	−8947.21	−1.01759
$427$	2510.37	0.284509
$428$	2295.94	0.259296
$429$	0	0
$430$	−32940.5	−3.69426
$431$	4476.22	0.500260	0.250130	−	0.968212i	$-0.419527\pi$
$431$	4476.22	0.500260	0.250130	+	0.968212i	$0.419527\pi$
$432$	−4410.85	−0.491244
$433$	1646.00	0.182683	0.0913416	−	0.995820i	$-0.470884\pi$
$433$	1646.00	0.182683	0.0913416	+	0.995820i	$0.470884\pi$
$434$	−325.434	−0.0359938
$435$	5460.74	0.601891
$436$	29858.4	3.27972
$437$	−11014.6	−1.20572
$438$	28503.5	3.10947
$439$	7241.05	0.787236	0.393618	−	0.919274i	$-0.371223\pi$
$439$	7241.05	0.787236	0.393618	+	0.919274i	$0.371223\pi$
$440$	0	0
$441$	793.695	0.0857029
$442$	−34211.1	−3.68158
$443$	5887.27	0.631405	0.315703	−	0.948858i	$-0.397760\pi$
$443$	5887.27	0.631405	0.315703	+	0.948858i	$0.397760\pi$
$444$	−956.695	−0.102258
$445$	−13173.3	−1.40332
$446$	535.721	0.0568770
$447$	22707.7	2.40276
$448$	−3687.85	−0.388917
$449$	6378.42	0.670415	0.335207	−	0.942144i	$-0.391194\pi$
$449$	6378.42	0.670415	0.335207	+	0.942144i	$0.391194\pi$
$450$	9256.84	0.969715
$451$	0	0
$452$	19224.8	2.00058
$453$	16222.1	1.68252
$454$	−14785.6	−1.52846
$455$	−8093.26	−0.833886
$456$	−34551.7	−3.54831
$457$	18368.0	1.88013	0.940064	−	0.340998i	$-0.110765\pi$
$457$	18368.0	1.88013	0.940064	+	0.340998i	$0.110765\pi$
$458$	−11145.3	−1.13709
$459$	−6679.67	−0.679260
$460$	20100.3	2.03735
$461$	−17510.6	−1.76909	−0.884546	−	0.466453i	$-0.845532\pi$
$461$	−17510.6	−1.76909	−0.884546	+	0.466453i	$0.845532\pi$
$462$	0	0
$463$	−12732.5	−1.27803	−0.639016	−	0.769193i	$-0.720658\pi$
$463$	−12732.5	−1.27803	−0.639016	+	0.769193i	$0.720658\pi$
$464$	3319.20	0.332091
$465$	−971.556	−0.0968921
$466$	9117.20	0.906323
$467$	−4997.23	−0.495170	−0.247585	−	0.968866i	$-0.579637\pi$
$467$	−4997.23	−0.495170	−0.247585	+	0.968866i	$0.579637\pi$
$468$	−19173.6	−1.89381
$469$	−4860.23	−0.478517
$470$	−4143.70	−0.406670
$471$	−10261.5	−1.00387
$472$	−20705.1	−2.01913
$473$	0	0
$474$	5967.30	0.578243
$475$	−15853.8	−1.53141
$476$	10485.8	1.00970
$477$	2005.78	0.192534
$478$	−6870.30	−0.657406
$479$	9075.53	0.865703	0.432851	−	0.901465i	$-0.357507\pi$
$479$	9075.53	0.865703	0.432851	+	0.901465i	$0.357507\pi$
$480$	−623.156	−0.0592563
$481$	679.688	0.0644306
$482$	−18973.7	−1.79300
$483$	3734.86	0.351847
$484$	0	0
$485$	15590.0	1.45960
$486$	−19683.4	−1.83716
$487$	−3867.63	−0.359875	−0.179937	−	0.983678i	$-0.557590\pi$
$487$	−3867.63	−0.359875	−0.179937	+	0.983678i	$0.557590\pi$
$488$	13894.9	1.28892
$489$	22731.6	2.10216
$490$	3727.01	0.343611
$491$	−7334.24	−0.674113	−0.337057	−	0.941484i	$-0.609431\pi$
$491$	−7334.24	−0.674113	−0.337057	+	0.941484i	$0.609431\pi$
$492$	35523.1	3.25509
$493$	5026.50	0.459193
$494$	49337.4	4.49351
$495$	0	0
$496$	−590.541	−0.0534598
$497$	−1948.31	−0.175843
$498$	3941.89	0.354699
$499$	−4365.93	−0.391675	−0.195838	−	0.980636i	$-0.562743\pi$
$499$	−4365.93	−0.391675	−0.195838	+	0.980636i	$0.562743\pi$
$500$	−2019.26	−0.180609
$501$	6391.94	0.570002
$502$	−10484.0	−0.932120
$503$	16352.6	1.44956	0.724779	−	0.688981i	$-0.241942\pi$
