LMFDB - Embedded newform 855.4.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,4,Mod(1,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("855.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	855.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,0,7,-15,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$50.4466330549$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3580.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 7 $ x^3 - x^2 - 15*x + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.18473$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.18473 q^{2} +9.51199 q^{4} -5.00000 q^{5} -11.3273 q^{7} +6.32725 q^{8} -20.9237 q^{10} +48.7207 q^{11} +19.2692 q^{13} -47.4015 q^{14} -49.6180 q^{16} -121.574 q^{17} -19.0000 q^{19} -47.5599 q^{20} +203.883 q^{22} -133.817 q^{23} +25.0000 q^{25} +80.6363 q^{26} -107.745 q^{28} +7.97030 q^{29} -122.116 q^{31} -258.256 q^{32} -508.754 q^{34} +56.6363 q^{35} +89.7616 q^{37} -79.5099 q^{38} -31.6363 q^{40} -124.738 q^{41} -270.378 q^{43} +463.430 q^{44} -559.988 q^{46} +387.769 q^{47} -214.693 q^{49} +104.618 q^{50} +183.288 q^{52} +157.361 q^{53} -243.603 q^{55} -71.6704 q^{56} +33.3536 q^{58} -35.1738 q^{59} -543.534 q^{61} -511.024 q^{62} -683.789 q^{64} -96.3458 q^{65} -617.006 q^{67} -1156.41 q^{68} +237.008 q^{70} -400.155 q^{71} -732.299 q^{73} +375.628 q^{74} -180.728 q^{76} -551.872 q^{77} -903.791 q^{79} +248.090 q^{80} -521.996 q^{82} +910.929 q^{83} +607.869 q^{85} -1131.46 q^{86} +308.268 q^{88} -739.460 q^{89} -218.267 q^{91} -1272.86 q^{92} +1622.71 q^{94} +95.0000 q^{95} +1770.18 q^{97} -898.434 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 7 q^{4} - 15 q^{5} - 24 q^{7} + 9 q^{8} - 5 q^{10} + 70 q^{11} + 20 q^{13} + 8 q^{14} - 69 q^{16} - 100 q^{17} - 57 q^{19} - 35 q^{20} + 98 q^{22} + 56 q^{23} + 75 q^{25} + 192 q^{26} - 204 q^{28}+ \cdots - 615 q^{98}+O(q^{100}) $ 3 * q + q^2 + 7 * q^4 - 15 * q^5 - 24 * q^7 + 9 * q^8 - 5 * q^10 + 70 * q^11 + 20 * q^13 + 8 * q^14 - 69 * q^16 - 100 * q^17 - 57 * q^19 - 35 * q^20 + 98 * q^22 + 56 * q^23 + 75 * q^25 + 192 * q^26 - 204 * q^28 + 140 * q^29 - 422 * q^31 + 33 * q^32 - 396 * q^34 + 120 * q^35 + 140 * q^37 - 19 * q^38 - 45 * q^40 + 130 * q^41 - 286 * q^43 + 666 * q^44 - 1068 * q^46 + 792 * q^47 - 673 * q^49 + 25 * q^50 - 176 * q^52 - 434 * q^53 - 350 * q^55 - 236 * q^56 + 36 * q^58 + 478 * q^59 - 650 * q^61 + 262 * q^62 - 1241 * q^64 - 100 * q^65 - 1060 * q^67 - 1652 * q^68 - 40 * q^70 - 796 * q^71 + 218 * q^73 - 800 * q^74 - 133 * q^76 - 988 * q^77 - 1314 * q^79 + 345 * q^80 - 870 * q^82 - 326 * q^83 + 500 * q^85 - 2254 * q^86 + 638 * q^88 - 502 * q^89 + 248 * q^91 - 856 * q^92 + 1292 * q^94 + 285 * q^95 + 968 * q^97 - 615 * 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 $

$2$	4.18473	1.47953	0.739763	−	0.672867i	$-0.234938\pi$
$2$	4.18473	1.47953	0.739763	+	0.672867i	$0.234938\pi$
$3$	0	0
$3$	0	0
$4$	9.51199	1.18900
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−11.3273	−0.611614	−0.305807	−	0.952093i	$-0.598926\pi$
$7$	−11.3273	−0.611614	−0.305807	+	0.952093i	$0.598926\pi$
$8$	6.32725	0.279628
$9$	0	0
$10$	−20.9237	−0.661664
$11$	48.7207	1.33544	0.667720	−	0.744413i	$-0.267270\pi$
$11$	48.7207	1.33544	0.667720	+	0.744413i	$0.267270\pi$
$12$	0	0
$13$	19.2692	0.411100	0.205550	−	0.978647i	$-0.434102\pi$
$13$	19.2692	0.411100	0.205550	+	0.978647i	$0.434102\pi$
$14$	−47.4015	−0.904899
$15$	0	0
$16$	−49.6180	−0.775282
$17$	−121.574	−1.73447	−0.867234	−	0.497900i	$-0.834104\pi$
$17$	−121.574	−1.73447	−0.867234	+	0.497900i	$0.834104\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−47.5599	−0.531736
$21$	0	0
$22$	203.883	1.97582
$23$	−133.817	−1.21316	−0.606581	−	0.795021i	$-0.707460\pi$
$23$	−133.817	−1.21316	−0.606581	+	0.795021i	$0.707460\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	80.6363	0.608234
$27$	0	0
$28$	−107.745	−0.727208
$29$	7.97030	0.0510361	0.0255181	−	0.999674i	$-0.491876\pi$
$29$	7.97030	0.0510361	0.0255181	+	0.999674i	$0.491876\pi$
$30$	0	0
$31$	−122.116	−0.707507	−0.353753	−	0.935339i	$-0.615095\pi$
$31$	−122.116	−0.707507	−0.353753	+	0.935339i	$0.615095\pi$
$32$	−258.256	−1.42668
$33$	0	0
$34$	−508.754	−2.56619
$35$	56.6363	0.273522
$36$	0	0
$37$	89.7616	0.398830	0.199415	−	0.979915i	$-0.436096\pi$
$37$	89.7616	0.398830	0.199415	+	0.979915i	$0.436096\pi$
$38$	−79.5099	−0.339427
$39$	0	0
$40$	−31.6363	−0.125053
$41$	−124.738	−0.475142	−0.237571	−	0.971370i	$-0.576351\pi$
$41$	−124.738	−0.475142	−0.237571	+	0.971370i	$0.576351\pi$
$42$	0	0
$43$	−270.378	−0.958890	−0.479445	−	0.877572i	$-0.659162\pi$
$43$	−270.378	−0.958890	−0.479445	+	0.877572i	$0.659162\pi$
$44$	463.430	1.58784
$45$	0	0
$46$	−559.988	−1.79491
$47$	387.769	1.20344	0.601722	−	0.798705i	$-0.294481\pi$
$47$	387.769	1.20344	0.601722	+	0.798705i	$0.294481\pi$
$48$	0	0
$49$	−214.693	−0.625928
$50$	104.618	0.295905
$51$	0	0
$52$	183.288	0.488797
$53$	157.361	0.407833	0.203917	−	0.978988i	$-0.434633\pi$
$53$	157.361	0.407833	0.203917	+	0.978988i	$0.434633\pi$
$54$	0	0
$55$	−243.603	−0.597227
$56$	−71.6704	−0.171024
$57$	0	0
$58$	33.3536	0.0755093
$59$	−35.1738	−0.0776142	−0.0388071	−	0.999247i	$-0.512356\pi$
$59$	−35.1738	−0.0776142	−0.0388071	+	0.999247i	$0.512356\pi$
$60$	0	0
$61$	−543.534	−1.14086	−0.570430	−	0.821346i	$-0.693223\pi$
$61$	−543.534	−1.14086	−0.570430	+	0.821346i	$0.693223\pi$
$62$	−511.024	−1.04678
$63$	0	0
$64$	−683.789	−1.33552
$65$	−96.3458	−0.183850
$66$	0	0
$67$	−617.006	−1.12506	−0.562532	−	0.826775i	$-0.690173\pi$
$67$	−617.006	−1.12506	−0.562532	+	0.826775i	$0.690173\pi$
$68$	−1156.41	−2.06228
$69$	0	0
$70$	237.008	0.404683
$71$	−400.155	−0.668869	−0.334434	−	0.942419i	$-0.608545\pi$
$71$	−400.155	−0.668869	−0.334434	+	0.942419i	$0.608545\pi$
$72$	0	0
$73$	−732.299	−1.17410	−0.587048	−	0.809552i	$-0.699710\pi$
$73$	−732.299	−1.17410	−0.587048	+	0.809552i	$0.699710\pi$
$74$	375.628	0.590079
$75$	0	0
$76$	−180.728	−0.272775
$77$	−551.872	−0.816774
$78$	0	0
$79$	−903.791	−1.28714	−0.643572	−	0.765385i	$-0.722548\pi$
$79$	−903.791	−1.28714	−0.643572	+	0.765385i	$0.722548\pi$
$80$	248.090	0.346716
$81$	0	0
$82$	−521.996	−0.702985
$83$	910.929	1.20467	0.602334	−	0.798244i	$-0.294238\pi$
$83$	910.929	1.20467	0.602334	+	0.798244i	$0.294238\pi$
$84$	0	0
$85$	607.869	0.775678
$86$	−1131.46	−1.41870
$87$	0	0
$88$	308.268	0.373426
$89$	−739.460	−0.880703	−0.440352	−	0.897825i	$-0.645146\pi$
$89$	−739.460	−0.880703	−0.440352	+	0.897825i	$0.645146\pi$
$90$	0	0
$91$	−218.267	−0.251435
$92$	−1272.86	−1.44245
$93$	0	0
$94$	1622.71	1.78053
$95$	95.0000	0.102598
$96$	0	0
$97$	1770.18	1.85293	0.926466	−	0.376379i	$-0.122831\pi$
$97$	1770.18	1.85293	0.926466	+	0.376379i	$0.122831\pi$
$98$	−898.434	−0.926077
$99$	0	0
$100$	237.800	0.237800
$101$	332.869	0.327938	0.163969	−	0.986466i	$-0.447570\pi$
$101$	332.869	0.327938	0.163969	+	0.986466i	$0.447570\pi$
$102$	0	0
$103$	570.264	0.545532	0.272766	−	0.962080i	$-0.412062\pi$
$103$	570.264	0.545532	0.272766	+	0.962080i	$0.412062\pi$
$104$	121.921	0.114955
$105$	0	0
$106$	658.513	0.603400
$107$	677.514	0.612129	0.306064	−	0.952011i	$-0.400988\pi$
$107$	677.514	0.612129	0.306064	+	0.952011i	$0.400988\pi$
$108$	0	0
$109$	−921.322	−0.809602	−0.404801	−	0.914405i	$-0.632659\pi$
$109$	−921.322	−0.809602	−0.404801	+	0.914405i	$0.632659\pi$
$110$	−1019.42	−0.883613
$111$	0	0
$112$	562.036	0.474173
$113$	1855.53	1.54472	0.772361	−	0.635184i	$-0.219076\pi$
$113$	1855.53	1.54472	0.772361	+	0.635184i	$0.219076\pi$
$114$	0	0
$115$	669.084	0.542543
$116$	75.8134	0.0606819
$117$	0	0
$118$	−147.193	−0.114832
$119$	1377.10	1.06083
$120$	0	0
$121$	1042.71	0.783400
$122$	−2274.55	−1.68793
$123$	0	0
$124$	−1161.57	−0.841224
$125$	−125.000	−0.0894427
$126$	0	0
$127$	963.401	0.673134	0.336567	−	0.941660i	$-0.390734\pi$
$127$	963.401	0.673134	0.336567	+	0.941660i	$0.390734\pi$
$128$	−795.424	−0.549267
$129$	0	0
$130$	−403.181	−0.272010
$131$	−1031.58	−0.688013	−0.344006	−	0.938967i	$-0.611784\pi$
$131$	−1031.58	−0.688013	−0.344006	+	0.938967i	$0.611784\pi$
$132$	0	0
$133$	215.218	0.140314
$134$	−2582.01	−1.66456
$135$	0	0
$136$	−769.228	−0.485005
$137$	1244.79	0.776277	0.388138	−	0.921601i	$-0.373118\pi$
$137$	1244.79	0.776277	0.388138	+	0.921601i	$0.373118\pi$
$138$	0	0
$139$	−2853.77	−1.74139	−0.870697	−	0.491820i	$-0.836332\pi$
$139$	−2853.77	−1.74139	−0.870697	+	0.491820i	$0.836332\pi$
$140$	538.723	0.325217
$141$	0	0
$142$	−1674.54	−0.989609
$143$	938.807	0.549000
$144$	0	0
$145$	−39.8515	−0.0228241
$146$	−3064.47	−1.73711
$147$	0	0
$148$	853.811	0.474208
$149$	2697.92	1.48337	0.741684	−	0.670749i	$-0.234027\pi$
$149$	2697.92	1.48337	0.741684	+	0.670749i	$0.234027\pi$
$150$	0	0
$151$	2001.45	1.07865	0.539324	−	0.842099i	$-0.318680\pi$
$151$	2001.45	1.07865	0.539324	+	0.842099i	$0.318680\pi$
$152$	−120.218	−0.0641510
$153$	0	0
$154$	−2309.43	−1.20844
$155$	610.581	0.316407
$156$	0	0
$157$	−2142.94	−1.08933	−0.544667	−	0.838652i	$-0.683344\pi$
$157$	−2142.94	−1.08933	−0.544667	+	0.838652i	$0.683344\pi$
$158$	−3782.12	−1.90436
$159$	0	0
$160$	1291.28	0.638029
$161$	1515.78	0.741988
$162$	0	0
$163$	3696.16	1.77611	0.888054	−	0.459738i	$-0.152057\pi$
$163$	3696.16	1.77611	0.888054	+	0.459738i	$0.152057\pi$
$164$	−1186.51	−0.564943
$165$	0	0
$166$	3811.99	1.78234
$167$	−1937.20	−0.897634	−0.448817	−	0.893624i	$-0.648155\pi$
$167$	−1937.20	−0.897634	−0.448817	+	0.893624i	$0.648155\pi$
$168$	0	0
$169$	−1825.70	−0.830997
$170$	2543.77	1.14764
$171$	0	0
$172$	−2571.83	−1.14012
$173$	3799.75	1.66988	0.834940	−	0.550341i	$-0.185502\pi$
$173$	3799.75	1.66988	0.834940	+	0.550341i	$0.185502\pi$
$174$	0	0
$175$	−283.181	−0.122323
$176$	−2417.42	−1.03534
$177$	0	0
$178$	−3094.44	−1.30302
$179$	−121.921	−0.0509095	−0.0254548	−	0.999676i	$-0.508103\pi$
$179$	−121.921	−0.0509095	−0.0254548	+	0.999676i	$0.508103\pi$
$180$	0	0
$181$	225.206	0.0924829	0.0462414	−	0.998930i	$-0.485276\pi$
$181$	225.206	0.0924829	0.0462414	+	0.998930i	$0.485276\pi$
$182$	−913.387	−0.372004
$183$	0	0
$184$	−846.693	−0.339234
$185$	−448.808	−0.178362
$186$	0	0
$187$	−5923.16	−2.31628
$188$	3688.45	1.43089
$189$	0	0
$190$	397.550	0.151796