$503$	16352.6	1.44956	0.724779	+	0.688981i	$0.241942\pi$
$504$	4393.09	0.388262
$505$	−17492.0	−1.54135
$506$	0	0
$507$	21888.5	1.91737
$508$	−18207.4	−1.59020
$509$	−12108.2	−1.05440	−0.527199	−	0.849742i	$-0.676758\pi$
$509$	−12108.2	−1.05440	−0.527199	+	0.849742i	$0.676758\pi$
$510$	47033.7	4.08370
$511$	6206.82	0.537326
$512$	−19635.7	−1.69489
$513$	9633.06	0.829065
$514$	35648.5	3.05912
$515$	15024.4	1.28555
$516$	−45319.7	−3.86645
$517$	0	0
$518$	−313.002	−0.0265492
$519$	7585.86	0.641584
$520$	−44796.1	−3.77777
$521$	−5625.67	−0.473062	−0.236531	−	0.971624i	$-0.576010\pi$
$521$	−5625.67	−0.473062	−0.236531	+	0.971624i	$0.576010\pi$
$522$	4232.58	0.354895
$523$	3280.32	0.274261	0.137130	−	0.990553i	$-0.456212\pi$
$523$	3280.32	0.274261	0.137130	+	0.990553i	$0.456212\pi$
$524$	−39460.3	−3.28976
$525$	5375.74	0.446889
$526$	−13061.5	−1.08271
$527$	−894.298	−0.0739207
$528$	0	0
$529$	−5576.93	−0.458365
$530$	9418.70	0.771929
$531$	−8656.06	−0.707422
$532$	−15122.1	−1.23238
$533$	−25237.6	−2.05096
$534$	−27230.4	−2.20670
$535$	2242.54	0.181221
$536$	−26901.3	−2.16784
$537$	1703.82	0.136918
$538$	5251.96	0.420870
$539$	0	0
$540$	−17579.2	−1.40091
$541$	7772.10	0.617650	0.308825	−	0.951119i	$-0.400064\pi$
$541$	7772.10	0.617650	0.308825	+	0.951119i	$0.400064\pi$
$542$	−17729.2	−1.40505
$543$	−13943.5	−1.10197
$544$	−573.602	−0.0452077
$545$	29163.8	2.29219
$546$	−16729.5	−1.31127
$547$	−10022.2	−0.783398	−0.391699	−	0.920094i	$-0.628112\pi$
$547$	−10022.2	−0.783398	−0.391699	+	0.920094i	$0.628112\pi$
$548$	−13305.1	−1.03717
$549$	5808.94	0.451584
$550$	0	0
$551$	−7248.95	−0.560464
$552$	20672.4	1.59398
$553$	1299.42	0.0999220
$554$	23992.0	1.83993
$555$	−934.441	−0.0714681
$556$	27496.7	2.09733
$557$	10849.3	0.825310	0.412655	−	0.910887i	$-0.364601\pi$
$557$	10849.3	0.825310	0.412655	+	0.910887i	$0.364601\pi$
$558$	−753.047	−0.0571308
$559$	32197.6	2.43616
$560$	6763.13	0.510347
$561$	0	0
$562$	13019.3	0.977200
$563$	−22019.9	−1.64836	−0.824180	−	0.566328i	$-0.808363\pi$
$563$	−22019.9	−1.64836	−0.824180	+	0.566328i	$0.808363\pi$
$564$	−5700.92	−0.425624
$565$	18777.7	1.39820
$566$	21360.7	1.58632
$567$	−6327.80	−0.468682
$568$	−10783.9	−0.796624
$569$	−13742.9	−1.01253	−0.506267	−	0.862377i	$-0.668975\pi$
$569$	−13742.9	−1.01253	−0.506267	+	0.862377i	$0.668975\pi$
$570$	−67829.5	−4.98432
$571$	−19382.4	−1.42054	−0.710269	−	0.703931i	$-0.751427\pi$
$571$	−19382.4	−1.42054	−0.710269	+	0.703931i	$0.751427\pi$
$572$	0	0
$573$	19180.4	1.39838
$574$	11622.1	0.845116
$575$	9485.37	0.687943
$576$	−8533.61	−0.617304
$577$	−565.348	−0.0407899	−0.0203949	−	0.999792i	$-0.506492\pi$
$577$	−565.348	−0.0407899	−0.0203949	+	0.999792i	$0.506492\pi$
$578$	19264.2	1.38631
$579$	−24696.0	−1.77260
$580$	13228.5	0.947040
$581$	858.371	0.0612930
$582$	32226.0	2.29521
$583$	0	0
$584$	34354.7	2.43426
$585$	−18727.6	−1.32358
$586$	−9744.30	−0.686917
$587$	−20727.4	−1.45743	−0.728714	−	0.684818i	$-0.759882\pi$
$587$	−20727.4	−1.45743	−0.728714	+	0.684818i	$0.759882\pi$
$588$	5127.63	0.359626
$589$	1289.71	0.0902233
$590$	−40646.9	−2.83628