$191$	2991.27	1.13320	0.566600	−	0.823993i	$-0.308259\pi$
$191$	2991.27	1.13320	0.566600	+	0.823993i	$0.308259\pi$
$192$	0	0
$193$	4282.60	1.59725	0.798623	−	0.601832i	$-0.205562\pi$
$193$	4282.60	1.59725	0.798623	+	0.601832i	$0.205562\pi$
$194$	7407.72	2.74146
$195$	0	0
$196$	−2042.16	−0.744227
$197$	−3085.51	−1.11591	−0.557954	−	0.829872i	$-0.688413\pi$
$197$	−3085.51	−1.11591	−0.557954	+	0.829872i	$0.688413\pi$
$198$	0	0
$199$	−4351.94	−1.55026	−0.775128	−	0.631804i	$-0.782315\pi$
$199$	−4351.94	−1.55026	−0.775128	+	0.631804i	$0.782315\pi$
$200$	158.181	0.0559255
$201$	0	0
$202$	1392.97	0.485193
$203$	−90.2816	−0.0312144
$204$	0	0
$205$	623.691	0.212490
$206$	2386.40	0.807129
$207$	0	0
$208$	−956.097	−0.318718
$209$	−925.693	−0.306371
$210$	0	0
$211$	−539.478	−0.176015	−0.0880076	−	0.996120i	$-0.528050\pi$
$211$	−539.478	−0.176015	−0.0880076	+	0.996120i	$0.528050\pi$
$212$	1496.81	0.484913
$213$	0	0
$214$	2835.22	0.905661
$215$	1351.89	0.428829
$216$	0	0
$217$	1383.24	0.432721
$218$	−3855.49	−1.19783
$219$	0	0
$220$	−2317.15	−0.710102
$221$	−2342.62	−0.713040
$222$	0	0
$223$	−559.515	−0.168017	−0.0840087	−	0.996465i	$-0.526772\pi$
$223$	−559.515	−0.168017	−0.0840087	+	0.996465i	$0.526772\pi$
$224$	2925.33	0.872576
$225$	0	0
$226$	7764.89	2.28546
$227$	458.145	0.133957	0.0669783	−	0.997754i	$-0.478664\pi$
$227$	458.145	0.133957	0.0669783	+	0.997754i	$0.478664\pi$
$228$	0	0
$229$	−2729.93	−0.787769	−0.393884	−	0.919160i	$-0.628869\pi$
$229$	−2729.93	−0.787769	−0.393884	+	0.919160i	$0.628869\pi$
$230$	2799.94	0.802706
$231$	0	0
$232$	50.4301	0.0142711
$233$	3642.55	1.02417	0.512085	−	0.858935i	$-0.328873\pi$
$233$	3642.55	1.02417	0.512085	+	0.858935i	$0.328873\pi$
$234$	0	0
$235$	−1938.84	−0.538197
$236$	−334.573	−0.0922832
$237$	0	0
$238$	5762.78	1.56952
$239$	−5074.58	−1.37342	−0.686709	−	0.726932i	$-0.740945\pi$
$239$	−5074.58	−1.37342	−0.686709	+	0.726932i	$0.740945\pi$
$240$	0	0
$241$	−1574.01	−0.420710	−0.210355	−	0.977625i	$-0.567462\pi$
$241$	−1574.01	−0.420710	−0.210355	+	0.977625i	$0.567462\pi$
$242$	4363.44	1.15906
$243$	0	0
$244$	−5170.09	−1.35648
$245$	1073.47	0.279924
$246$	0	0
$247$	−366.114	−0.0943129
$248$	−772.660	−0.197839
$249$	0	0
$250$	−523.092	−0.132333
$251$	−1174.41	−0.295331	−0.147665	−	0.989037i	$-0.547176\pi$
$251$	−1174.41	−0.295331	−0.147665	+	0.989037i	$0.547176\pi$
$252$	0	0
$253$	−6519.65	−1.62011
$254$	4031.57	0.995919
$255$	0	0
$256$	2141.68	0.522870
$257$	−781.504	−0.189684	−0.0948422	−	0.995492i	$-0.530235\pi$
$257$	−781.504	−0.189684	−0.0948422	+	0.995492i	$0.530235\pi$
$258$	0	0
$259$	−1016.75	−0.243930
$260$	−916.440	−0.218597
$261$	0	0
$262$	−4316.89	−1.01793
$263$	4299.03	1.00795	0.503973	−	0.863720i	$-0.331871\pi$
$263$	4299.03	1.00795	0.503973	+	0.863720i	$0.331871\pi$
$264$	0	0
$265$	−786.804	−0.182389
$266$	900.629	0.207598
$267$	0	0
$268$	−5868.95	−1.33770
$269$	2095.26	0.474907	0.237454	−	0.971399i	$-0.423687\pi$
$269$	2095.26	0.474907	0.237454	+	0.971399i	$0.423687\pi$
$270$	0	0
$271$	833.929	0.186928	0.0934641	−	0.995623i	$-0.470206\pi$
$271$	833.929	0.186928	0.0934641	+	0.995623i	$0.470206\pi$
$272$	6032.25	1.34470
$273$	0	0
$274$	5209.13	1.14852
$275$	1218.02	0.267088
$276$	0	0
$277$	3902.66	0.846528	0.423264	−	0.906006i	$-0.360884\pi$
$277$	3902.66	0.846528	0.423264	+	0.906006i	$0.360884\pi$
$278$	−11942.3	−2.57644
$279$	0	0
$280$	358.352	0.0764844
$281$	2292.03	0.486587	0.243293	−	0.969953i	$-0.421772\pi$
$281$	2292.03	0.486587	0.243293	+	0.969953i	$0.421772\pi$
$282$	0	0
$283$	−4393.23	−0.922792	−0.461396	−	0.887194i	$-0.652651\pi$
$283$	−4393.23	−0.922792	−0.461396	+	0.887194i	$0.652651\pi$
$284$	−3806.27	−0.795284
$285$	0	0
$286$	3928.65	0.812259
$287$	1412.94	0.290604
$288$	0	0
$289$	9867.17	2.00838
$290$	−166.768	−0.0337688
$291$	0	0
$292$	−6965.61	−1.39600
$293$	−5036.77	−1.00427	−0.502135	−	0.864789i	$-0.667452\pi$
$293$	−5036.77	−1.00427	−0.502135	+	0.864789i	$0.667452\pi$
$294$	0	0
$295$	175.869	0.0347101
$296$	567.944	0.111524
$297$	0	0
$298$	11290.1	2.19468
$299$	−2578.54	−0.498731
$300$	0	0
$301$	3062.64	0.586471
$302$	8375.53	1.59589
$303$	0	0
$304$	942.742	0.177862
$305$	2717.67	0.510208
$306$	0	0
$307$	7911.97	1.47088	0.735439	−	0.677590i	$-0.236976\pi$
$307$	7911.97	1.47088	0.735439	+	0.677590i	$0.236976\pi$
$308$	−5249.39	−0.971143
$309$	0	0
$310$	2555.12	0.468132
$311$	2572.75	0.469091	0.234545	−	0.972105i	$-0.424640\pi$
$311$	2572.75	0.469091	0.234545	+	0.972105i	$0.424640\pi$
$312$	0	0
$313$	1444.77	0.260905	0.130452	−	0.991455i	$-0.458357\pi$
$313$	1444.77	0.260905	0.130452	+	0.991455i	$0.458357\pi$
$314$	−8967.64	−1.61170
$315$	0	0
$316$	−8596.85	−1.53041
$317$	−8509.32	−1.50767	−0.753835	−	0.657064i	$-0.771798\pi$
$317$	−8509.32	−1.50767	−0.753835	+	0.657064i	$0.771798\pi$
$318$	0	0
$319$	388.319	0.0681557
$320$	3418.94	0.597265
$321$	0	0
$322$	6343.12	1.09779
$323$	2309.90	0.397914
$324$	0	0
$325$	481.729	0.0822200
$326$	15467.4	2.62780
$327$	0	0
$328$	−789.250	−0.132863
$329$	−4392.36	−0.736044
$330$	0	0
$331$	7671.09	1.27384	0.636920	−	0.770930i	$-0.280208\pi$
$331$	7671.09	1.27384	0.636920	+	0.770930i	$0.280208\pi$
$332$	8664.74	1.43235
$333$	0	0
$334$	−8106.66	−1.32807
$335$	3085.03	0.503144
$336$	0	0
$337$	−3317.95	−0.536321	−0.268161	−	0.963374i	$-0.586416\pi$
$337$	−3317.95	−0.536321	−0.268161	+	0.963374i	$0.586416\pi$
$338$	−7640.06	−1.22948
$339$	0	0
$340$	5782.04	0.922280
$341$	−5949.58	−0.944833
$342$	0	0
$343$	6317.13	0.994441
$344$	−1710.75	−0.268132
$345$	0	0
$346$	15900.9	2.47063
$347$	−6378.17	−0.986738	−0.493369	−	0.869820i	$-0.664235\pi$
$347$	−6378.17	−0.986738	−0.493369	+	0.869820i	$0.664235\pi$
$348$	0	0
$349$	9217.85	1.41381	0.706905	−	0.707308i	$-0.250091\pi$
$349$	9217.85	1.41381	0.706905	+	0.707308i	$0.250091\pi$
$350$	−1185.04	−0.180980
$351$	0	0
$352$	−12582.4	−1.90524
$353$	−3047.35	−0.459473	−0.229737	−	0.973253i	$-0.573786\pi$
$353$	−3047.35	−0.459473	−0.229737	+	0.973253i	$0.573786\pi$
$354$	0	0
$355$	2000.78	0.299127
$356$	−7033.73	−1.04715
$357$	0	0
$358$	−510.207	−0.0753220
$359$	−12092.0	−1.77769	−0.888844	−	0.458210i	$-0.848491\pi$
$359$	−12092.0	−1.77769	−0.888844	+	0.458210i	$0.848491\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	942.425	0.136831
$363$	0	0
$364$	−2076.15	−0.298955
$365$	3661.49	0.525072
$366$	0	0
$367$	−10108.3	−1.43774	−0.718869	−	0.695146i	$-0.755340\pi$
$367$	−10108.3	−1.43774	−0.718869	+	0.695146i	$0.755340\pi$
$368$	6639.73	0.940543
$369$	0	0
$370$	−1878.14	−0.263892
$371$	−1782.47	−0.249437
$372$	0	0
$373$	−805.021	−0.111749	−0.0558745	−	0.998438i	$-0.517795\pi$
$373$	−805.021	−0.111749	−0.0558745	+	0.998438i	$0.517795\pi$
$374$	−24786.8	−3.42699
$375$	0	0
$376$	2453.51	0.336517
$377$	153.581	0.0209810
$378$	0	0
$379$	−11059.9	−1.49896	−0.749482	−	0.662025i	$-0.769697\pi$
$379$	−11059.9	−1.49896	−0.749482	+	0.662025i	$0.769697\pi$
$380$	903.639	0.121989
$381$	0	0
$382$	12517.7	1.67660
$383$	6847.47	0.913549	0.456774	−	0.889583i	$-0.349005\pi$
$383$	6847.47	0.913549	0.456774	+	0.889583i	$0.349005\pi$
$384$	0	0
$385$	2759.36	0.365272
$386$	17921.5	2.36317
$387$	0	0
$388$	16837.9	2.20313
$389$	−898.568	−0.117119	−0.0585594	−	0.998284i	$-0.518651\pi$
$389$	−898.568	−0.117119	−0.0585594	+	0.998284i	$0.518651\pi$
$390$	0	0
$391$	16268.6	2.10419
$392$	−1358.42	−0.175027
$393$	0	0
$394$	−12912.1	−1.65101
$395$	4518.95	0.575629
$396$	0	0
$397$	−10647.3	−1.34603	−0.673014	−	0.739630i	$-0.735001\pi$
$397$	−10647.3	−1.34603	−0.673014	+	0.739630i	$0.735001\pi$
$398$	−18211.7	−2.29364
$399$	0	0
$400$	−1240.45	−0.155056
$401$	1337.87	0.166608	0.0833041	−	0.996524i	$-0.473453\pi$
$401$	1337.87	0.166608	0.0833041	+	0.996524i	$0.473453\pi$
$402$	0	0
$403$	−2353.08	−0.290856
$404$	3166.25	0.389917
$405$	0	0
$406$	−377.804	−0.0461826
$407$	4373.24	0.532613
$408$	0	0
$409$	−10307.7	−1.24617	−0.623086	−	0.782153i	$-0.714121\pi$
$409$	−10307.7	−1.24617	−0.623086	+	0.782153i	$0.714121\pi$
$410$	2609.98	0.314385
$411$	0	0
$412$	5424.34	0.648636
$413$	398.423	0.0474700
$414$	0	0
$415$	−4554.64	−0.538744
$416$	−4976.38	−0.586507
$417$	0	0
$418$	−3873.78	−0.453284
$419$	10471.2	1.22088	0.610442	−	0.792061i	$-0.290992\pi$
$419$	10471.2	1.22088	0.610442	+	0.792061i	$0.290992\pi$
$420$	0	0
$421$	−10698.7	−1.23854	−0.619268	−	0.785180i	$-0.712571\pi$
$421$	−10698.7	−1.23854	−0.619268	+	0.785180i	$0.712571\pi$
$422$	−2257.57	−0.260419
$423$	0	0
$424$	995.662	0.114042
$425$	−3039.34	−0.346894
$426$	0	0
$427$	6156.75	0.697766
$428$	6444.51	0.727820
$429$	0	0
$430$	5657.30	0.634463
$431$	12717.5	1.42130	0.710649	−	0.703547i	$-0.248401\pi$
$431$	12717.5	1.42130	0.710649	+	0.703547i	$0.248401\pi$
$432$	0	0
$433$	−14511.6	−1.61058	−0.805290	−	0.592881i	$-0.797991\pi$
$433$	−14511.6	−1.61058	−0.805290	+	0.592881i	$0.797991\pi$
$434$	5788.49	0.640223
$435$	0	0
$436$	−8763.60	−0.962616
$437$	2542.52	0.278319
$438$	0	0
$439$	1878.49	0.204226	0.102113	−	0.994773i	$-0.467440\pi$
$439$	1878.49	0.204226	0.102113	+	0.994773i	$0.467440\pi$
$440$	−1541.34	−0.167001
$441$	0	0
$442$	−9803.25	−1.05496
$443$	−18219.3	−1.95401	−0.977004	−	0.213220i	$-0.931605\pi$
$443$	−18219.3	−1.95401	−0.977004	+	0.213220i	$0.931605\pi$
$444$	0	0
$445$	3697.30	0.393863
$446$	−2341.42	−0.248586
$447$	0	0
$448$	7745.45	0.816826
$449$	−8127.70	−0.854276	−0.427138	−	0.904186i	$-0.640478\pi$
$449$	−8127.70	−0.854276	−0.427138	+	0.904186i	$0.640478\pi$
$450$	0	0
$451$	−6077.33	−0.634524
$452$	17649.8	1.83667
$453$	0	0
$454$	1917.21	0.198192
$455$	1091.33	0.112445
$456$	0	0
$457$	6526.93	0.668089	0.334045	−	0.942557i	$-0.391586\pi$
$457$	6526.93	0.668089	0.334045	+	0.942557i	$0.391586\pi$
$458$	−11424.0	−1.16552
$459$	0	0
$460$	6364.32	0.645082
$461$	−8800.09	−0.889070	−0.444535	−	0.895762i	$-0.646631\pi$
$461$	−8800.09	−0.889070	−0.444535	+	0.895762i	$0.646631\pi$
$462$	0	0
$463$	−12267.1	−1.23131	−0.615657	−	0.788014i	$-0.711109\pi$
$463$	−12267.1	−1.23131	−0.615657	+	0.788014i	$0.711109\pi$
$464$	−395.471	−0.0395674
$465$	0	0
$466$	15243.1	1.51529
$467$	−1041.33	−0.103184	−0.0515920	−	0.998668i	$-0.516430\pi$
$467$	−1041.33	−0.103184	−0.0515920	+	0.998668i	$0.516430\pi$
$468$	0	0
$469$	6988.99	0.688105