$591$	−10571.1	−0.735767
$592$	−567.981	−0.0394322
$593$	2787.10	0.193006	0.0965030	−	0.995333i	$-0.469234\pi$
$593$	2787.10	0.193006	0.0965030	+	0.995333i	$0.469234\pi$
$594$	0	0
$595$	10241.9	0.705675
$596$	55008.7	3.78061
$597$	18833.7	1.29114
$598$	−29518.8	−2.01858
$599$	18935.5	1.29162	0.645812	−	0.763497i	$-0.276519\pi$
$599$	18935.5	1.29162	0.645812	+	0.763497i	$0.276519\pi$
$600$	29754.7	2.02455
$601$	−15821.3	−1.07382	−0.536908	−	0.843641i	$-0.680408\pi$
$601$	−15821.3	−1.07382	−0.536908	+	0.843641i	$0.680408\pi$
$602$	−14827.2	−1.00384
$603$	−11246.5	−0.759521
$604$	39297.5	2.64734
$605$	0	0
$606$	−36157.4	−2.42375
$607$	2084.63	0.139394	0.0696972	−	0.997568i	$-0.477797\pi$
$607$	2084.63	0.139394	0.0696972	+	0.997568i	$0.477797\pi$
$608$	827.218	0.0551778
$609$	2458.00	0.163552
$610$	27277.5	1.81055
$611$	4050.25	0.268176
$612$	24263.9	1.60263
$613$	4395.15	0.289590	0.144795	−	0.989462i	$-0.453748\pi$
$613$	4395.15	0.289590	0.144795	+	0.989462i	$0.453748\pi$
$614$	−37335.5	−2.45397
$615$	34696.8	2.27498
$616$	0	0
$617$	−98.5856	−0.00643259	−0.00321629	−	0.999995i	$-0.501024\pi$
$617$	−98.5856	−0.00643259	−0.00321629	+	0.999995i	$0.501024\pi$
$618$	31056.8	2.02150
$619$	−16533.9	−1.07359	−0.536796	−	0.843712i	$-0.680366\pi$
$619$	−16533.9	−1.07359	−0.536796	+	0.843712i	$0.680366\pi$
$620$	−2353.57	−0.152454
$621$	−5763.50	−0.372434
$622$	7109.12	0.458280
$623$	−5929.60	−0.381323
$624$	−30357.8	−1.94757
$625$	−16577.9	−1.06099
$626$	9470.27	0.604645
$627$	0	0
$628$	−24858.1	−1.57953
$629$	−860.135	−0.0545243
$630$	8624.22	0.545392
$631$	22032.9	1.39004	0.695022	−	0.718989i	$-0.255395\pi$
$631$	22032.9	1.39004	0.695022	+	0.718989i	$0.255395\pi$
$632$	7192.27	0.452679
$633$	−5402.57	−0.339231
$634$	−7504.54	−0.470100
$635$	−17783.8	−1.11139
$636$	12958.3	0.807908
$637$	−3642.95	−0.226592
$638$	0	0
$639$	−4508.35	−0.279104
$640$	−39313.4	−2.42812
$641$	434.281	0.0267598	0.0133799	−	0.999910i	$-0.495741\pi$
$641$	434.281	0.0267598	0.0133799	+	0.999910i	$0.495741\pi$
$642$	4635.52	0.284968
$643$	10963.1	0.672381	0.336190	−	0.941794i	$-0.390861\pi$
$643$	10963.1	0.672381	0.336190	+	0.941794i	$0.390861\pi$
$644$	9047.59	0.553610
$645$	−44265.5	−2.70225
$646$	−62435.7	−3.80263
$647$	−452.083	−0.0274702	−0.0137351	−	0.999906i	$-0.504372\pi$
$647$	−452.083	−0.0274702	−0.0137351	+	0.999906i	$0.504372\pi$
$648$	−35024.3	−2.12328
$649$	0	0
$650$	−42487.6	−2.56385
$651$	−437.318	−0.0263285
$652$	55066.6	3.30763
$653$	−10438.7	−0.625572	−0.312786	−	0.949824i	$-0.601262\pi$
$653$	−10438.7	−0.625572	−0.312786	+	0.949824i	$0.601262\pi$
$654$	60284.2	3.60443
$655$	−38542.5	−2.29920
$656$	21089.8	1.25521
$657$	14362.4	0.852865
$658$	−1865.17	−0.110504
$659$	8738.79	0.516563	0.258281	−	0.966070i	$-0.416844\pi$
$659$	8738.79	0.516563	0.258281	+	0.966070i	$0.416844\pi$
$660$	0	0
$661$	12849.4	0.756102	0.378051	−	0.925785i	$-0.376594\pi$
$661$	12849.4	0.756102	0.378051	+	0.925785i	$0.376594\pi$
$662$	18622.0	1.09330
$663$	−45972.9	−2.69297
$664$	4751.08	0.277677
$665$	−14770.3	−0.861305
$666$	−724.279	−0.0421400
$667$	4337.08	0.251773
$668$	15484.3	0.896864
$669$	719.903	0.0416040
$670$	−52810.8	−3.04517
$671$	0	0
$672$	−280.496	−0.0161017
$673$	22926.5	1.31315	0.656577	−	0.754259i	$-0.272004\pi$
$673$	22926.5	1.31315	0.656577	+	0.754259i	$0.272004\pi$
$674$	23105.5	1.32046
$675$	−8295.66	−0.473037
$676$	53024.4	3.01686
$677$	−13633.4	−0.773965	−0.386983	−	0.922087i	$-0.626483\pi$
$677$	−13633.4	−0.773965	−0.386983	+	0.922087i	$0.626483\pi$
$678$	38815.1	2.19865
$679$	7017.41	0.396618
$680$	56688.8	3.19694
$681$	−19868.9	−1.11803
$682$	0	0
$683$	33307.5	1.86600	0.932998	−	0.359882i	$-0.117183\pi$
$683$	33307.5	1.86600	0.932998	+	0.359882i	$0.117183\pi$
$684$	−34992.1	−1.95608
$685$	−12995.7	−0.724874
$686$	1677.61	0.0933693
$687$	−14977.1	−0.831748
$688$	−26905.9	−1.49096
$689$	−9206.28	−0.509044
$690$	40582.7	2.23907
$691$	−10077.3	−0.554786	−0.277393	−	0.960757i	$-0.589470\pi$
$691$	−10077.3	−0.554786	−0.277393	+	0.960757i	$0.589470\pi$
$692$	18376.5	1.00950
$693$	0	0
$694$	−29944.6	−1.63787
$695$	26857.1	1.46582
$696$	13605.0	0.740942
$697$	31937.7	1.73562
$698$	−33155.6	−1.79794
$699$	12251.7	0.662950
$700$	13022.6	0.703153
$701$	4621.74	0.249017	0.124508	−	0.992219i	$-0.460265\pi$
$701$	4621.74	0.249017	0.124508	+	0.992219i	$0.460265\pi$
$702$	25816.3	1.38800
$703$	1240.44	0.0665492
$704$	0	0
$705$	−5568.31	−0.297468
$706$	−17054.8	−0.909157
$707$	−7873.51	−0.418831
$708$	−55922.1	−2.96848
$709$	17746.6	0.940038	0.470019	−	0.882656i	$-0.344247\pi$
$709$	17746.6	0.940038	0.470019	+	0.882656i	$0.344247\pi$
$710$	−21170.2	−1.11902
$711$	3006.82	0.158600
$712$	−32820.3	−1.72752
$713$	−771.638	−0.0405302
$714$	21170.9	1.10966
$715$	0	0
$716$	4127.45	0.215433
$717$	−9232.31	−0.480875
$718$	−15798.6	−0.821170
$719$	−31652.1	−1.64176	−0.820879	−	0.571103i	$-0.806516\pi$
$719$	−31652.1	−1.64176	−0.820879	+	0.571103i	$0.806516\pi$
$720$	15649.7	0.810044
$721$	6762.83	0.349322
$722$	56494.1	2.91204
$723$	−25496.9	−1.31153
$724$	−33777.7	−1.73389
$725$	6242.54	0.319782
$726$	0	0
$727$	21610.8	1.10248	0.551239	−	0.834348i	$-0.314155\pi$
$727$	21610.8	1.10248	0.551239	+	0.834348i	$0.314155\pi$
$728$	−20163.7	−1.02653
$729$	−2043.39	−0.103815
$730$	67442.8	3.41941
$731$	−40745.5	−2.06160
$732$	37528.5	1.89493
$733$	7665.43	0.386261	0.193130	−	0.981173i	$-0.438136\pi$
$733$	7665.43	0.386261	0.193130	+	0.981173i	$0.438136\pi$
$734$	−40057.9	−2.01439
$735$	5008.36	0.251342
$736$	−494.928	−0.0247871
$737$	0	0
$738$	26893.3	1.34140
$739$	13841.5	0.688996	0.344498	−	0.938787i	$-0.388049\pi$
$739$	13841.5	0.688996	0.344498	+	0.938787i	$0.388049\pi$
$740$	−2263.66	−0.112451
$741$	66299.7	3.28688
$742$	4239.56	0.209756
$743$	12136.3	0.599243	0.299621	−	0.954058i	$-0.403140\pi$
$743$	12136.3	0.599243	0.299621	+	0.954058i	$0.403140\pi$
$744$	−2420.55	−0.119277
$745$	53729.1	2.64226
$746$	−24496.6	−1.20226
$747$	1986.25	0.0972867
$748$	0	0
$749$	1009.42	0.0492433
$750$	−4076.91	−0.198490
$751$	28866.7	1.40261	0.701305	−	0.712861i	$-0.252601\pi$
$751$	28866.7	1.40261	0.701305	+	0.712861i	$0.252601\pi$
$752$	−3384.59	−0.164127