$470$	−8113.55	−0.796276
$471$	0	0
$472$	−222.554	−0.0217031
$473$	−13173.0	−1.28054
$474$	0	0
$475$	−475.000	−0.0458831
$476$	13098.9	1.26132
$477$	0	0
$478$	−21235.7	−2.03201
$479$	6828.11	0.651324	0.325662	−	0.945486i	$-0.394413\pi$
$479$	6828.11	0.651324	0.325662	+	0.945486i	$0.394413\pi$
$480$	0	0
$481$	1729.63	0.163959
$482$	−6586.82	−0.622451
$483$	0	0
$484$	9918.20	0.931461
$485$	−8850.89	−0.828656
$486$	0	0
$487$	−4134.06	−0.384666	−0.192333	−	0.981330i	$-0.561605\pi$
$487$	−4134.06	−0.384666	−0.192333	+	0.981330i	$0.561605\pi$
$488$	−3439.08	−0.319016
$489$	0	0
$490$	4492.17	0.414154
$491$	3776.58	0.347117	0.173559	−	0.984824i	$-0.444473\pi$
$491$	3776.58	0.347117	0.173559	+	0.984824i	$0.444473\pi$
$492$	0	0
$493$	−968.979	−0.0885206
$494$	−1532.09	−0.139538
$495$	0	0
$496$	6059.16	0.548517
$497$	4532.66	0.409090
$498$	0	0
$499$	−4548.80	−0.408081	−0.204040	−	0.978962i	$-0.565407\pi$
$499$	−4548.80	−0.408081	−0.204040	+	0.978962i	$0.565407\pi$
$500$	−1189.00	−0.106347
$501$	0	0
$502$	−4914.58	−0.436949
$503$	18001.2	1.59569	0.797847	−	0.602861i	$-0.205972\pi$
$503$	18001.2	1.59569	0.797847	+	0.602861i	$0.205972\pi$
$504$	0	0
$505$	−1664.35	−0.146658
$506$	−27283.0	−2.39699
$507$	0	0
$508$	9163.85	0.800355
$509$	565.749	0.0492660	0.0246330	−	0.999697i	$-0.492158\pi$
$509$	565.749	0.0492660	0.0246330	+	0.999697i	$0.492158\pi$
$510$	0	0
$511$	8294.93	0.718094
$512$	15325.7	1.32287
$513$	0	0
$514$	−3270.39	−0.280643
$515$	−2851.32	−0.243969
$516$	0	0
$517$	18892.4	1.60713
$518$	−4254.83	−0.360901
$519$	0	0
$520$	−609.604	−0.0514094
$521$	13187.9	1.10897	0.554483	−	0.832195i	$-0.312916\pi$
$521$	13187.9	1.10897	0.554483	+	0.832195i	$0.312916\pi$
$522$	0	0
$523$	−9251.41	−0.773491	−0.386745	−	0.922187i	$-0.626401\pi$
$523$	−9251.41	−0.773491	−0.386745	+	0.922187i	$0.626401\pi$
$524$	−9812.39	−0.818046
$525$	0	0
$526$	17990.3	1.49128
$527$	14846.1	1.22715
$528$	0	0
$529$	5739.95	0.471764
$530$	−3292.57	−0.269849
$531$	0	0
$532$	2047.15	0.166833
$533$	−2403.60	−0.195331
$534$	0	0
$535$	−3387.57	−0.273752
$536$	−3903.95	−0.314599
$537$	0	0
$538$	8768.09	0.702638
$539$	−10460.0	−0.835889
$540$	0	0
$541$	−5305.50	−0.421629	−0.210814	−	0.977526i	$-0.567612\pi$
$541$	−5305.50	−0.421629	−0.210814	+	0.977526i	$0.567612\pi$
$542$	3489.77	0.276565
$543$	0	0
$544$	31397.2	2.47453
$545$	4606.61	0.362065
$546$	0	0
$547$	2840.53	0.222034	0.111017	−	0.993819i	$-0.464589\pi$
$547$	2840.53	0.222034	0.111017	+	0.993819i	$0.464589\pi$
$548$	11840.5	0.922992
$549$	0	0
$550$	5097.08	0.395164
$551$	−151.436	−0.0117085
$552$	0	0
$553$	10237.5	0.787236
$554$	16331.6	1.25246
$555$	0	0
$556$	−27145.0	−2.07051
$557$	5667.19	0.431107	0.215553	−	0.976492i	$-0.430844\pi$
$557$	5667.19	0.431107	0.215553	+	0.976492i	$0.430844\pi$
$558$	0	0
$559$	−5209.96	−0.394200
$560$	−2810.18	−0.212057
$561$	0	0
$562$	9591.52	0.719918
$563$	9405.63	0.704086	0.352043	−	0.935984i	$-0.385487\pi$
$563$	9405.63	0.704086	0.352043	+	0.935984i	$0.385487\pi$
$564$	0	0
$565$	−9277.65	−0.690820
$566$	−18384.5	−1.36530
$567$	0	0
$568$	−2531.88	−0.187034
$569$	−10443.6	−0.769452	−0.384726	−	0.923031i	$-0.625704\pi$
$569$	−10443.6	−0.769452	−0.384726	+	0.923031i	$0.625704\pi$
$570$	0	0
$571$	892.063	0.0653795	0.0326897	−	0.999466i	$-0.489593\pi$
$571$	892.063	0.0653795	0.0326897	+	0.999466i	$0.489593\pi$
$572$	8929.91	0.652760
$573$	0	0
$574$	5912.78	0.429956
$575$	−3345.42	−0.242633
$576$	0	0
$577$	−10933.3	−0.788838	−0.394419	−	0.918931i	$-0.629054\pi$
$577$	−10933.3	−0.788838	−0.394419	+	0.918931i	$0.629054\pi$
$578$	41291.5	2.97145
$579$	0	0
$580$	−379.067	−0.0271378
$581$	−10318.3	−0.736792
$582$	0	0
$583$	7666.73	0.544637
$584$	−4633.44	−0.328310
$585$	0	0
$586$	−21077.5	−1.48584
$587$	−16737.3	−1.17687	−0.588434	−	0.808545i	$-0.700255\pi$
$587$	−16737.3	−1.17687	−0.588434	+	0.808545i	$0.700255\pi$
$588$	0	0
$589$	2320.21	0.162313
$590$	735.965	0.0513546
$591$	0	0
$592$	−4453.79	−0.309206
$593$	−18886.9	−1.30792	−0.653958	−	0.756531i	$-0.726893\pi$
$593$	−18886.9	−1.30792	−0.653958	+	0.756531i	$0.726893\pi$
$594$	0	0
$595$	−6885.48	−0.474416
$596$	25662.5	1.76372
$597$	0	0
$598$	−10790.5	−0.737886
$599$	25827.3	1.76173	0.880864	−	0.473369i	$-0.156962\pi$
$599$	25827.3	1.76173	0.880864	+	0.473369i	$0.156962\pi$
$600$	0	0
$601$	−440.830	−0.0299199	−0.0149599	−	0.999888i	$-0.504762\pi$
$601$	−440.830	−0.0299199	−0.0149599	+	0.999888i	$0.504762\pi$
$602$	12816.3	0.867699
$603$	0	0
$604$	19037.8	1.28251
$605$	−5213.53	−0.350347
$606$	0	0
$607$	21991.1	1.47050	0.735249	−	0.677797i	$-0.237065\pi$
$607$	21991.1	1.47050	0.735249	+	0.677797i	$0.237065\pi$
$608$	4906.87	0.327302
$609$	0	0
$610$	11372.7	0.754866
$611$	7471.98	0.494736
$612$	0	0
$613$	−20231.6	−1.33303	−0.666514	−	0.745493i	$-0.732214\pi$
$613$	−20231.6	−1.33303	−0.666514	+	0.745493i	$0.732214\pi$
$614$	33109.5	2.17620
$615$	0	0
$616$	−3491.83	−0.228393
$617$	7126.86	0.465019	0.232509	−	0.972594i	$-0.425306\pi$
$617$	7126.86	0.465019	0.232509	+	0.972594i	$0.425306\pi$
$618$	0	0
$619$	−16585.6	−1.07695	−0.538474	−	0.842642i	$-0.680999\pi$