$753$	−14088.4	−0.681820
$754$	−19427.0	−0.938314
$755$	38383.4	1.85022
$756$	−7912.78	−0.380668
$757$	8370.48	0.401889	0.200945	−	0.979603i	$-0.435599\pi$
$757$	8370.48	0.401889	0.200945	+	0.979603i	$0.435599\pi$
$758$	7445.33	0.356763
$759$	0	0
$760$	−81753.5	−3.90199
$761$	−30458.1	−1.45086	−0.725430	−	0.688296i	$-0.758359\pi$
$761$	−30458.1	−1.45086	−0.725430	+	0.688296i	$0.758359\pi$
$762$	−36760.7	−1.74764
$763$	13127.3	0.622856
$764$	46463.9	2.20027
$765$	23699.5	1.12008
$766$	41671.5	1.96561
$767$	39730.1	1.87037
$768$	−53563.2	−2.51666
$769$	32239.5	1.51182	0.755908	−	0.654678i	$-0.227196\pi$
$769$	32239.5	1.51182	0.755908	+	0.654678i	$0.227196\pi$
$770$	0	0
$771$	47904.5	2.23766
$772$	−59825.5	−2.78907
$773$	−7524.47	−0.350112	−0.175056	−	0.984559i	$-0.556011\pi$
$773$	−7524.47	−0.350112	−0.175056	+	0.984559i	$0.556011\pi$
$774$	−34309.9	−1.59334
$775$	−1110.65	−0.0514784
$776$	38841.3	1.79681
$777$	−420.612	−0.0194200
$778$	7770.96	0.358101
$779$	−46058.8	−2.11839
$780$	−120989.	−5.55398
$781$	0	0
$782$	37355.5	1.70822
$783$	−3793.09	−0.173121
$784$	3044.23	0.138677
$785$	−24279.9	−1.10393
$786$	−79670.7	−3.61547
$787$	−33196.6	−1.50360	−0.751798	−	0.659394i	$-0.770813\pi$
$787$	−33196.6	−1.50360	−0.751798	+	0.659394i	$0.770813\pi$
$788$	−25608.3	−1.15769
$789$	−17552.0	−0.791976
$790$	14119.4	0.635879
$791$	8452.23	0.379932
$792$	0	0
$793$	−26662.3	−1.19395
$794$	−7255.59	−0.324296
$795$	12656.9	0.564645
$796$	45624.1	2.03154
$797$	8817.06	0.391865	0.195932	−	0.980617i	$-0.437227\pi$
$797$	8817.06	0.391865	0.195932	+	0.980617i	$0.437227\pi$
$798$	−30531.5	−1.35439
$799$	−5125.52	−0.226944
$800$	−712.371	−0.0314827
$801$	−13721.0	−0.605252
$802$	49836.4	2.19425
$803$	0	0
$804$	−72657.3	−3.18710
$805$	8837.14	0.386917
$806$	3456.38	0.151049
$807$	7057.59	0.307855
$808$	−43579.8	−1.89744
$809$	28930.3	1.25727	0.628637	−	0.777699i	$-0.283613\pi$
$809$	28930.3	1.25727	0.628637	+	0.777699i	$0.283613\pi$
$810$	−68757.4	−2.98258
$811$	45923.3	1.98839	0.994195	−	0.107594i	$-0.0343146\pi$
$811$	45923.3	1.98839	0.994195	+	0.107594i	$0.0343146\pi$
$812$	5954.42	0.257339
$813$	−23824.5	−1.02775
$814$	0	0
$815$	53785.7	2.31169
$816$	38417.3	1.64813
$817$	58761.0	2.51626
$818$	−16952.0	−0.724586
$819$	−8429.71	−0.359656
$820$	84052.0	3.57954
$821$	−16382.8	−0.696423	−0.348211	−	0.937416i	$-0.613211\pi$
$821$	−16382.8	−0.696423	−0.348211	+	0.937416i	$0.613211\pi$
$822$	−26863.2	−1.13985
$823$	−32395.3	−1.37209	−0.686045	−	0.727559i	$-0.740655\pi$
$823$	−32395.3	−1.37209	−0.686045	+	0.727559i	$0.740655\pi$
$824$	37432.2	1.58254
$825$	0	0
$826$	−18296.0	−0.770703
$827$	14329.2	0.602508	0.301254	−	0.953544i	$-0.402595\pi$
$827$	14329.2	0.602508	0.301254	+	0.953544i	$0.402595\pi$
$828$	20935.9	0.878712
$829$	16445.8	0.689005	0.344503	−	0.938785i	$-0.388048\pi$
$829$	16445.8	0.689005	0.344503	+	0.938785i	$0.388048\pi$
$830$	9326.99	0.390054
$831$	32240.4	1.34586
$832$	39168.1	1.63210
$833$	4610.10	0.191753
$834$	55515.9	2.30499
$835$	15124.1	0.626816
$836$	0	0
$837$	674.854	0.0278690
$838$	13473.8	0.555424
$839$	40514.7	1.66713	0.833565	−	0.552422i	$-0.186296\pi$
$839$	40514.7	1.66713	0.833565	+	0.552422i	$0.186296\pi$
$840$	27721.2	1.13866
$841$	−21534.7	−0.882967
$842$	−15358.6	−0.628614
$843$	17495.4	0.714795
$844$	−13087.6	−0.533760
$845$	51791.0	2.10848
$846$	−4315.96	−0.175397
$847$	0	0
$848$	7693.22	0.311540
$849$	28704.5	1.16035
$850$	53767.4	2.16966
$851$	−742.160	−0.0298953
$852$	−29126.0	−1.17118
$853$	27962.7	1.12242	0.561211	−	0.827673i	$-0.310336\pi$
$853$	27962.7	1.12242	0.561211	+	0.827673i	$0.310336\pi$
$854$	12278.2	0.491980
$855$	−34178.2	−1.36710
$856$	5587.10	0.223088
$857$	4639.52	0.184928	0.0924639	−	0.995716i	$-0.470526\pi$
$857$	4639.52	0.184928	0.0924639	+	0.995716i	$0.470526\pi$
$858$	0	0
$859$	18521.2	0.735663	0.367831	−	0.929892i	$-0.380100\pi$
$859$	18521.2	0.735663	0.367831	+	0.929892i	$0.380100\pi$
$860$	−107232.	−4.25183
$861$	15617.8	0.618179
$862$	21893.1	0.865061
$863$	40491.4	1.59715	0.798577	−	0.601893i	$-0.205587\pi$
$863$	40491.4	1.59715	0.798577	+	0.601893i	$0.205587\pi$
$864$	432.851	0.0170439
$865$	17949.1	0.705534
$866$	8050.57	0.315900
$867$	25887.3	1.01405
$868$	−1059.39	−0.0414263
$869$	0	0
$870$	26708.4	1.04080
$871$	51619.8	2.00812
$872$	72659.3	2.82174
$873$	16238.1	0.629528
$874$	−53872.1	−2.08496
$875$	−887.772	−0.0342996
$876$	92788.0	3.57878
$877$	48530.3	1.86859	0.934295	−	0.356502i	$-0.116031\pi$
$877$	48530.3	1.86859	0.934295	+	0.356502i	$0.116031\pi$
$878$	35415.9	1.36131
$879$	−13094.4	−0.502461
$880$	0	0
$881$	−11590.0	−0.443219	−0.221609	−	0.975135i	$-0.571131\pi$
$881$	−11590.0	−0.443219	−0.221609	+	0.975135i	$0.571131\pi$
$882$	3881.95	0.148199
$883$	41900.7	1.59691	0.798455	−	0.602054i	$-0.205651\pi$
$883$	41900.7	1.59691	0.798455	+	0.602054i	$0.205651\pi$
$884$	−111368.	−4.23723
$885$	−54621.3	−2.07466
$886$	28794.5	1.09184
$887$	17136.8	0.648700	0.324350	−	0.945937i	$-0.394854\pi$
$887$	17136.8	0.648700	0.324350	+	0.945937i	$0.394854\pi$
$888$	−2328.08	−0.0879790
$889$	−8004.89	−0.301997
$890$	−64430.6	−2.42665
$891$	0	0
$892$	1743.94	0.0654614
$893$	7391.75	0.276994
$894$	111063.	4.15492
$895$	4031.44	0.150566
$896$	−17695.8	−0.659794
$897$	−39667.4	−1.47654
$898$	31196.7	1.15930
$899$	−507.832	−0.0188400
$900$	30134.0	1.11607
$901$	11650.4	0.430778
$902$	0	0
$903$	−19924.8	−0.734283
$904$	46783.0	1.72122
$905$	−32992.0	−1.21181
$906$	79341.9	2.90944
$907$	40350.7	1.47720	0.738601	−	0.674143i	$-0.235487\pi$
$907$	40350.7	1.47720	0.738601	+	0.674143i	$0.235487\pi$
$908$	−48131.8	−1.75915
$909$	−18219.1	−0.664786
$910$	−39584.0	−1.44197
$911$	−32322.7	−1.17552	−0.587760	−	0.809035i	$-0.699990\pi$
$911$	−32322.7	−1.17552	−0.587760	+	0.809035i	$0.699990\pi$
$912$	−55403.3	−2.01161
$913$	0	0
$914$	89837.5	3.25116
$915$	36655.5	1.32437
$916$	−36281.5	−1.30871
$917$	−17348.8	−0.624763
$918$	−32670.2	−1.17459
$919$	−2431.28	−0.0872694	−0.0436347	−	0.999048i	$-0.513894\pi$
$919$	−2431.28	−0.0872694	−0.0436347	+	0.999048i	$0.513894\pi$
$920$	48913.5	1.75286
$921$	−50171.4	−1.79501
$922$	−85644.2	−3.05916
$923$	20692.7	0.737930
$924$	0	0