$619$	−16585.6	−1.07695	−0.538474	+	0.842642i	$0.680999\pi$
$620$	5807.84	0.376207
$621$	0	0
$622$	10766.3	0.694032
$623$	8376.05	0.538651
$624$	0	0
$625$	625.000	0.0400000
$626$	6045.97	0.386016
$627$	0	0
$628$	−20383.6	−1.29522
$629$	−10912.6	−0.691758
$630$	0	0
$631$	5112.66	0.322554	0.161277	−	0.986909i	$-0.448439\pi$
$631$	5112.66	0.322554	0.161277	+	0.986909i	$0.448439\pi$
$632$	−5718.51	−0.359921
$633$	0	0
$634$	−35609.2	−2.23064
$635$	−4817.00	−0.301035
$636$	0	0
$637$	−4136.96	−0.257319
$638$	1625.01	0.100838
$639$	0	0
$640$	3977.12	0.245640
$641$	11167.5	0.688127	0.344064	−	0.938946i	$-0.388196\pi$
$641$	11167.5	0.688127	0.344064	+	0.938946i	$0.388196\pi$
$642$	0	0
$643$	−17105.3	−1.04909	−0.524547	−	0.851381i	$-0.675765\pi$
$643$	−17105.3	−1.04909	−0.524547	+	0.851381i	$0.675765\pi$
$644$	14418.1	0.882222
$645$	0	0
$646$	9666.32	0.588725
$647$	−17613.1	−1.07024	−0.535118	−	0.844777i	$-0.679733\pi$
$647$	−17613.1	−1.07024	−0.535118	+	0.844777i	$0.679733\pi$
$648$	0	0
$649$	−1713.69	−0.103649
$650$	2015.91	0.121647
$651$	0	0
$652$	35157.8	2.11179
$653$	10574.2	0.633690	0.316845	−	0.948477i	$-0.397376\pi$
$653$	10574.2	0.633690	0.316845	+	0.948477i	$0.397376\pi$
$654$	0	0
$655$	5157.91	0.307689
$656$	6189.26	0.368369
$657$	0	0
$658$	−18380.8	−1.08900
$659$	−32462.1	−1.91888	−0.959440	−	0.281912i	$-0.909031\pi$
$659$	−32462.1	−1.91888	−0.959440	+	0.281912i	$0.909031\pi$
$660$	0	0
$661$	9572.61	0.563285	0.281642	−	0.959519i	$-0.409121\pi$
$661$	9572.61	0.563285	0.281642	+	0.959519i	$0.409121\pi$
$662$	32101.5	1.88468
$663$	0	0
$664$	5763.68	0.336858
$665$	−1076.09	−0.0627503
$666$	0	0
$667$	−1066.56	−0.0619151
$668$	−18426.6	−1.06729
$669$	0	0
$670$	12910.0	0.744415
$671$	−26481.4	−1.52355
$672$	0	0
$673$	18179.6	1.04127	0.520634	−	0.853780i	$-0.325696\pi$
$673$	18179.6	1.04127	0.520634	+	0.853780i	$0.325696\pi$
$674$	−13884.7	−0.793501
$675$	0	0
$676$	−17366.0	−0.988053
$677$	−23894.9	−1.35651	−0.678255	−	0.734827i	$-0.737263\pi$
$677$	−23894.9	−1.35651	−0.678255	+	0.734827i	$0.737263\pi$
$678$	0	0
$679$	−20051.3	−1.13328
$680$	3846.14	0.216901
$681$	0	0
$682$	−24897.4	−1.39791
$683$	−854.033	−0.0478458	−0.0239229	−	0.999714i	$-0.507616\pi$
$683$	−854.033	−0.0478458	−0.0239229	+	0.999714i	$0.507616\pi$
$684$	0	0
$685$	−6223.97	−0.347161
$686$	26435.5	1.47130
$687$	0	0
$688$	13415.6	0.743410
$689$	3032.21	0.167660
$690$	0	0
$691$	−25872.2	−1.42435	−0.712173	−	0.702004i	$-0.752289\pi$
$691$	−25872.2	−1.42435	−0.712173	+	0.702004i	$0.752289\pi$
$692$	36143.1	1.98548
$693$	0	0
$694$	−26690.9	−1.45990
$695$	14268.9	0.778775
$696$	0	0
$697$	15164.9	0.824119
$698$	38574.2	2.09177
$699$	0	0
$700$	−2693.62	−0.145442
$701$	9573.42	0.515810	0.257905	−	0.966170i	$-0.416968\pi$
$701$	9573.42	0.515810	0.257905	+	0.966170i	$0.416968\pi$
$702$	0	0
$703$	−1705.47	−0.0914979
$704$	−33314.7	−1.78351
$705$	0	0
$706$	−12752.3	−0.679803
$707$	−3770.49	−0.200571
$708$	0	0
$709$	−13262.4	−0.702511	−0.351256	−	0.936280i	$-0.614245\pi$
$709$	−13262.4	−0.702511	−0.351256	+	0.936280i	$0.614245\pi$
$710$	8372.71	0.442567
$711$	0	0
$712$	−4678.75	−0.246269
$713$	16341.2	0.858321
$714$	0	0
$715$	−4694.03	−0.245520
$716$	−1159.71	−0.0605313
$717$	0	0
$718$	−50601.7	−2.63014
$719$	−12144.1	−0.629900	−0.314950	−	0.949108i	$-0.601988\pi$
$719$	−12144.1	−0.629900	−0.314950	+	0.949108i	$0.601988\pi$
$720$	0	0
$721$	−6459.53	−0.333655
$722$	1510.69	0.0778698
$723$	0	0
$724$	2142.15	0.109962
$725$	199.258	0.0102072
$726$	0	0
$727$	34831.7	1.77694	0.888471	−	0.458934i	$-0.151768\pi$
$727$	34831.7	1.77694	0.888471	+	0.458934i	$0.151768\pi$
$728$	−1381.03	−0.0703081
$729$	0	0
$730$	15322.4	0.776858
$731$	32870.9	1.66316
$732$	0	0
$733$	11266.2	0.567704	0.283852	−	0.958868i	$-0.408388\pi$
$733$	11266.2	0.567704	0.283852	+	0.958868i	$0.408388\pi$
$734$	−42300.6	−2.12717
$735$	0	0
$736$	34559.0	1.73079
$737$	−30061.0	−1.50246
$738$	0	0
$739$	33847.1	1.68482	0.842412	−	0.538834i	$-0.181135\pi$
$739$	33847.1	1.68482	0.842412	+	0.538834i	$0.181135\pi$
$740$	−4269.05	−0.212072
$741$	0	0
$742$	−7459.14	−0.369048
$743$	6060.32	0.299235	0.149618	−	0.988744i	$-0.452196\pi$
$743$	6060.32	0.299235	0.149618	+	0.988744i	$0.452196\pi$
$744$	0	0
$745$	−13489.6	−0.663383
$746$	−3368.80	−0.165336
$747$	0	0
$748$	−56341.0	−2.75405
$749$	−7674.38	−0.374387
$750$	0	0
$751$	−8585.76	−0.417176	−0.208588	−	0.978004i	$-0.566887\pi$
$751$	−8585.76	−0.417176	−0.208588	+	0.978004i	$0.566887\pi$
$752$	−19240.3	−0.933009
$753$	0	0
$754$	642.695	0.0310419
$755$	−10007.3	−0.482386
$756$	0	0
$757$	1170.67	0.0562071	0.0281035	−	0.999605i	$-0.491053\pi$
$757$	1170.67	0.0562071	0.0281035	+	0.999605i	$0.491053\pi$
$758$	−46282.6	−2.21776
$759$	0	0
$760$	601.089	0.0286892
$761$	−1595.31	−0.0759923	−0.0379961	−	0.999278i	$-0.512097\pi$
$761$	−1595.31	−0.0759923	−0.0379961	+	0.999278i	$0.512097\pi$
$762$	0	0
$763$	10436.0	0.495164
$764$	28453.0	1.34737
$765$	0	0
$766$	28654.8	1.35162
$767$	−677.770	−0.0319072
$768$	0	0
$769$	29719.0	1.39362	0.696810	−	0.717255i	$-0.254602\pi$
$769$	29719.0	1.39362	0.696810	+	0.717255i	$0.254602\pi$