$925$	−1068.22	−0.0379708
$926$	−62274.4	−2.21000
$927$	15649.0	0.554457
$928$	−325.724	−0.0115220
$929$	40221.0	1.42046	0.710230	−	0.703970i	$-0.248591\pi$
$929$	40221.0	1.42046	0.710230	+	0.703970i	$0.248591\pi$
$930$	−4751.86	−0.167548
$931$	−6648.43	−0.234042
$932$	29679.4	1.04311
$933$	9553.25	0.335219
$934$	−24441.4	−0.856260
$935$	0	0
$936$	−46658.4	−1.62936
$937$	−41503.4	−1.44702	−0.723510	−	0.690314i	$-0.757472\pi$
$937$	−41503.4	−1.44702	−0.723510	+	0.690314i	$0.757472\pi$
$938$	−23771.3	−0.827463
$939$	12726.2	0.442282
$940$	−13489.1	−0.468048
$941$	9032.78	0.312923	0.156461	−	0.987684i	$-0.449991\pi$
$941$	9032.78	0.312923	0.156461	+	0.987684i	$0.449991\pi$
$942$	−50188.6	−1.73592
$943$	27557.2	0.951629
$944$	−33200.5	−1.14469
$945$	−7728.72	−0.266048
$946$	0	0
$947$	5565.96	0.190992	0.0954960	−	0.995430i	$-0.469556\pi$
$947$	5565.96	0.190992	0.0954960	+	0.995430i	$0.469556\pi$
$948$	19425.5	0.665517
$949$	−65921.7	−2.25491
$950$	−77540.5	−2.64815
$951$	−10084.6	−0.343865
$952$	25516.8	0.868703
$953$	−32657.4	−1.11005	−0.555024	−	0.831834i	$-0.687291\pi$
$953$	−32657.4	−1.11005	−0.555024	+	0.831834i	$0.687291\pi$
$954$	9810.25	0.332934
$955$	45383.1	1.53776
$956$	−22365.0	−0.756628
$957$	0	0
$958$	44388.3	1.49699
$959$	−5849.63	−0.196970
$960$	−53848.7	−1.81037
$961$	−29700.6	−0.996967
$962$	3324.34	0.111415
$963$	2335.76	0.0781609
$964$	−61765.4	−2.06362
$965$	−58433.9	−1.94928
$966$	18267.1	0.608422
$967$	610.079	0.0202883	0.0101442	−	0.999949i	$-0.496771\pi$
$967$	610.079	0.0202883	0.0101442	+	0.999949i	$0.496771\pi$
$968$	0	0
$969$	−83901.2	−2.78152
$970$	76250.6	2.52398
$971$	−45371.4	−1.49952	−0.749762	−	0.661708i	$-0.769832\pi$
$971$	−45371.4	−1.49952	−0.749762	+	0.661708i	$0.769832\pi$
$972$	−64075.9	−2.11444
$973$	12088.9	0.398308
$974$	−18916.5	−0.622304
$975$	−57094.9	−1.87539
$976$	22280.3	0.730713
$977$	27229.0	0.891639	0.445820	−	0.895123i	$-0.352912\pi$
$977$	27229.0	0.891639	0.445820	+	0.895123i	$0.352912\pi$
$978$	111180.	3.63511
$979$	0	0
$980$	12132.6	0.395471
$981$	30376.2	0.988622
$982$	−35871.6	−1.16569
$983$	7458.49	0.242003	0.121002	−	0.992652i	$-0.461389\pi$
$983$	7458.49	0.242003	0.121002	+	0.992652i	$0.461389\pi$
$984$	86444.2	2.80055
$985$	−25012.6	−0.809104
$986$	24584.5	0.794048
$987$	−2506.42	−0.0808310
$988$	160609.	5.17171
$989$	−35156.9	−1.13036
$990$	0	0
$991$	−24357.8	−0.780778	−0.390389	−	0.920650i	$-0.627660\pi$
$991$	−24357.8	−0.780778	−0.390389	+	0.920650i	$0.627660\pi$
$992$	57.9516	0.00185480
$993$	25024.3	0.799721
$994$	−9529.17	−0.304071
$995$	44562.9	1.41984
$996$	12832.1	0.408234
$997$	−19954.3	−0.633862	−0.316931	−	0.948449i	$-0.602652\pi$
$997$	−19954.3	−0.633862	−0.316931	+	0.948449i	$0.602652\pi$
$998$	−21353.7	−0.677294
$999$	649.074	0.0205563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.4.a.d.1.4		4
11.10	odd	2		77.4.a.d.1.1	✓	4
33.32	even	2		693.4.a.l.1.4		4
44.43	even	2		1232.4.a.s.1.2		4
55.54	odd	2		1925.4.a.p.1.4		4
77.76	even	2		539.4.a.g.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.1	✓	4	11.10	odd	2
539.4.a.g.1.1		4	77.76	even	2
693.4.a.l.1.4		4	33.32	even	2
847.4.a.d.1.4		4	1.1	even	1	trivial
1232.4.a.s.1.2		4	44.43	even	2
1925.4.a.p.1.4		4	55.54	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q+4.89098 q^{2} +6.57251 q^{3} +15.9217 q^{4} +15.5514 q^{5} +32.1460 q^{6} +7.00000 q^{7} +38.7449 q^{8} +16.1978 q^{9} +76.0614 q^{10} +104.646 q^{12} -74.3459 q^{13} +34.2369 q^{14} +102.211 q^{15} +62.1271 q^{16} +94.0836 q^{17} +79.2234 q^{18} -135.682 q^{19} +247.604 q^{20} +46.0075 q^{21} +81.1793 q^{23} +254.651 q^{24} +116.845 q^{25} -363.625 q^{26} -70.9972 q^{27} +111.452 q^{28} +53.4259 q^{29} +499.914 q^{30} -9.50536 q^{31} -6.09673 q^{32} +460.161 q^{34} +108.860 q^{35} +257.897 q^{36} -9.14224 q^{37} -663.620 q^{38} -488.639 q^{39} +602.536 q^{40} +339.461 q^{41} +225.022 q^{42} -433.078 q^{43} +251.899 q^{45} +397.046 q^{46} -54.4784 q^{47} +408.331 q^{48} +49.0000 q^{49} +571.486 q^{50} +618.365 q^{51} -1183.71 q^{52} +123.830 q^{53} -347.246 q^{54} +271.215 q^{56} -891.773 q^{57} +261.305 q^{58} -534.396 q^{59} +1627.38 q^{60} +358.624 q^{61} -46.4905 q^{62} +113.385 q^{63} -526.836 q^{64} -1156.18 q^{65} -694.318 q^{67} +1497.97 q^{68} +533.551 q^{69} +532.430 q^{70} -278.330 q^{71} +627.585 q^{72} +886.688 q^{73} -44.7145 q^{74} +767.963 q^{75} -2160.29 q^{76} -2389.93 q^{78} +185.631 q^{79} +966.162 q^{80} -903.972 q^{81} +1660.30 q^{82} +122.624 q^{83} +732.519 q^{84} +1463.13 q^{85} -2118.18 q^{86} +351.142 q^{87} -847.086 q^{89} +1232.03 q^{90} -520.422 q^{91} +1292.51 q^{92} -62.4740 q^{93} -266.453 q^{94} -2110.04 q^{95} -40.0708 q^{96} +1002.49 q^{97} +239.658 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} - 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} + 2 q^{10} + 70 q^{12} - 58 q^{13} + 14 q^{14} + 284 q^{15} + 2 q^{16} - 4 q^{17} + 62 q^{18} - 258 q^{19} + 182 q^{20}+ \cdots + 98 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 - 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 + 2 * q^10 + 70 * q^12 - 58 * q^13 + 14 * q^14 + 284 * q^15 + 2 * q^16 - 4 * q^17 + 62 * q^18 - 258 * q^19 + 182 * q^20 + 98 * q^21 + 8 * q^23 + 498 * q^24 + 80 * q^25 - 482 * q^26 + 428 * q^27 + 182 * q^28 + 396 * q^29 + 628 * q^30 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 - 418 * q^36 + 84 * q^37 - 942 * q^38 + 412 * q^39 + 1026 * q^40 - 52 * q^41 - 98 * q^42 - 408 * q^43 + 826 * q^45 - 368 * q^46 + 8 * q^47 + 982 * q^48 + 196 * q^49 + 1642 * q^50 + 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 126 * q^56 - 48 * q^57 + 864 * q^58 - 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 14 * q^70 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 - 996 * q^74 - 806 * q^75 - 3042 * q^76 - 2632 * q^78 + 1324 * q^79 + 2342 * q^80 + 1444 * q^81 + 1984 * q^82 - 450 * q^83 + 490 * q^84 + 1736 * q^85 - 4380 * q^86 - 588 * q^87 - 3072 * q^89 + 218 * q^90 - 406 * q^91 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 - 24 * q^95 - 862 * q^96 - 652 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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