$770$	11547.2	0.540430
$771$	0	0
$772$	40736.0	1.89912
$773$	27460.3	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$773$	27460.3	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$774$	0	0
$775$	−3052.90	−0.141501
$776$	11200.4	0.518131
$777$	0	0
$778$	−3760.27	−0.173280
$779$	2370.03	0.109005
$780$	0	0
$781$	−19495.8	−0.893234
$782$	68079.8	3.11321
$783$	0	0
$784$	10652.7	0.485271
$785$	10714.7	0.487165
$786$	0	0
$787$	−30349.6	−1.37464	−0.687322	−	0.726353i	$-0.741214\pi$
$787$	−30349.6	−1.37464	−0.687322	+	0.726353i	$0.741214\pi$
$788$	−29349.4	−1.32681
$789$	0	0
$790$	18910.6	0.851658
$791$	−21018.0	−0.944773
$792$	0	0
$793$	−10473.4	−0.469008
$794$	−44556.1	−1.99148
$795$	0	0
$796$	−41395.6	−1.84325
$797$	−20761.2	−0.922711	−0.461355	−	0.887215i	$-0.652637\pi$
$797$	−20761.2	−0.922711	−0.461355	+	0.887215i	$0.652637\pi$
$798$	0	0
$799$	−47142.5	−2.08734
$800$	−6456.40	−0.285335
$801$	0	0
$802$	5598.62	0.246501
$803$	−35678.1	−1.56794
$804$	0	0
$805$	−7578.89	−0.331827
$806$	−9846.99	−0.430329
$807$	0	0
$808$	2106.15	0.0917005
$809$	−14067.6	−0.611362	−0.305681	−	0.952134i	$-0.598884\pi$
$809$	−14067.6	−0.611362	−0.305681	+	0.952134i	$0.598884\pi$
$810$	0	0
$811$	−22623.9	−0.979571	−0.489786	−	0.871843i	$-0.662925\pi$
$811$	−22623.9	−0.979571	−0.489786	+	0.871843i	$0.662925\pi$
$812$	−858.757	−0.0371139
$813$	0	0
$814$	18300.9	0.788016
$815$	−18480.8	−0.794300
$816$	0	0
$817$	5137.18	0.219984
$818$	−43135.1	−1.84374
$819$	0	0
$820$	5932.54	0.252650
$821$	−21609.9	−0.918626	−0.459313	−	0.888275i	$-0.651904\pi$
$821$	−21609.9	−0.918626	−0.459313	+	0.888275i	$0.651904\pi$
$822$	0	0
$823$	−25.9937	−0.00110095	−0.000550476	−	1.00000i	$-0.500175\pi$
$823$	−25.9937	−0.00110095	−0.000550476		1.00000i	$0.500175\pi$
$824$	3608.20	0.152546
$825$	0	0
$826$	1667.29	0.0702331
$827$	12550.9	0.527735	0.263867	−	0.964559i	$-0.415002\pi$
$827$	12550.9	0.527735	0.263867	+	0.964559i	$0.415002\pi$
$828$	0	0
$829$	−27280.3	−1.14293	−0.571463	−	0.820628i	$-0.693624\pi$
$829$	−27280.3	−1.14293	−0.571463	+	0.820628i	$0.693624\pi$
$830$	−19060.0	−0.797086
$831$	0	0
$832$	−13176.0	−0.549035
$833$	26101.1	1.08565
$834$	0	0
$835$	9686.00	0.401434
$836$	−8805.18	−0.364274
$837$	0	0
$838$	43819.1	1.80633
$839$	−36581.3	−1.50528	−0.752638	−	0.658434i	$-0.771219\pi$
$839$	−36581.3	−1.50528	−0.752638	+	0.658434i	$0.771219\pi$
$840$	0	0
$841$	−24325.5	−0.997395
$842$	−44771.3	−1.83245
$843$	0	0
$844$	−5131.51	−0.209282
$845$	9128.50	0.371633
$846$	0	0
$847$	−11811.0	−0.479139
$848$	−7807.93	−0.316186
$849$	0	0
$850$	−12718.8	−0.513238
$851$	−12011.6	−0.483846
$852$	0	0
$853$	24834.5	0.996855	0.498427	−	0.866931i	$-0.333911\pi$
$853$	24834.5	0.996855	0.498427	+	0.866931i	$0.333911\pi$
$854$	25764.4	1.03236
$855$	0	0
$856$	4286.80	0.171168
$857$	−37743.1	−1.50441	−0.752205	−	0.658930i	$-0.771009\pi$
$857$	−37743.1	−1.50441	−0.752205	+	0.658930i	$0.771009\pi$
$858$	0	0
$859$	18336.6	0.728330	0.364165	−	0.931334i	$-0.381354\pi$
$859$	18336.6	0.728330	0.364165	+	0.931334i	$0.381354\pi$
$860$	12859.2	0.509877
$861$	0	0
$862$	53219.3	2.10285
$863$	36481.1	1.43897	0.719484	−	0.694509i	$-0.244378\pi$
$863$	36481.1	1.43897	0.719484	+	0.694509i	$0.244378\pi$
$864$	0	0
$865$	−18998.7	−0.746793
$866$	−60727.0	−2.38290
$867$	0	0
$868$	13157.4	0.514505
$869$	−44033.3	−1.71890
$870$	0	0
$871$	−11889.2	−0.462514
$872$	−5829.44	−0.226387
$873$	0	0
$874$	10639.8	0.411780
$875$	1415.91	0.0547044
$876$	0	0
$877$	24426.3	0.940498	0.470249	−	0.882534i	$-0.344164\pi$
$877$	24426.3	0.940498	0.470249	+	0.882534i	$0.344164\pi$
$878$	7860.97	0.302158
$879$	0	0
$880$	12087.1	0.463019
$881$	−50770.3	−1.94154	−0.970768	−	0.240018i	$-0.922847\pi$
$881$	−50770.3	−1.94154	−0.970768	+	0.240018i	$0.922847\pi$
$882$	0	0
$883$	−760.639	−0.0289893	−0.0144946	−	0.999895i	$-0.504614\pi$
$883$	−760.639	−0.0289893	−0.0144946	+	0.999895i	$0.504614\pi$
$884$	−22283.0	−0.847804
$885$	0	0
$886$	−76243.0	−2.89101
$887$	15947.7	0.603688	0.301844	−	0.953357i	$-0.402398\pi$
$887$	15947.7	0.603688	0.301844	+	0.953357i	$0.402398\pi$
$888$	0	0
$889$	−10912.7	−0.411698
$890$	15472.2	0.582730
$891$	0	0
$892$	−5322.10	−0.199772
$893$	−7367.61	−0.276089
$894$	0	0
$895$	609.605	0.0227674
$896$	9009.96	0.335939
$897$	0	0
$898$	−34012.2	−1.26392
$899$	−973.303	−0.0361084
$900$	0	0
$901$	−19130.9	−0.707374
$902$	−25432.0	−0.938795
$903$	0	0
$904$	11740.4	0.431947
$905$	−1126.03	−0.0413596
$906$	0	0
$907$	36524.4	1.33712	0.668562	−	0.743656i	$-0.266910\pi$
$907$	36524.4	1.33712	0.668562	+	0.743656i	$0.266910\pi$
$908$	4357.87	0.159274
$909$	0	0
$910$	4566.94	0.166365
$911$	−27341.7	−0.994370	−0.497185	−	0.867645i	$-0.665633\pi$
$911$	−27341.7	−0.994370	−0.497185	+	0.867645i	$0.665633\pi$
$912$	0	0
$913$	44381.1	1.60876
$914$	27313.4	0.988455
$915$	0	0
$916$	−25967.1	−0.936655
$917$	11685.0	0.420798
$918$	0	0
$919$	1790.82	0.0642804	0.0321402	−	0.999483i	$-0.489768\pi$
$919$	1790.82	0.0642804	0.0321402	+	0.999483i	$0.489768\pi$
$920$	4233.47	0.151710
$921$	0	0
$922$	−36826.0	−1.31540
$923$	−7710.65	−0.274972
$924$	0	0
$925$	2244.04	0.0797660
$926$	−51334.4	−1.82176
$927$	0	0
$928$	−2058.38	−0.0728121
$929$	−38812.1	−1.37070	−0.685352	−	0.728212i	$-0.740352\pi$
$929$	−38812.1	−1.37070	−0.685352	+	0.728212i	$0.740352\pi$
$930$	0	0
$931$	4079.17	0.143598
$932$	34647.9	1.21774
$933$	0	0
$934$	−4357.68	−0.152663
$935$	29615.8	1.03587
$936$	0	0
$937$	−29469.3	−1.02745	−0.513725	−	0.857955i	$-0.671735\pi$
$937$	−29469.3	−1.02745	−0.513725	+	0.857955i	$0.671735\pi$
$938$	29247.0	1.01807
$939$	0	0
$940$	−18442.3	−0.639915
$941$	9329.66	0.323207	0.161604	−	0.986856i	$-0.448333\pi$
$941$	9329.66	0.323207	0.161604	+	0.986856i	$0.448333\pi$
$942$	0	0
$943$	16692.1	0.576425
$944$	1745.25	0.0601729
$945$	0	0
$946$	−55125.5	−1.89459
$947$	38365.3	1.31648	0.658240	−	0.752808i	$-0.271301\pi$
$947$	38365.3	1.31648	0.658240	+	0.752808i	$0.271301\pi$
$948$	0	0
$949$	−14110.8	−0.482671
$950$	−1987.75	−0.0678853
$951$	0	0
$952$	8713.24	0.296636
$953$	−39507.7	−1.34290	−0.671448	−	0.741052i	$-0.734327\pi$
$953$	−39507.7	−1.34290	−0.671448	+	0.741052i	$0.734327\pi$
$954$	0	0
$955$	−14956.4	−0.506782
$956$	−48269.3	−1.63299
$957$	0	0
$958$	28573.8	0.963651
$959$	−14100.1	−0.474782
$960$	0	0
$961$	−14878.6	−0.499434
$962$	7238.04	0.242582
$963$	0	0
$964$	−14972.0	−0.500223
$965$	−21413.0	−0.714310
$966$	0	0
$967$	−19476.8	−0.647707	−0.323854	−	0.946107i	$-0.604979\pi$
$967$	−19476.8	−0.647707	−0.323854	+	0.946107i	$0.604979\pi$
$968$	6597.46	0.219060
$969$	0	0
$970$	−37038.6	−1.22602
$971$	22849.2	0.755166	0.377583	−	0.925976i	$-0.376755\pi$
$971$	22849.2	0.755166	0.377583	+	0.925976i	$0.376755\pi$
$972$	0	0
$973$	32325.4	1.06506
$974$	−17299.9	−0.569123
$975$	0	0
$976$	26969.1	0.884488
$977$	52540.4	1.72049	0.860243	−	0.509884i	$-0.170312\pi$
$977$	52540.4	1.72049	0.860243	+	0.509884i	$0.170312\pi$
$978$	0	0
$979$	−36027.0	−1.17613
$980$	10210.8	0.332829
$981$	0	0
$982$	15804.0	0.513569
$983$	−14871.4	−0.482527	−0.241263	−	0.970460i	$-0.577562\pi$
$983$	−14871.4	−0.482527	−0.241263	+	0.970460i	$0.577562\pi$
$984$	0	0
$985$	15427.6	0.499049
$986$	−4054.92	−0.130969
$987$	0	0
$988$	−3482.47	−0.112138
$989$	36181.2	1.16329
$990$	0	0
$991$	−2772.58	−0.0888738	−0.0444369	−	0.999012i	$-0.514149\pi$
$991$	−2772.58	−0.0888738	−0.0444369	+	0.999012i	$0.514149\pi$
$992$	31537.3	1.00938
$993$	0	0
$994$	18968.0	0.605259
$995$	21759.7	0.693296
$996$	0	0
$997$	−10476.8	−0.332803	−0.166402	−	0.986058i	$-0.553215\pi$
$997$	−10476.8	−0.332803	−0.166402	+	0.986058i	$0.553215\pi$
$998$	−19035.5	−0.603766
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.4.a.h.1.3		3
3.2	odd	2		285.4.a.c.1.1	✓	3
15.14	odd	2		1425.4.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.4.a.c.1.1	✓	3	3.2	odd	2
855.4.a.h.1.3		3	1.1	even	1	trivial
1425.4.a.h.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+4.18473 q^{2} +9.51199 q^{4} -5.00000 q^{5} -11.3273 q^{7} +6.32725 q^{8} -20.9237 q^{10} +48.7207 q^{11} +19.2692 q^{13} -47.4015 q^{14} -49.6180 q^{16} -121.574 q^{17} -19.0000 q^{19} -47.5599 q^{20} +203.883 q^{22} -133.817 q^{23} +25.0000 q^{25} +80.6363 q^{26} -107.745 q^{28} +7.97030 q^{29} -122.116 q^{31} -258.256 q^{32} -508.754 q^{34} +56.6363 q^{35} +89.7616 q^{37} -79.5099 q^{38} -31.6363 q^{40} -124.738 q^{41} -270.378 q^{43} +463.430 q^{44} -559.988 q^{46} +387.769 q^{47} -214.693 q^{49} +104.618 q^{50} +183.288 q^{52} +157.361 q^{53} -243.603 q^{55} -71.6704 q^{56} +33.3536 q^{58} -35.1738 q^{59} -543.534 q^{61} -511.024 q^{62} -683.789 q^{64} -96.3458 q^{65} -617.006 q^{67} -1156.41 q^{68} +237.008 q^{70} -400.155 q^{71} -732.299 q^{73} +375.628 q^{74} -180.728 q^{76} -551.872 q^{77} -903.791 q^{79} +248.090 q^{80} -521.996 q^{82} +910.929 q^{83} +607.869 q^{85} -1131.46 q^{86} +308.268 q^{88} -739.460 q^{89} -218.267 q^{91} -1272.86 q^{92} +1622.71 q^{94} +95.0000 q^{95} +1770.18 q^{97} -898.434 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 7 q^{4} - 15 q^{5} - 24 q^{7} + 9 q^{8} - 5 q^{10} + 70 q^{11} + 20 q^{13} + 8 q^{14} - 69 q^{16} - 100 q^{17} - 57 q^{19} - 35 q^{20} + 98 q^{22} + 56 q^{23} + 75 q^{25} + 192 q^{26} - 204 q^{28}+ \cdots - 615 q^{98}+O(q^{100}) \) 3 * q + q^2 + 7 * q^4 - 15 * q^5 - 24 * q^7 + 9 * q^8 - 5 * q^10 + 70 * q^11 + 20 * q^13 + 8 * q^14 - 69 * q^16 - 100 * q^17 - 57 * q^19 - 35 * q^20 + 98 * q^22 + 56 * q^23 + 75 * q^25 + 192 * q^26 - 204 * q^28 + 140 * q^29 - 422 * q^31 + 33 * q^32 - 396 * q^34 + 120 * q^35 + 140 * q^37 - 19 * q^38 - 45 * q^40 + 130 * q^41 - 286 * q^43 + 666 * q^44 - 1068 * q^46 + 792 * q^47 - 673 * q^49 + 25 * q^50 - 176 * q^52 - 434 * q^53 - 350 * q^55 - 236 * q^56 + 36 * q^58 + 478 * q^59 - 650 * q^61 + 262 * q^62 - 1241 * q^64 - 100 * q^65 - 1060 * q^67 - 1652 * q^68 - 40 * q^70 - 796 * q^71 + 218 * q^73 - 800 * q^74 - 133 * q^76 - 988 * q^77 - 1314 * q^79 + 345 * q^80 - 870 * q^82 - 326 * q^83 + 500 * q^85 - 2254 * q^86 + 638 * q^88 - 502 * q^89 + 248 * q^91 - 856 * q^92 + 1292 * q^94 + 285 * q^95 + 968 * q^97 - 615 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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